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GMAT TEST 1 – Keys for Maths Section

(37 questions, 75 minutes)

1. Roy is now 4 years older than Erik and half of that amount older than Iris. If in 2

years, Roy will be twice as old as Erik, then in 2 years what would be Erik’s age

multiplied by Iris’s age? [Roy hiện nay nhiều hơn Erik 4 tuổi và nhiều hơn Iris một

nửa con số đó. Nếu sau 2 năm nữa Roy sẽ nhiều tuổi gấp đôi Erik thì sau 2 năm tiếp

nữa tuổi Roy nhân với tuổi Iris sẽ là bao nhiêu?]

(a) 8

(b) 28

(d) 50

(e) 52

Translate piece by piece into numbers. R (Roy) = Erik (E) + 4. The second equation: R =

I (Iris)+ 2. The third equation: R +2 = 2(E + 2). We have three equations with three

variables. à Roy is 6, Iris is 4 and Erik is 2. In four years Erik would be 6 and Iris 8, the

answer is 48. The correct answer is C.

[Chuyển từng câu một thành các con số ta có: R = E + 4; R = I + 2; R + 2 = 2(E+2). Ta

có hệ 3 phương trình 3 biến. Giải ra được R=6, I=4, E=2. Sau 4 năm thì Erik 6 tuổi và Iris

8 tuổi, nhân với nhau là 48.]

2. An investment yields an interest payment of $228 each month. If the simple

annual interest rate is 9%, what is the amount of the investment? [Một món tiền đầu

tư sinh lãi mỗi tháng 228$. Nếu tỷ xuất lợi nhuận hàng năm là 9% thì số tiền đầu tư

là bao nhiêu?]

(a) $28,300

(b) $30,400

(d) $32,500

(e) $35,100

Principal × percent interest × time = interest earned [Tiền gốc x tỷ xuất lợi nhuận x thời

gian = tiền lãi]

Principle × (0.09)× 1/12 = $228. [Gốc x 0.09 x 1/12 = 228$]

Solve to find the principal (228 × 12)/0.09= $30,400. [Giải ra ta được vốn gốc là

30.400$]

The correct answer is B.

3. X, Y, Z, and W are integers. The expression X-Y-Z is even and the expression Y-

Z-W is odd. If X is even what must be true? [X, Y, Z và W là các số nguyên. Biểu

thức X-Y-Z là chẵn còn biểu thức Y-Z-W là lẻ. Nếu X chẵn thì trong các câu sau câu

nào đúng?]

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(a) Y-Z must be odd.

(b) W must be even.

(d) W must be even.

(e) Z must be odd

The first expression is even and the second is odd, the only difference between the

expressions is that the first expression has X and the second has W. So, if X is even W

must be odd and the correct answer is C.

[Biểu thức đầu là chẵn còn biểu thức sau là lẻ, sự khác nhau duy nhất giữa hai BT là BT

đầu chứa X còn BT sau chứa W. Vậy nếu X chẵn thì W phải lẻ; đáp án đúng là C]

X and Y are two sides of a triangle, is the area of the triangle an integer?[X và Y là 2

cạnh của một tam giác, liệu diện tích tam giác có là một số nguyên nếu:]

(1) X is a prime number. [X là số nguyên tố]

(2) Y is an odd integer. [Y là số nguyên lẻ]

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not. [Chỉ riêng điều kiện (1) là đủ trả lời câu hỏi, nhưng chỉ riêng ĐK (2) thì

không]

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not. [Chỉ riêng điều kiện (2) là đủ trả lời câu hỏi, nhưng chỉ riêng ĐK (1) thì

không]

even though NEITHER statement BY ITSELF is sufficient. [Kết hợp cả hai ĐK thì

đủ trả lời câu hỏi, tuy nhiên chỉ riêng mỗi ĐK lại không đủ để trả lời]

(d) Either statement BY ITSELF is sufficient to answer the question. [Mỗi ĐK riêng rẽ

đều đủ để trả lời câu hỏi]

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem. [Kết hợp cả hai ĐK cũng

không đủ để trả lời câu hỏi. Cần có thêm dữ liệu.]

Chú ý: Từ các câu sau, chúng tôi sẽ không dịch những phương án cần phải suy luận

giống như câu này. Các bạn hãy lấy câu này làm ví dụ để tham khảo cho các câu

tương tự.

The area of the triangle is XY/2. Statement (1) tells us that X is a prime number, that can

be even (2) or odd (3, 5, 7, etc.).

Statement (2) tells us that Y is an odd integer. The multiplication of X and Y can be an

odd number or an even number, thus we cannot determine if the area of the triangle is an

integer or not. The correct answer is E.

[Diện tích tam giác là XY/2. (??? Theo chúng tôi đề bài cho chưa chính xác. Nếu cho

chính xác thì phải là tam giác vuông). ĐK (1) cho biết X là số nguyên tố nên có thể là

chẵn (là 2) hoặc lẻ (là 3,5,7,...). ĐK(2) cho biết Y là số nguyên lẻ. Tích số của X và Y có

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thể là chẵn cũng có thể là lẻ, vì vậy ta không thể biết diện tích tam giác có là số nguyên

hay không. Đáp án đúng là E]

5. Fuel tanker A can fill the underground reservoir in 12 minutes. How long will it

take fuel tanker A and fuel tanker B to fill up the same reservoir together? [Một xe

xì-tẹc xăng A đổ đầy một bồn xăng ngầm trong 12 phút. Mất bao nhiêu lâu để cả xe

A và xe B đổ đầy cũng bồn ngầm đó cùng nhau?]

(1) The reservoir contains 3000 liters of fuel. [Bồn ngầm chứa được 3000 lít]

(2) Fuel tanker B alone will require the same number of hours to fill the same

reservoir. [Xe B riêng nó cũng cần một thời lượng tương tự để đổ đầy bồn ngầm]

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) is insufficient since the size of the reservoir is irrelevant. Statement (2) is

sufficient since it tells us that the second tanker has the same work rate as the first. So, it

will take them both half the time it took the first tanker alone. The correct answer is B.

[ĐK (1) là không đủ vì dung tích bồn ngầm chẳng liên quan gì đến bài toán (nếu ta không

biết lưu tốc bơm xăng). ĐK (2) là đủ vì nó cho biết xe B cũng có cùng tốc độ bơm xăng

như xe A. Vì vậy nếu cả hai xe cùng làm việc thì chỉ mất một nửa thời gian so với một

xe.]

6. Q is a prime number bigger than 10. What is the smallest positive number (except

1) that 3Q can be divided by equally? [Q là số nguyên tố lớn hơn 10. Số nào là số

dương nhỏ nhất (khác 1) mà có thể chia hết 3Q]

(a) 3Q

(b) Q

(d) Q+3

(e) 2Q

Q is a prime number so 3Q can be divide equally by 3Q, by 1 and by the components 3

and Q. The smallest number therefore is 3. The correct answer is C.

[Q là số nguyên tố nên 3Q chỉ có 3 ước số là 3Q, 1, 3 và Q. Do vậy ước số nhỏ nhất khác

1 chính là 3]

7. In a box there are A green balls, 3A + 6 red balls and 2 yellow ones. If there are

no other colors, what is the probability of taking out a green or a yellow ball?

[Trong 1 cái hộp có A bóng xanh, 3A+6 bóng đỏ và 2 bóng vàng. Nếu không có màu

nào khác thì xác xuất lấy được 1 quả vàng hoặc xanh là bao nhiêu?]

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(a) 1/5.

(b) 1/2.

(d) 1/4.

(e) 2/3.

The number of green and yellow balls in the box is A+2. The total number of balls is 4A

+8. The probability of taking out a green or a yellow ball is: (A+2)/(4A+8)=1/4. The

correct answer is D.

[Tổng số quả xanh và vàng là A+2. Tổng số bóng là 4A+8. Xác xuất lấy được 1 quả xanh

hoặc vàng là (A+2)/(4A+8)=1/4]

8. Kelly used to get a 30% discount on movie tickets. When the price of the movie

ticket increased by 50%, the amount of discount in dollars remained the same.

What is Kelly's discount with the new Ticket price in percent terms? [Kelly thường

được giảm 30% giá vé xem phim. Khi giá vé tăng 50% thì số tiền được giảm vẫn

không đổi. Vậy Kelly được giảm bao nhiêu % khi vé tăng giá?]

(a) 10%

(b) 20%

(d) 35%

(e) 38%

The price of the ticket is unknown. It would be most convenient to plug in 100 as the

price of the ticket. A 30% discount of 100 is $30, that amount remained the same after

the price of a ticket increased by 50%. The new price of a ticket is $150, so 30/150 is

20%. The correct answer is B.

[Giá vé là chưa biết. Để đơn giản ta coi giá vé là 100. Được giảm 30% là 30$. Số tiền này

không đổi khi giá vé tăng 50%. Giá mới là 150$ nên tỷ lệ giảm giá Kelly được hưởng là

30/150 = 20%]

9. Is the square root of A an integer? [Liệu căn bậc 2 của A có là số nguyên nếu:]

(1) The last digit of A is 8 .[Chữ số cuối cùng của A là 8]

(2) A is divisible by 6 .[A chia hết cho 6]

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

If you square each digit {0, 1, 2,..., 8, 9}, you will see that the possible last digits for a

square are 0, 1, 4, 5, 6 and 9. Thus, if the last digit of A is 8, A cannot be a square and the

square root of A is not an integer. Statement (1) by itself is sufficient. Statement (2) by

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itself is insufficient since there are numbers that are divisible by 6 and have an integer

square root and numbers divisible by 6 that do not have an integer square root. The

correct answer is A.

[Nếu bạn bình phương mỗi chữ số {0,1,2...8,9} bạn sẽ được các số có chữ số tận cùng là

0,1,4,5,6 và 9. Vậy nếu chữ số tận cùng của A là 8 thì A không thể là một số chính

phương và căn bậc 2 của A không thể nguyên. ĐK (1) riêng nó là đủ để khẳng định. ĐK

(2) riêng nó là không đủ vì có những số chia hết cho 6 và có căn bậc 2 là nguyên (vd 36)

nhưng cũng có số chia hết cho 6 nhưng căn bậc 2 lại không nguyên (vd 18). Đáp án đúng

là A]

10. If Q and T are integers, what is the value of Q? [Nếu Q và T là 2 số nguyên thì

Q=? nếu:]

(1) Q = 2T/7.

(2) (T+7)/2=7(Q+2)/4

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

We want to find the number value of Q. In statement (1) we are given the ratio between

Q and T, which is not sufficient. Statement (2) can be simplified: (T+7)/2=7(Q+2)/4 =>

T+7=(7Q+14)/2 = 7Q/2 + 7 => T=7Q/2. We can see that the same ratio is given in

statement (2) also and more data is required to answer the question. Both statements give

the same information. The correct answer is E.

[ĐK (1) chỉ đưa ra tỷ số giữa P và Q nên không đủ để tìm Q. ĐK (2) có thể đơn giản hoá

như sau: (T+7)/2=7(Q+2)/4 => T+7=(7Q+14)/2 = 7Q/2 + 7 => T=7Q/2 là tỷ số đã biết ở

ĐK (1). Cần có thêm dữ liệu để trả lời bài toán. Đáp án đúng là đáp án E].

11. In a psychology school the grade of the students is determined by the following

method: At the end of the first year the grade equals to twice the age of the student.

From then on, the grade is determined by twice the age of the student plus half of

his grade from the previous year. If Joey’s grade at the end of the first year is 40,

what will be his grade at the end of the third year? [Trong một trường tâm lý học,

điểm của các sinh viên được xác định như sau: Vào cuối năm thứ nhất, điểm của

mỗi sinh viên bằng 2 lần tuổi của sinh viên đó. Từ đó về sau, điểm được tính bằng 2

lần tuổi của sinh viên cộng thêm một nửa số điểm của năm trước đó. Nếu điểm của

Joe cuối năm thứ nhất là 40 thì cuối năm thứ 3 cậu có bao nhiêu điểm?]

(a) 44.

(b) 56.

(d) 75.

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(e) 80.

From the grade 40 at the end of the first year we learn that his age is 20. At the end of the

second year, he will be 21 and his grade will be (21 x 2 + ½ x 40 = 62). At the end of the

third year, he will be 22 and his grade will be (22 x 2 + ½ x 62 = 75). The correct answer

is D.

[Năm đầu tiên Joe 20 tuổi. Cuối năm thứ 2, cậu 21 tuổi và có số điểm (21 x 2 + 40/2 =

62). Cuối năm thứ 3 cậu 22 tuổi và có số điểm (22 x 2 + 62/2 = 75).]

12. A is a prime number (A>2). If C = A3, by how many different integers can C be

equally divided? [A là một số nguyên tố (A>2). Nếu C=A3 thì có bao nhiêu số nguyên

khác nhau có thể chia hết C?]

(a) 3.

(b) 4.

(d) 6.

(e) 7.

Factorize C: C = A x A x A. C can be equally divided into 1, A, A2, and A3 =Cà 4

numbers total. The correct answer is B.

[Phân tích C thành thừa số nguyên tố: C = A x A x A. C có thể chia hết cho 1, A, A2, A3.

Do vậy C có tổng cộng 4 ước số]

13. If X is a positive integer and (405) 4 is a multiple of 3X, what is the largest

possible value of X? [Nếu X là một số nguyên dương và 4054 là bội số của 3x thì giá

trị lớn nhất có thể có của X là bao nhiêu?]

(a) 5.

(b) 12.

(d) 20.

(e) 26.

Find the factors of (405) 4 and see what the largest value of X can be. 405 = 81 x 5 = 9 x

9 x 5 = 3 x 3 x 3 x 3 x 5 è (405) 4 = (3 x 3 x 3 x 3 x 5) 4 = 316 x 54. The largest possible

value of 3X that is still a factor of (405) 4 is the largest possible value of X and that is 316.

X = 16. The correct answer is C.

[Phân tích thành thừa số nguyên tố thì: 405 = 81x5 = 9 x 9 x 5 = 3 x 3 x 3 x 3 x 5. Nên

4054 = 316 x 54. Giá trị lớn nhất của 3x vẫn là ước số của 4054 là 316. Do đó X=16 là giá trị

lớn nhất]

14. X, Y>0, If X3 = Y, is Y a fraction? [X, Y > 0. Nếu X3 = Y thì liệu Y có là phân số

khi:]

(1) X2 is a fraction. [X2 là một phân số]

(2) X > Y.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

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(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) is sufficient. If X2 is a fraction, X must also be a fraction. Meaning that X3

and Y will also be fractions. Statement (2) is sufficient. If X3 is a fraction then X must be

greater than X3, which is also equal to Y. We get that from X > Y, so X3 and Y are

fractions. The correct answer is D.

[ĐK (1) là đủ để kết luận. Nếu X2 là phân số thì X cũng phải là phân số.??? Nghĩa là X3

hay Y cũng là phân số. ĐK (2) cũng đủ vì nếu X > Y = X3 thì 1 > X2 > 0 => 1 > X > 0 =>

X là phân số] ??? Theo người dịch thì lời giải này sai. Ví dụ nếu X = 1/√2 thì X2=1/2

là phân số hữu tỷ nhưng Y là vô tỷ. Vậy (1) không đủ. Cũng với giá trị trên thì X>Y

nhưng X và Y đều vô tỷ chứ không phải phân số hữu tỷ. (2) cũng không đủ. Đáp án

đúng phải là E.

15. A turtle is crossing a field, how many meters total did he pass? [Một con rùa bò

qua cánh đồng. Nó đi được bao nhiêu mét nếu:]

(1) The average speed of the turtle is 2 meters per minute. [Tốc độ trung bình của

rùa là 2m/phút]

(2) Had the turtle walked 1 meter per minute faster than his average speed it would

have finished the same path 40 minutes earlier. [Nếu nó bò nhanh hơn vận tốc trung

bình là 1m/phút thì nó đã có thể hoàn thành quãng đường sớm hơn 40 phút]

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) gives us the average speed of the turtle; this statement is insufficient since

the time is not given. Statement (2) is also insufficient by itself since we don’t know what

the average speed is. Both statements combined are sufficient since we can calculate the

distance 2*T=D and 3(T-40)=D, solve to get D=240 meters. The correct answer is C.

[ĐK (1) cho ta biết vận tốc trung bình của rùa mà không biết thời gian nên chưa đủ để kết

luận. ĐK (2) cũng không đủ. Kết hợp cả hai ĐK ta được hệ 2 phương trình 2 ẩn: 2T=D

và 3(T-40)=D. Giải ra được D=240m.]

16. The net value of a certain stock increased at a constant rate during the ten-year

period between 1990 and 2000. What was the value of the stock in the year 1998?

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[Giá trị của một cổ phiếu tăng đều trong quãng thời gian 10 năm từ 1990 đến 2000.

Giá trị của cổ phiếu đó năm 1998 là bao nhiêu nếu:]

(1) In 1991, the value of the stock was 130 U.S dollars. [Năm 1991 nó có giá 130$]

(2) In 1992, the value of the stock was 149.5 U.S dollars. [Năm 1992 giá của nó là

149,5$]

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

We are told that the stock increases its value by a constant rate and therefore we need to

find some kind of pattern in order to know the value every year. Statements (1) and (2)

taken together are sufficient since they give us the percent increase of the stock from

1991 to 1992, which is 15%. The value of the stock in 1998 can be easily calculated, add

15% every year until 1998. The correct answer is C.

[Chúng ta biết rằng cổ phiếu tăng giá đều qua từng năm do vậy ta cần tìm giá trị của từng

năm. Kết hợp (1) và (2) ta tìm được tỷ lệ tăng giá cổ phiếu từ 1991 đến 1992 là 15%. Giá

trị của cổ phiếu năm 1998 có thể dễ dàng tính được bằng cách cộng thêm 15% và mỗi

năm sau đó. Đáp án đúng là C]

17. N is a prime number bigger than 5. Which of the following expressions must be

even? [N là một số nguyên tố lớn hơn 5. Giá trị nào dưới đây sẽ chẵn?]

(a) (N+2)2.

(b) N2+2.

(d) (N+1)(N+2).

(e) (N – 2)2.

Answer D is a multiplication of two consecutive numbers, therefore one of them must be

even, and an even number multiplied by a different number is an even number. The

correct answer is D.

[D là tích của 2 số nguyên liên tiếp, 1 trong 2 số đó phải là số chẵn]

18. The original price of a car was $25,200. Because the car owner thought he could

get more money for the car, he increased the price of the car to 110% of its original

price. After a week, the car had not sold, so the owner then discounted the price by

10%, and the car was finally sold. What price was the car sold for? [Giá gốc của 1

chiếc ô tô là 25.200$. Người chủ xe nghĩ anh ta có thể kiếm được chút ít khi bán xe

nên đã phát giá 110% giá gốc. Sau 1 tuần vẫn không bán được xe, anh ta đành giảm

giá 10% và chiếc xe đã được bán. Nó được bán với giá bao nhiêu?]

(a) $25,200

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(b) $25,000

(d) $24,542

(e) $23,658

25,200 x 1.1 = 27,720. 27,720 x 0.9 = 24,948. The correct answer is C.

19. On a map, 1 inch represents 28 miles. How many inches would be necessary to

represent a distance of 383.6 miles? [Trên bản đồ, 1 inxơ thể hiện 28 dặm. Cần bao

nhiêu inxơ để biểu diễn 1 khoảng cách là 383,6 dặm?]

(a) 5.2

(b) 7.4

(d) 21.2

(e) 28.7

Dividing the requested amount of miles by the reference amount would give us the

answer in inches. 383.6 / 28 = 13.7 inches. The correct answer is C.

20. 15 Java programmers, working in a constant pace, finish a web page in 3 days. If

after one day, 9 programmers quit, how many more days are needed to finish the

remainder of the job? [15 lập trình viên bằng ngôn ngữ Java làm việc với năng suất

như nhau hoàn thành 1 tran web trong 3 ngày. Nếu sau 1 ngày đầu có 9 người bỏ đi

làm việc khác thì mất thêm bao nhiêu ngày để hoàn thành phần công việc còn lại?]

(a) 2.

(b) 4.

(d) 6.

(e) 8.

The total working days for finishing a web page are (15 x 3) 45. If after one day 9

programmers quit, only 15 working days are done and the rest of the programmers (6)

Need to finish (45 – 15) 30 days of work. It will take them 5 more days. The correct

answer is C.

[Tổng số ngày công cần thiết để làm xong công việc là 15 x 3 = 45 ngày. Sau 1 ngày thì

mới chỉ có 15 ngày công được hoàn thành. 6 người còn lại sẽ hoàn thành 30 ngày công

còn lại trong 5 ngày.]

21. Tim and Élan are 90 miles away from one another. They are starting to move

towards each other simultaneously, Tim at a speed of 10 Mph and Élan at a speed of

5 Mph. If every hour they double their speeds, what is the distance that Tim will

pass until he meets Élan? [T và E đang ở cách nhau 90 dặm. Họ bắt đầu khởi hành

đi về phía người kia cùng 1 lúc, T có vận tốc 10 dặm/giờ, E là 5 dặm/giờ. Nếu cứ sau

mỗi giờ họ lại tăng vận tốc gấp đôi thì cho đến khi gặp nhau T đi được bao nhiêu

dặm?]

(a) 30 miles.

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(b) 35 miles.

(d) 60 miles.

(e) 65 miles.

Tim is traveling at twice the speed of Élan, and so will be after they double their speeds.

The ratio between their velocities will always be 2:1 and the ratio between the distances

they will pass will also be 2:1 or 60 miles to 30 miles. Tim will go through 60 miles. The

correct answer is D.

[T đi nhanh gấp 2 lần E, sau khi tăng tốc cũng vậy. Tỷ lệ vận tốc giữa họ luôn là 2:1 và tỷ

lệ về quãng đường đi được cũng thế. Trong số 90 dặm, T đi 60 dặm còn E đi 30 dặm]

22. A spaceship in orbit rotates around the planet Pluto. How many full rotations

can the spaceship complete in 20 hours. [Một tàu không gian bay trên quỹ đạo quay

quanh sao Diêm Vương. Trong 20 giờ nó quay được bao nhiêu vòng nếu:]

(1) The radius of a single rotation is 21,000 miles. [Bán kính mỗi vòng là 21.000

dặm]

(2) The spaceship travels at 35 miles per second. [Tàu di chuyển với vận tốc 35

dặm/giây]

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

In order to calculate the time it would take to complete one rotation, you need the radius

and the velocity. Statement (1) provides the radius of rotation and (2) provides the

velocity. The distance the spaceship has to travel in order to complete one rotation is

2 ⋅ π ⋅ R = 2 ⋅ π ⋅ ( 21,000) and the speed is 35 miles per second. Divide the first by the

second and you’ll get the time it would take the spaceship to complete one rotation. The

correct answer is C.

[Để tính được thời gian cần thiết để quay được 1 vòng, ta cần bán kính và vận tốc. ĐK (1)

cho ta bán kính quỹ đạo còn (2) cho biết vận tốc. Quãng đường mà con tàu đi được khi

quay 1 vòng là 2 ⋅ π ⋅ R = 2 ⋅ π ⋅ ( 21,000) còn vận tốc là 35 dặm/giây. Chia ra ta sẽ được

thời gian cần để quay hết 1 vòng.]

23. Is X > Y?

(1) 12X = 4C.

(2) C = 3Y4.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

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(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Since we need to compare between X and Y, look at both statements together. (1) and (2)

state that: 12X = 4C = 12Y4 è X = Y4. Take Y=-1, X=1: X is bigger than Y. Take X=1

and Y=1: X is not bigger than Y. The answer is not distinct and therefore more data is

required in order to solve the question. The correct answer E.

[Kết hợp (1) và (2) suy ra X = Y4. Nếu Y = -1 thì X = 1, X>Y. Nếu lấy Y=1 thì X=1, X

không lớn hơn Y. Cần thêm dữ kiện để kết luận.]

24. What is the circumference of circle O? [Chu vi đường tròn O là bao nhiêu nếu:]

(1) The circle inscribes a square. [Đường tròn ngoại tiếp một hình vuông]

(2) The perimeter of the square is 10. [Chu vi của hình vuông là 10]

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

If a circle inscribers a square then the diagonal of the square is the diameter of the circle,

which is sufficient to find the perimeter. Statement (1) tells us about the square that is

inscribed with out any further data. Statement (2) gives us the perimeter, which is equal

to 4 times the side of the square. If we know the side of the square, we know its diagonal.

Both statements, taken together, are sufficient. The correct answer is C.

[Nếu một đường trong ngoại tiếp một hình vuông thì đường chéo của hình vuông cũng là

đường kính của hình tròn, chính là đại lượng đủ để tính ra chu vi của nó. ĐK (1) chỉ cho

biết về hình vuông nội tiếp trong đường tròn mà không cho thông tin gì hơn. ĐK (2) cho

biết chu vi hình vuông, tức là 4 lần của mỗi cạnh. Nếu ta biết cạnh hình vuông, ta sẽ tính

được đường chéo. Do vậy kết hợp cả 2 ĐK ta sẽ có đủ dữ liệu để kết luận. Đáp án đúng là

25. In a rectangular coordinate system, what is the area of a triangle whose vertices

have the coordinates (4, 0), (6, 3), and (6, -3)? [Trong hệ toạ độ vuông (toạ độ Đề-

các), một hình tam giác sẽ có diện tích bao nhiêu nếu các đỉnh cua nó là (4,0),(6,3) và

(6,-3)]

(a) 7.5

(b) 7

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(d) 6

(e) 5.5

Draw the x and y-axes, then plot the points and connect them. The length of the base is 6

units [from (6, 3) to (6, -3)] and the height is 2 units [from (6, 0) to (4, 0)]. Area of a

triangle = (base × height) / 2, so (6 × 2)/2 is 6. The correct answer is D.

[Vẽ hai trục toạ độ x và y sau đó chấm các điểm đã cho và nối chúng lại. Cạnh nằm

ngang dài 6 đơn vị (từ (6,3) đến (6,-3)) và chiều cao là 2 đơn vị (từ (6,0) đến (4,0). Diện

tích tam giác = đáy x chiều cao /2 = 6x2/2 = 6]

26. For every X, the action [X] is defined: [X] is the greatest integer less than or

equal to X. What is the value of [6.5] x [2/3] + [2] x 7.2 + [8.4] – 6.6? [Với mỗi số X,

toán tử [X] được định nghĩa như sau: [X] là số nguyên lớn nhất không vượt quá X

(toán tử phần nguyên). Tính giá trị của [6.5] x [2/3] + [2] x 7.2 + [8.4] – 6.6?]

(a) 12.6.

(b) 14.4.

(d) 16.2.

(e) 16.4.

[6.5] x [2/3] + [2] x 7.2 + [8.4] – 6.6 = 6 x 0 + 2 x 7.2 + 8 - 6.6 = 15.8. The correct

answer is C.

27. What is the decimal equivalent of (1/5)4? [Tìm số thập phân tương ứng của

(1/5)4?]

(a) 0.0032

(b) 0.032

(d) 0.003125

(e) 0.0016

444122 = 16 × − 4 = 0.0016==4

. The correct answer is E.10

51010

28. How many four-digit numbers that do not contain the digits 3 or 6 are there?

[Có bao nhiêu số có 4 chữ số mà không chứa chữ số 3 và chữ số 6]

(a) 2401

(b) 3584

(d) 5040

(e) 7200

The first digit has 7 possibilities (10 – 0,3 and 6). The other three digits have 8

possibilities each. 7*8*8*8= 3584. The correct answer is B.

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[Có 7 khả năng lựa chọn chữ số đầu tiên (trong con số có 4 chữ số): 1,2,4,5,7,8,9. Các

chữ số tiếp theo có 8 khả năng lựa chọn: 0,1,2,4,5,7,8,9. Tổng cộng có 7x8x8x8=3584

cách chọn 1 số có 4 chữ số theo yêu cầu của đầu bài.]

29. How many of the girls in a group of 200 children have an average score of 80 in

their final exams? [Có bao nhiêu nữ sinh đạt điểm trung bình 80 trong kỳ thi tốt

nghiệp trong một nhóm gồm 200 học sinh nếu]

(1) 45% of the children have an average score of 80 in their final exams. [45% số

học sinh đạt điểm trung bình 80]

(2) 50% of the children in the group are girls. [50% số học sinh là nữ]

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

From (1) by itself we can only learn that 90 kids have good grades. This statement is

insufficient. From (2) we can learn that there are 100 girls in the group. Combining the

statements doesn’t help much since we know nothing about the overlapping of (1) and (2)

and more data is required. The correct answer is E.

[Từ (1) suy ra có 90 em có điểm tốt (80). Đk này là không đủ. Từ (2) suy ra có 100 em là

nữ. Kết hợp cả 2 ĐK cũng không tìm được lời giải bởi ta không biết phần chung nhau của

hai tập hợp này. Đáp án đúng là E]

30. A paint shop sells spray cans at a flat charge of 50 cents per can. If a customer

bought 10 cans and the owner decided to give that customer a special discount on

the last two cans, what was the price of the two discounted cans? [Một cửa hàng sơn

bán loại sơn phun với giá chung là 50 xu một thùng. Nếu một khách hàng mua 10

thùng và chủ cửa hàng quyết định giảm giá cho 2 thùng cuối cùng thì giá của 2

thùng được giảm giá là bao nhiêu nếu:]

(1) The customer paid four dollars and twenty cents total for the ten cans. [khách

hàng phải trả 4,2$ cho 10 thùng]

(2) The customer bought the ten cans for an average price of 42 cents per can.

[Khách hàng mua 10 thùng với giá trung bình 42 xu/thùng]

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

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(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) tells us that 10 cans cost 4.2 dollars instead of 5 dollars and therefore the

last two cans were sold for 20 cents. This statement is sufficient. Statement (2) also tells

us that the entire amount of cans cost 4.2 and therefore this statement is also sufficient.

The correct answer is D.

[(1) cho ta biết rằng 10 thùng giá 4,2$ thay vì 5$ do vậy 2 thùng cuối cùng giá 20 xu. Vậy

(1) là điều kiện đủ. (2) cũng cho biết 10 thùng giá 4,2$, tương đương với ĐK (1)]

31. Does the product of XYZW = 16? [Tích của XYZW có bằng 16 nếu:]

(1) Y = 1.

(2) X = 4Y and ZW = 4Y2

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

From (1) we know the value of Y only, which is 1. From (2) and (1) we know the value

of all the other parameters, X = 4 and ZW = 4. Therefore, (1) and (2) together are

sufficient. The correct answer is C.

[(1) => Y = 1. Từ (1) & (2) => X = 4 và ZW = 4. Vậy (1) kết hợp với (2) là đủ.]

32. The telephone company wants to add an area code composed of 2 letters to every

phone number. In order to do so, the company chose a special sign language

containing 124 different signs. If the company used 122 of the signs fully and two

remained unused, how many additional area codes can be created if the company

uses all 124 signs? [Công ty bưu điện muốn thêm một mã vùng gồm 2 ký tự vào mỗi

số ĐT. Để làm như vậy, công ty chọn một ngôn ngữ kí hiệu đặc biệt có 124 kí tự

khác nhau. Nếu công ty đã sử dụng 122 ký tự, còn 2 ký tự chưa sử dụng thì khi sử

dụng hết 124 ký tự sẽ tạo được thêm bao nhiêu mã vùng?]

(a) 246

(b) 248

(d) 15,128

(e) 30,256

The phone company already created 122*122 area codes, now it can create 124*124.

1242-1222=(124+122)(124-122) = 246*2 = 492 additional codes. The correct answer is C.

[Công ty đã sử dụng 122 ký tự tức là tạo được 122*122 mã vùng (mỗi mã vùng là 2 ký

tự). Khi sử dụng hết 124 ký tự thì tạo được 124*124 mã vùng. Số mã vùng mới được tạo

thêm là 124*124-122*122 = 492]

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33. The average (arithmetic mean) of seven numbers is 12.2. If the sum of four of

these numbers is 42.8, what is the average of the other 3 numbers? [Trung bình

cộng của 7 con số là 12,2. Nếu tổng của 4 số trong số đó là 42,8 thì trung bình cộng

của 3 số còn lại là bao nhiêu?]

(a) 12.4

(b) 14.2

(d) 18.6

(e) 19.2

If the average of 7 numbers is 12.2, we can solve for their sum: 7 × 12.2 = 85.4. If four of

these numbers total 42.8, then by subtracting 42.8 from 85.4, we get the sum of the other

three numbers, 42.6. To find the average of these three numbers, we divide their sum by

their number: 42.6/3 = 14.2. The correct answer is B.

[Do trung bình cộng của 7 số là 12,2 nên tổng của chúng là 12,2x7=85,4. Nếu tổng của 4

số là 42,8 thì tổng của 3 số còn lại là 42,6. Suy ra trung bình cộng của 3 số còn lại là 42,6

: 3=14,2]

34. If A = 2B, is A4 > B4?

(1) A2 = 4B2.

(2) 2A + B < A/2 + B.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) is insufficient. Take A=0 and B=0, (1) is correct yet A4 is not bigger than

B4. Take different numbers, A=6 and B=3 A4 is larger than B4. Statement (2) is sufficient.

The only possible way that A will not be larger than B is if they are both zero. (2) Claims

that A < 0 and therefore A cannot be zero and this statement is sufficient, A4 is bigger

than B4. The correct answer is B.

[(1) là không đủ. Lấy A=0, B=0 thì (1) thoả mãn nhưng A4 không lớn hơn B4. Lấy 1 cặp

khác ví dụ A=6, B=3 thì (1) cũng đúng nhưng A4 lớn hơn B4. Vậy (1) là ĐK không đủ để

kết luận. Ngược lại, ĐK (2) là đủ. Do A=2B nên A4 = 16B4. Chỉ có 1 khả năng để A4

không lớn hơn B4 là cả hai đều bằng 0. (2) ó 2A<A/2 ó A<0.]

35. 5 numbers are randomly chosen. If their average is 20, how many of the

numbers are larger than 15? [Có 5 số được chọn ra ngẫu nhiên, biết trung bình

cộng của chúng là 20 thì có bao nhiêu số lớn hơn 15 nếu:]

(1) One of the numbers is 15. [Một trong các số đó là 15]

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(2) The average of three of the numbers is 15. [Trung bình cộng của 3 trong số đó là

15]

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Lets look at two cases, where each one will give out different results. One of the numbers

is 15 and three more are 15, 15 and 15 (with an average of 15). The sum of all the

numbers should be (20 x 5 = 100). The sum of the numbers we picked up is (15 x 4 = 60)

and therefore the last number should be 40 which is greater than 15. The answer in this

case is 1. Take another case: One of the numbers is 15 and three more are 14, 15 and 16

(with an average of 15). In this case there will be 2 numbers over 15. More data is

required. The correct answer is E.

[Ta sẽ lấy 2 ví dụ để thấy kết quả của chúng khác nhau. Thứ nhất lấy 5 số

(15,15,15,15,40) thảo mãn cả (2) và (1). Trường hợp này chỉ có 1 số 40 là >15. Trường

hợp thứ hai là (14,15,16,15,40) cũng thoả mãn cả (1) và (2). Lúc này lại có 2 số là 16 và

40 > 15. Vậy kết hợp cả (1) và (2) cũng chưa đủ để kết luận]

36. A drawer holds 4 red hats and 4 blue hats. What is the probability of getting

exactly three red hats or exactly three blue hats when taking out 4 hats randomly

out of the drawer and immediately returning every hat to the drawer before taking

out the next? [Một ngăn tủ chứa 4 mũ đỏ và 4 mũ xanh. Xác xuất lấy được chính

xác 3 mũ đỏ hoặc 3 mũ xanh là bao nhiêu khi lấy ra 4 lần một cách ngẫu nhiên và

mỗi chiếc lấy ra được trả lại ngăn tủ trước khi lấy chiếc tiếp theo?]

(a) 1/8

(b) 1/4

(d) 3/8

(e) 7/12

Getting three red out of 4 that are taken out has 4 options (4!/(3!*1!)) each option has a

probability of (1/2)4 since drawing a red or blue has a 50% chance. 4*1/16= ¼ to get

three red hats. The same goes for three blue hats so ¼+¼ =1/2. The correct answer is C.

[Lấy ra được 3 trong số 4 mũ đỏ thì có 4 cơ hội (=4!/(3!x1!)). Mỗi cơ hội lại có xác xuất

(1/2)4 do ngăn tủ có số mũ đỏ và xanh như nhau. Vậy xác xuất lấy được 3 mũ đỏ là 4 x

1/16 = ¼. Xác xuất lấy được 3 mũ xanh cũng tương tự. Vậy xác xuất lấy được 3 mũ xanh

hoặc 3 mũ đỏ là ¼+ ¼ =1/2.]

37. If 4XZ + YW = 3 and XW + YZ = 6, what is the value of the expression (2X +

Y)(2Z + W)?

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(a) 9.

(b) 12.

(d) 16.

(e) 18.

(2X + Y)(2Z + W) = 4XZ + 2XW + 2ZY + WY = (4XZ+WY) + 2(XW+YZ) = 3 + 2 x 6

= 15.

The correct answer is C.

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GMAT TEST 2 – Key for MATH SECTION

(37 questions, 75 minutes)

1. X is an even number and Y is a positive odd number. Which of the following

expressions cannot be even?

(a) (XY) Y

(b) X3Y3

(d) XY

(e) Y2

The fastest way to solve this problem is by plugging in some numbers.

Lets say: X = 2, Y = 1.

According to answer e: 1 x 1 = 1 and that must be an odd number.

The correct answer is E.

2. How much interest will $2,400 earn at an annual rate of 8% in one year if the

interest is compounded every 4 months?

(a) $141

(b) $150

(d) $234

(e) $312

Here, it is enough to calculate the simple interest of 8% per year.

$2,400 x 8/100 x 1 = $192. Since we are calculating as a compounded rate, the interest

should be a bit higher, or C as the correct answer.

3. What is the value of P?

(1) P is even.

(2) P is a square of a prime number.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) alone is insufficient since there is infinite number of possibilities.

Statement (2) alone is insufficient since there is an infinite number of primes.

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The combination of the statements is sufficient since the only even square of a prime

number is can be 22, which is 4. Both statements, taken together, are sufficient. The

correct answer is C.

4. If AB = 40, what is the value of AB(A + 2B)?

(1) A – B = -18.

(2) A2B = 80.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) alone is insufficient since we need the value of A + 2B.

Statement (2) is sufficient. A2B = A(AB) = A(40) = 80 è A=2 and B=20.

Now, we know that A + 2B = 42 and we can calculate the required expression.

The correct answer is B.

5. If X3Y = 24, what is the value of (X3Y3 – X2Y2)?

(1) X2Y2 = 36.

(2) X3Y2 = 72.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) is insufficient since the expression given is only one of two needed.

Statement (2) by itself is sufficient since X3Y2 = (X3Y)Y = 24Y = 72 è Y = 3.

If Y=3 then X3=8 and so X=2.

The expression required is a combination of X and Y and is then calculable.

The correct answer is B.

6. X is a prime number bigger than 10. Also, Y = X+X3+X5+X7.

What is definitely true about Y?

(a) Y is a prime number.

(b) Y is odd.

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(d) Y is divisible by 3.

(e) Y is divisible by 7.

Because X is a prime number bigger than 10, it must be odd. Ignoring the powers of X in

the expression of Y, you’ll see that Y is a sum of 4 odd numbers therefore it must be

even. The correct answer is C.

7. What is the least integer that is a sum of four different primes each greater than

20?

(a) 79

(b) 83

(d) 133

(e) 169

Find the sum of the first 4 primes larger than 20: 23+29+31+37=120. The correct answer

is C.

8. If X>X, the average of X and Y is Z, and the average of Z and X is W, what is the

value of (x-w) / (w-y) = ?

(a) 1/4

(b) 1/3

(d) 3

(e) 4

The best way is to take an example: Assume x=2 and y=10, then their average z=6. The

average of x=2 and z=6 is w=4. Therefore:

x−w 2−4 −2 1

===.

w − y 4 − 10 − 6 3

The correct answer is B.

9. One quarter of the workers at the factory are clerical, one fifth are technical, half

are administrative and the other 25 are managerial. How many workers total are

there in the factory?

(a) 250

(b) 366

(d) 500

(e) 2500

First, calculate the fraction of the 25 managerial workers out of the total number of

workers by subtracting the fractions given from 1.

1115 + 4 + 1019 1

1− − − = 1−= 1−=.

4522020 20

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The 25 managerial workers are 1/20 of the workers, therefore, 25 x 20 = 500, a total

number of 500 workers. The correct answer is D.

10. The price of a product is $a. Bill bought s products and then sold 80 percents of

them $b. which of the following represents the whole deal if Bill’s profit was three

times the cost?

(a) 0.8sb = 4sa.

(b) (1-0.8)s –ab = 3s.

(d) (s-0.8s)(b-a) = 3.

(e) (s0.8s) / (b-a) = 3.

This statement is equivalent to 0.8sb – sa = 3sa. The income is 0.8sb, the cost is sa, so the

profit is three times the cost.

The correct answer is A.

11. X equals to Y% of what number?

(1) X = 3Y.

(2) 6Y+2X = 56X/14.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

From (1) we have X and Y and therefore we can find A easily, A = 300.

(2) Is identical to (1), simplify it and see that it can be written as X = 3Y.

Either statement by itself is sufficient. The correct answer is D.

12. Which expression is larger 1/(5 - X) or X/5?

(1) X < 8.

(2) X > -8.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

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The easiest way to solve such a problem is to plug in numbers.

Use both statements to see that they are both insufficient even together.

Take X=0: 1/(5 – X) = 1/5 and X/5 = 0, in this case the first expression is larger.

Take X=7: 1/(5 – X) = -1/2 and X/5 = 7/5, in this case the second expression is larger.

We can see that the answer is dependent on which numbers we choose and more data is

required in order to determine the answer. The correct answer is E.

13. X is a two-digit number. If the ratio between the units digit and the tens digit is 1

to 2, what is the value of X?

(1) The sum of the digits multiplied by the tens digit is 54.

(2) The product of the digits divided by 2 is 9.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

There are a limited number of possibilities: 21, 42, 63 and 84.

According to statement (1), the only number that is compatible is 63.

According to statement (2), the only number that is compatible is also 63.

Either statement alone is sufficient. The correct answer is D.

14. A grocer bought 24Kg of coffee beans at price X. After a while one third of the

stock got spoiled so he sold the rest for 200$ per Kilo and made a total profit of

twice the cost. What must be price X?

(a) 66 2/3$.

(b) 50 1/3$.

(d) 33 1/3$.

(e) 24 1/2$.

The total income is 16x200. The cost is 24X, and the total profit is 3200 – 24X = 48X.

The correct answer is C.

15. Kenny is three times older than Bob. In P years he will be twice older than Bob

will be Q years later. Which of the following represents Kenny’s age comparing to

Bob’s? (If X = Kenny’s age)

(a)

(c)

(d)

X + P = 6X(P+Q).

2(X+P) = 3X + Q.

(X+P)/2 = X/3 + P + Q.

3(X+P+Q) = 2X.

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(e) 3X = 2(P + Q).

X is Kenny’s age. X/3 is Bob’s age. Answer C is equivalent to (X+P) = 2(X/3+P+Q),

which means that in P years, X+P is 2 times (X/3+P) plus Q years. The correct answer is

16. A Cuban cigar would cost 1 dollar less than 1.5 times a French cigar, had the

French cigar cost 0.7 dollar less than it does now. An Arabian cigar costs 50 cents

more than 1.5 times the Cuban cigar. The three cigars together cost 74.7 dollars.

What is the price of the French cigar?

(a) 16.7$.

(b) 23$.

(d) 35$.

(e) 37.4$.

Sign the French cigar as X. The Cuban cigar is 1.5(X-0.7) - 1.

The Arabian cigar is 1.5[1.5(X-0.7)-1] + 0.5.

The sum of all the three is 74.7. The correct answer is A.

17. Ashley paid 5 dollars for 1 notebook and 1 pencil. If both prices were integers,

how many pencils did Ashley buy if she paid 93 dollars for the pencils and for 15

notebooks?

(a) 6.

(b) 16.

(d) 21.

(e) 26.

One notebook can cost 1, 2, 3 or 4 dollars. Subtract 15 times each price from 93, and

check if what you get is divisible by 5 minus the price of the notebook. The number could

be 24, 21 or 33. The correct answer is D.

18. What percent is X of Y?

(1) Y is bigger than 2X by 54.

(2) X is smaller than 3Y by 72.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) can be written as: Y = 2X + 54.

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Statement (2) can be written as: X = 3Y – 72.

Combining both statements, we have two different equations containing X and Y and so

we can solve and find the value of X and Y and calculate what is X percent of Y.

The correct answer is C.

19. A store bought Q windows at $150 per window and W shelves at $75 per shelve.

What is the total price of the windows and the shelves?

(1) The Q windows cost $600.

(2) Q + W/2 = 12.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) is all about the windows and therefore it is not sufficient by itself.

Statement (2) tells us that Q + W/2 = 12, this is sufficient because the total price of

windows and shelves is 150Q + 75W.

Multiply the data in statement (2) be 150 to get: 150Q + 75W = 18,000.

Statement (2) is sufficient by itself.

The correct answer is B.

20. What is the perimeter of a rectangle having an area of 60?

(1) The length and width of the rectangle are even integers smaller than 25.

(2) The length of the rectangle is larger than three times the width.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

The question tells you that the area is 60, the area of a rectangle is equal to length x

width, in order to find the perimeter, one should find the value of the length and the value

of the width.

Statement (1) tells us that both the length and the width are even integers and therefore

we know their exact values because 60 can be factorized to: 1x60, 2x30, 3x20, 4x15,

5x12 and 6x10.

The only even integers smaller than 25 are (6x10) and we know the perimeter.

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The correct answer is A.

21. X is a positive integer, is X even?

(1) 9X2 is divisible by 4.

(2) 3X + 2 is divisible by 8.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) tells us that 9X2 is an even number (it’s divisible by 4) and therefore X2

must be an even number. An odd number squared is also odd and therefore X must be

even, this statement is sufficient.

Statement (2) tells us that 3X + 2 is an even number (it’s divisible by 8).

If you subtract 2 from (3X+2) the result will still be even. If 3X is even than X must be

even. This statement is also sufficient to answer the question. The correct answer is D.

22. The sum of the ages of 22 boys and 24 girls is 160.What is the sum of ages of one

boy plus one girl, if all the boys are of the same age and all the girls are of the same

age, and only full years are counted?

(a) 5.

(b) 6.

(d) 8.

(e) 9.

Changing the equality: 22X + 24Y = 160, we get: X = (80-12Y) / 11. Since X must be a

positive integer, Y must be smaller than 7. The numerator will be 11, 22, 33, 44 etc. Y

must be 3 and X=3. X+Y = 7. The correct answer is C.

23. X percents of the rooms are suits, and Y Percents of the rooms are painted light

blue?

Which of the following best represents the least Percentage of the light blue painted

suits?

(a)

(c)

(d)

X-Y.

Y-X + 100.

100X-Y.

X+Y-100.

100-XY.

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D shows us the difference between the sum of the percentages and 100 percent. If

X+Y>100, this percent must be subtracted from the sum X+Y in order to get 100 percent.

Sum of sets minus the total equals the congruence. The correct answer is D.

24. David bought 13 BMW cars for a total price of 1,105,000 dollars. If he wants to

make a profit of 39,000 dollars in the deal, at what price should he sell one car?

(a) 85,000$.

(b) 88,000$.

(d) 124,000$.

(e) 139,000$.

The buying price for one car is 1,105,000 / 13 = 85,000.The wished for a car is 39,000 /

13 = 3,000. So the selling price must be 85,000 + 3,000. The correct answer is B.

25. Loren bought a roll of cloth and sold it for a 5% profit based on the selling price.

If Loren‘s profit was $45.5 total on the cloth, how much did it cost her to buy the

cloth?

(a) $455.

(b) $525.5.

(d) $810.5.

(e) $864.5.

5% of the total price of the cloths is 45.5 dollars, multiply this number to get the entire

100% of the total selling price: (45.5 x 20 = $910). Now subtract the profit $45.5 to get

the cost: 910-45.5=$864.5. The correct answer is E.

26. X is a positive integer, is X even?

(1) 9X2 is divisible by 4.

(2) 3X + 2 is divisible by 8.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) tells us that 9X2 is an even number (it’s divisible by 4) and therefore X2

must be an even number. An odd number squared is also odd and therefore X must be

even, this statement is sufficient.

Statement (2) tells us that 3X + 2 is an even number (it’s divisible by 8).

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If you subtract 2 from (3X+2) the result will still be even. If 3X is even then X must be

even. This statement is also sufficient to answer the question.

27. If 10% of the employees of the state fare are police officers, what is the number

of employees who are not police officers?

(1) 5% of the police officers employed in the fare are woman.

(2) 45% of the employees at the state fare are woman.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

In order to know how many people aren’t officers you need a fix value, in other words

you need to translate percentage into real values.

Statement (1) doesn’t give any real numbers and therefore it’s not sufficient.

Statement (2) is the same as one in that matter, if we wanted to know the percentage of

the woman officers, the statements would have been satisfying. More data is required.

The correct answer is E.

28. In a recent tender, X people participated. 35% of the X people, who made an

offer won the specific tender they participated in. 70% of the rest, were

disappointed from the result of the tender. Which of the following expressions

represents the number of people who weren’t disappointed although they didn’t win

the tender?

(a) 39X/200.

(b) 25X/50.

(d) 35X/250.

(e) 90X/200.

65X/100 is the number of people who didn’t win the tender. 30% of that number is the

number of people who weren’t deeply disappointed about the fact that they didn’t win. 30

x (65X/100) /100 = 19.5/100 = 39/200. The correct answer is A.

29. What is 0.01 x 5 x 0.03 in terms of percent?

(a)

(c)

(d)

15%

1.5%

0.15%

0.015%

0.0015%

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Since 0.01 and 0.03 each have 2 decimal places, their product must have 4 (2 + 2)

decimal places. Because 1 x 3 x 5 is 15, you need to add 2 zeros to get the correct number

of decimal places, so the product of 0.01, 5 and 0.03 is 0.0015. To change a decimal to a

percentage you multiply by 100 and move the decimal point 2 places to the right, so

0.0015 is 0.15%. The correct answer is C.

30. In a barrel of juice there is 30 liters; in a barrel of beer there are 80 liters. If the

price ratio between a barrel of juice to a barrel of beer is 3:4, what is the price ratio

between one liter of juice and one liter of beer?

(a) 3:2.

(b) 2:1.

(d) 4:3.

(e) 3:4.

Pick numbers: a barrel of beer costs $40 and a barrel of juice costs $30 according to the

given ratio. One liter of beer will cost $0.5 and one liter of juice will cost $1. Therefore

the price ratio is 2:1. The correct answer is B.

31. Is X greater than 1?

(1) X > X2.

(2) –X < -X2.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) tells you that X is greater than X2, that is true only if X is between zero and

one and so the answer to the question is no. Statement two is identical to one, multiply

both sides by (-1) and don’t forget to change the sign of the inequality. Either statement is

sufficient by itself. The correct answer is D.

32. What is the sum of 7 consecutive integers?

(1) The median of the seven integers is 8.

(2) One of the integers is 7.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

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even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

From statement (1) we have only one set of numbers. We have 5, 6, 7, 8, 9, 10 and 11.

Statement (2) provides no additional data because we can see several sets containing 7.

The correct answer is A.

33. M is a positive integer, is M odd?

(1) 2M3 + 2M is divisible by 8.

(2) M + 10 is divisible by 10.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) tells us that 2M3 + 2M is divisible by 8 and so M4 + M is divisible by 4 and

is even. We have two choices: M3 and M are either odd or even. This statement is

insufficient.

Statement (2) is sufficient, if M + 10 is divisible by 10 then M + 10 is an even number.

This statement is sufficient and the answer is B.

34. From the starting point in a boat race, one competitor started to sail north at a

speed of 1.6 Km/h, the other competitor started to sail west at a speed of 1.2 Km/h.

What is the distance in Km between the two competitors after 5 hours?

(a) 10.

(b) 12.

(d) 14.

(e) 15.4.

One is going north and the other is going west. The distance can be calculated using the

Pythagorean theorem. One made a distance of 1.6 x 5 = 8 Km, The second one did 1.2 x

5 = 6 Km. The distance between them is the square root of (64 + 36) = 10 Km. The best

answer is A.

35. George can fill Q cans of paint in 3 minutes. If there are R cans of paint in one

gallon, how many gallons can George fill in 45 minutes?

(a) 30R/Q.

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(b) 15R/Q.

(d) 5Q/R.

(e) 15Q/R.

George can fill Q/3 cans of paint in one minute à There are R cans in one gallon, so

R/(Q/3) = 3R/QIs the time it takes George to fill one gallon (in minutes). In 45 minutes

George can fill up 45/(3R/Q) = 15Q/R. The correct answer is E.

36. If (4 # 2 = 14) and (2 # 3 = 6), what can replace (a # b)?

(a) ab.

(b) (a+3)b

(d) ab – 2.

(e) ba + 1.

Plug in the answers.

(a) (4 # 2) = 8. The answer should be 14.

(b) (2 # 3) = (2 + 3)3 = 15. The answer should be 6.

(d) (4 # 2) = (42 – 2) = 14. This is the right answer; check (2 # 3) also.

The correct answer is D.

37. In a rectangular coordinate system, what is the square root of the area of a

trapezoid whose vertices have the coordinates (2, -2), (2, 3), (20, 2), (20, -2)?

(a) 7.5

(b) 9

(d) 12.25

(e) 14

Draw the x and y-axes, then plot the points and connect them.

The area of a trapezoid is (base1 + base2) x (height) / 2.

Base1 = 5, base2 = 4, height = 18 thus the area is 9 x 9 = 81.

The answer to the question is the square root of 81, meaning 9.

The correct answer is B.

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GMAT Test 3

37 questions – 75 minutes

1. An investment yields an interest payment of $68 each week. If the simple annual

interest rate is 7.5%, what is the amount of the investment assuming there only 48

weeks are calculated in a year?

(a) $28,600

(b) $30,430

(d) $37,860

(e) $43,520

Principal × percent interest = interest earned

Principle × (0.075)× 1/(12 x 4) = $68. Solve to find the principal (68 × 12 x 4)/0.075=

$43,520. The correct answer is D.

2. The flying acrobatic team is made up of 120 airplanes. The team wants to form a

rectangular formation with X planes in a row and Y planes in a column. If the

number of airplanes in a row is no less than 4 and no more than 30, how many

different combinations of rectangular shapes are possible?

(a) 4.

(b) 5.

(d) 8.

(e) 10.

Use the factors of 120 are: 1x120, 2x60, 3x40, 10x12, 4x30, 5x24, 6x20 and 8x15.

We are looking for combinations of (row x column) that are all between 8 and 30.

The possibilities are: 8x15, 15x8, 4x30, 30x4, 5x24, 24x5, 6x20, 20x6, 10x12 and 12x10.

That is 10 possibilities total. The correct answer is E.

3. A storeowner estimates that the average price of type A products will increase by

25% next year and that the price of type B products will increase by 10% next year.

This year, the total amount paid for type A products was $4500 and the total price

paid for type B products was $8300. According to the store owner’s estimate, and

assuming the number of products purchased next year remains the same as that of

this year, how much will be spent for both products next year?

(a) $14,755

(b) $15,325

(d) $16,225

(e) $17,155

The total estimated amount that will be spent on type A product next year:

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125

= $5625

100

The total estimated amount that will be spent on type B product next year:

110

$8300 ×= $9130

100

Giving a total spending of $14,755. The correct answer is A.

$4500 ×

4. If a cube has a volume of 64 cubic feet, what is its lateral area?

(a) 16

(b) 24

(d) 64

(e) 96

The side of a cube equals the cubic root of its volume; in this question it is 4. The lateral

area of a cube equals the sum of four faces; in this question each face is 16, and the

lateral area is four times that, giving 64. The correct answer is B.

5. Is the integer X even?

(1) X is divisible by 7.

(2) X is divisible by 11.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

The fact that X is divisible by 7 does not help figure out whether it is odd or even, both

even and odd numbers could be divisible by 7. The same applies for 11; both even and

odd numbers could be divisible by 11. Both statements taken together do not shed a new

light on the matter, there could be even and odd numbers that are divisible by 7 and 11.

The correct answer is E.

6. Is the product XY divisible by 22?

(1) X is divisible by 4.

(2) Y is divisible by 11.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

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even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

The prime factors of 22 are 2 and 11. Hence, if X is divisible by 4, it is divisible by 2 and

if Y is divisible by 11, surely XY is divisible by 22. The correct answer is C.

7. A seven-digit combination lock on a safe has zero exactly three times, does not

have the digit 1 at all. What is the probability that exactly 3 of its digits are odd?

(a) 1/2

(b) 1/3

(d) 4/16

(e) 9/16

Since three digits are zero, only 4 digits are left for consideration (of which, none is

zero). Since 1 does not appear in the numbers, there are 4 even numbers (without 0), and

4 odd numbers (without 1) to choose from. The probability for every digit to be odd is

1/2. There are 4 different ways to arrange 3 odd numbers and one even number in 4

places. Each of these ways has a probability of (1/2)4. And together: 4 x (1/2)4 = 4/16.

The correct answer is D.

8. Pipe A fills a swimming pool in 4 hours. Pipe B empties the pool in 6 hours. If

pipe A was opened at 8:00 am and Pipe B at 9:00 am, at what time will the pool be

full?

(a) 15:00

(b) 17:00

(d) 19:00

(e) 20:00

From 8:00 am to 9:00 am, Pipe A, which fills the pool in 4 hours, was open for one hour,

filling one quarter of the pool. From 9:00 am, the two Pipes worked together at the rate

of: 1/4 - 1/6 = 1/12, one pool in 12 hours. Since the pool was already one quarter full at

9:00 am, it will take only 9 hours to fill the remaining three quarters of the pool. 9 hours

from 9:00 am is 18:00. The correct answer is C.

9. What is the value of (a+b)?

(1) a2-b2=133.

(2) a-b=7.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

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even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Since a2-b2=(a+b)(a-b), 133=(a+b)7, and (a+b)=19.

Both statements are needed to solve the question. The correct answer is C.

10. What is the value of X+Z?

(1) X+Y=11

(2) Z+Y=13

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Each statement alone leaves out one of the terms x or z, so we cannot find their sum

using any statement alone. Moreover, even combining both statements does not help:

x + y = 11 z + y = 13

y = 11 − x y = 13 − z

11 − x = 13 − z

z − x = 13 − 11

It is only possible to find z-x. The correct answer is E.

11. What was the total amount John earned on his two investments?

(1) John received an annual interest of 5% on one investment and 13% on the other.

(2) John invested a total of $15,000 on both investments.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Knowing the interest alone is not enough to calculate the profit.

Knowing the total amount invested is not enough to calculate the profit, unless we have

the interest rate.

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Since we have no knowledge of the amount invested in each investment, there is no way

to know how much was earned. The correct answer is E.

12. What is the sum of the two smallest integers in a set of different positive

integers?

(1) There are 4 integers in the set.

(2) The average of the integers in the set is 3.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

The only two possible sets that have 4 different positive integers and an average of 3 are:

[1, 2, 3, 6] and [1, 2, 4, 5]. The sum of the two smallest integers in both sets is always 3.

The correct answer is C.

13. In a school with 5 classes, each class has 2 students less then the previous class.

How many students are there in the largest class if the total number of students at

school is 95?

(a) 17

(b) 19

(d) 23

(e) 25

If X is the number of students in the largest class, then the numbers of students in the

other classes are: X-2, X-4, X-6 and X-8. The total number of students is:

X + (X − 2) + (X − 4) + (X − 6) + (X − 8) = 95

and

5X − 20 = 95 ⇒ 5X = 115 ⇒ X = 23

The correct answer is D.

14. A cylindrical ice cream container is half filled with ice cream. A second ice

cream container, half the size of the first one is filled to three quarters of its volume

with ice cream. What fraction of the total volume of the two containers is filled with

ice cream?

(a)

(c)

(d)

9/12

7/12

2/3

5/6

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(e) 15/24

Take 40 litters as the volume of the large container to ease the calculations. Of this

volume, 20 litters are ice cream. The volume of the second is 20 litters and 15 litters are

ice cream. So, the total volume of both containers is 40+20=60, and the ice cream is 20 +

15=35. Now, 35/60 = 7/12. The correct answer is B.

15. A 48 gallon solution of salt and water is 10% salt. How many gallons of water

must be added to the solution in order to decrease the salt to 8% of the volume?

(a) 8

(b) 12

(d) 14

(e) 16

Solve a combined average problem:

48 × 10% + X × 0%

= 8%

(48 + X)

The correct answer is B.

16. Five years ago, the average age of the X students in the class was n. What is their

average age now, if Y more students whose average age was also n five years ago,

joined the class?

(a) n + 5

(b) n - 5

(d) (x + y)/2 - 5

(e) None of the above

The average age of both groups, x and y was n five years ago. Therefore, it is possible to

treat them as one group with that average. Five years later, each student added 5 years to

his age, adding 5 to the average of the whole class, and making it n+5. The correct

answer is: A.

17. The “Racing magic” takes 120 seconds to circle the racing track once. The

“Charging bull” makes 40 rounds of the track in an hour. If they left the starting

point together, how many minutes will it take for them to meet at the starting point

for the second time?

(a) 3

(b) 6

(d) 12

(e) 15

The rate of the “racing magic” is 40 rounds per hour, or 1 round every 1.5 minutes. The

rate of the “Charging bull” is 1 round every 120 seconds, or 1 round every 2 minutes.

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The best way to solve such a question is to find the least common denominator between

the two rates. At that point, they will meet for the first time, and when multiplied by 2,

we find the second time they meet:

1134

↔=↔

21.5 66

They will meet for the first time after 6 minutes and for the second time after 12 minutes.

The correct answer is D.

18. If cookies are put in a jar and the jars of cookies are packed in a carton box, how

many cookies does one carton box contain?

(1) Every carton box is filed to half its volume.

(2) Twenty cookies are put in each jar, and 12 jars are put in each carton box.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

The first statement gives no information of the number of cookies. The volume of the

carton box or the percent of its volume filled, do not help in finding any number. The

second statement gives all the information needed in order to find the number of cookies

packed in each carton box. The number of cookies per jar, and the number of jars per box

are sufficient to calculate real numbers. The correct answer is B.

19. If X and Y are integers, what is the value of X?

(1) 6X = 4 – 9Y

(2) 4/3 – 3Y = 2X

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

In order to find the value of a term in an equation with two unknowns, two different

equations are needed. Since both equations given in statements 1 and 2 are, in fact, the

same equation in a different shape, it is impossible to find the value of X or Y. (6X = 4 –

9Y) ó (4/3 –3Y = 2X). The correct answer is E.

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20. A restaurant spends one quarter of its monthly budget for rent and half of the

rest for food and beverages. What percentage of the budget does the restaurant

spend for food and beverages?

(a) 23.5%

(b) 32.5%

(d) 37.5%

(e) 75%

After spending one quarter of the budget on rent, 75% is left, of that half = 37.5% is spent

for food and beverages. The correct answer is D.

21. If x oranges cost as much as y peaches do, and peaches cost 24 cents each, how

many dollars does each orange cost?

(a) 2400 / (xy)

(b) 24y / x

(d) 24y / 100x

(e) y / 24x

Compare x oranges to y peaches at the price of 24 cents per peach. Divide by 100 to

receive the amount in dollar:

24y24y

x × Oranges = y × Peaches ⇒ x × O =⇒O=

100100x

The correct answer is D.

22. The sum of the volumes of three spheres, each with radius r, equals to the

volume of a single sphere with radius R. What is the ratio between r and R?

(a) 1: 3 3

(b) 1: 3

(d) 1: 3 2

(e) 1:3

Compare the volumes:

44r1

3 × π r3 = π R3 ⇒ 3 × r3 = R3 ⇒ = 3

33R3

The correct answer is A.

23. If x and y are primes, and x@y=(1/x + 1/y), is x@y < 1?

(1) x=y

(2) x+y=4

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

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(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Since x=y and both are primes, (1/x + 1/y) could equal 2/3 when x=y=3, or equal 1 when

x=y=2. Therefore, the first statement is not sufficient to solve.

According to the second statement, x+y=4, the only primes to match are x=y=2. Now we

know that (1/x + 1/y) =1 and the question is solved. The correct answer is B.

24. A ball with a diameter of 10 cm is inscribed inside a rectangular box so that it

touches all internal faces of the box. What is the volume trapped between the box

and the ball?

(1) The box is a cube

(2) The surface area of the box is 600 cm2.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) tells us that the diameter of the ball is the height of cube, so we can find its

volume.

Statement (2) actually tells us that the box is a cube, and the rest is similar to statement

one. The correct answer is D.

25. A rectangular box is inscribed inside a cylinder. Both the width and length of the

box is 2 cm. What is the volume of the cylinder?

(1) The volume of the box is 20 cm3.

(2) The radius of the cylinder is 2 .

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

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Statement (1) can provide us the height of the cylinder while statement (2) gives us data

we already can conclude from the question. The correct answer is A.

26. A car travels a distance of 160 miles in 2 hours and 40 minutes, what is the speed

of the car in miles per hour?

(a) 54

(b) 60

(d) 116

(e) 120

The car travels 160 miles in 160 minutes, that is 1 mile per minute. So, it travels 60 miles

in 60 minutes, or 60 miles per hour. The correct answer is B.

27. If 16X+7 = 26 X+18 , what is x?

(a) 9

(b) 8

(d) 6

(e) 5

X+716 = 26 X +18 ⇒ 24( X + 7) = 26 X +18 ⇒ 4( X + 7) = 6X + 18

And x=5. The correct answer is: E.

28. In the following sequence: [x, y, z, 5, 7, 16, 28] each number is equal to the sum

of the three numbers preceding it. What is x+y?

(a) -5

(b) -1

(d) 5

(e) 6

First find z: z+5+7=16, therefore, z=4. Now, y+4+5=7 making y=(-2). Now we can find

x: x+(-2)+4=5, so, x=3. Hence, x+y=3+(-2)=1. The correct answer is: C.

29. John has $1,600 at the beginning of his trip, after spending money, he still has

exactly $800 less than he spent on the trip. How much money does John still have?

(a) $200

(b) $400

(d) $800

(e) $1,200

John spends a sum of m dollars and still has m-800 left. So, m+m-800=1,600, and

m=$1,200. Therefore, John still has $1,600-$1,200=$400. The correct answer is B.

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30. The odds of winning first prize at the casino are 1 to 200. The odds of winning

second prize are 1 to 100 and of winning third prize are 1 to 50. If no one person can

win more than one of the prizes, what is the probability of not winning the first,

second or third prize?

(a) 14/200

(b) 47/50

(d) 7/200

(e) 97/200

Calculate the odds of not winning first, second or third prize by deducting the chances of

winning first, second or third from 1:

11  1931

1−++ =

 200 100 50  200

The correct answer is C.

31. A cube has three of its faces painted half red, half white. The other three faces

are completely painted white. What is the ratio between the total red painted and

the total white painted areas of the cube?

(a) 1:6

(b) 3:6

(d) 1:4.5

(e) 1:3

The red painted area is 3 half faces and the white painted area is a total of 9 half faces.

The ratio is, therefore, 1 to 3. The correct answer is: E.

32. There are X watermelons of 10 Kg each, and Y Watermelons of R Kg each. The

average weight of a watermelon is 12 Kg. What is the value of R?

(1) There are five heavier watermelons more than lighter watermelons.

(2) The weight of the heavier watermelons in Kg is equal to their number.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) defines the relationship between X and Y.

Statement (2) defines the relationship between Y and R.

We get the following equations:

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10 x X + R x Y = 12 (X+Y), Y=X+5, and Y=R. Now solve and find R. The correct

answer is C.

33. The average height of a group of children is 125 cm. If one of the children leaves,

the average height drops by 2 cm. how many kids were there originally?

(1) The height of the child who left is twice greater than the height of the shortest

child.

(2) The height of the child who left is 130 cm.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Since we don’t know how many children there are at all, it won’t help us to know the

relationship between the child who left and the shortest child.

Statement (2) defines the connection between the sum of the heights before and after the

departure and using the change in the sum divided by the new number of children we can

find the number of children. The correct answer is B.

34. P is the standard deviation of the heights of NBA basketball players. R is the

standard deviation of the heights of 8 weight lifters. Is the average height of the

NBA players greater than the average height of the weight lifters?

(1) R>P

(2) There are 5 basketball players that are taller than the tallest weight lifter, and 2

players that are shorter than the shortest weight lifter.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Since we have no concrete info about the exact heights, or about the value of P and R, we

cannot determine whether the NBA players or the weight lifters are taller.

We need more data. The correct answer is E.

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35. The average monthly income of 14 younger workers is X, Together with 60 older

workers, the average monthly income of the workers rose to Y. what is the average

monthly income of the older workers?

(1) The factory’s monthly budget for all salaries is 300,000$, which is 10 times the

salary budget for the younger workers.

(2) The monthly income of all the workers is 10 times the younger workers income.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) can tell us the value of Y, and the value of X, which is enough to solve this

problem.

Statement (2) alone only defines the connection between X and Y. The correct answer is

36. One out of every 500 light bulbs are defected. If 2 out of every 10 defected light

bulbs have a broken glass and there are 20 broken glass light bulbs in the shipment,

how many light bulbs total are there in the shipment?

(a) 2,000

(b) 5,000

(d) 50,000

(e) 52,000

Two out of every 10 damaged light bulbs have a broken glass, so if 20 bulbs have a

broken glass, there are 100 damaged light bulbs. Since one out of every 500 is damaged,

there are 100x500 = 50,000 light bulbs in the shipment. The correct answer is D.

37. If a is an even integer and b is an odd integer, what must the expression

be?

(a) Always even

(b) Always odd

(d) Could be a fraction

(e) Always an integer

Since the even number a is raised to the power of 3, it is always divisible by 8. Therefore,

the whole expression must be an integer, an even or an odd one. Of course, the

expression cannot be a fraction. The correct answer is E.

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GMAT Test 4

37 questions – 75 minutes

1. A salesperson gets a 15% commission on each sale he makes. How many sales of

$270 each must he make in order to reach a salary of at least $1000?

(a) 15

(b) 24

(d) 26

(e) 52

The commission per sale is $40.5, multiply it by 25 sales and get to $1012.5. The correct

answer is C.

2. In how many different ways can five people be seated on a five-seat bench if two

of them must sit next to each other?

(a) 24

(b) 48

(d) 240

(e) 480

Treat the two that seat together as one person, now there are only 4 people to be seated in

4 places. This task has 4! Ways = 24. Remember that the two people sitting together can

switch places on any of the 24 different ways, so we have 48 ways total. The correct

answer is B.

3. Liqueur A contains 24% of alcohol. What is the alcohol concentration of the

mixed cocktail of liqueur A and B?

(1) The mixing ratio of liqueur A and B is 1:4

(2) The alcohol concentration of liqueur B is 1.5 times greater than the alcohol

concentration of the mixed cocktail.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) gives us the quantity of each of the liqueurs.

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Statement (2) gives us the connection between the alcohol concentration of liqueur B and

the mixed cocktail. The equation looks like this: 0.2*24 + 0.8(1.5X) = X. The correct

answer is C.

4. What is the average height of X buildings, each with height P meters, and 4X

buildings each with height P/2?

(1) X = 5, P = 35.

(2) P = 40.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Of course, statement (1) gives of the values of all the parameters, but the value of X is

unnecessary. (XP+2XP)/5X is equivalent to3P/5. Statement (2) also gives us the answer.

The correct answer is D.

5. A, B, C and D are four consecutive points on a straight line. What is the distance

between A to D?

(1) AC = 6.

(2) BD = 8.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

First, draw the line with the points marked.

We know AC and BD but it’s not sufficient to know the length of AD.

If the question said the points are evenly spaced than the answer would be solvable. The

correct answer is E.

6. The area of an isosceles trapezoid with sides of length 5 and bases of length 8 and

14 is?

(a) 22

(b) 32

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(d) 55

(e) 56

The height of the trapezoid can be calculated using a right angle triangle of 3,4 and 5

created with the height, the side and half of the difference between the two bases. When

finding the height of 4, the area is calculated as the product of the average of the bases

and the height:

 8 + 14 

 × 4 = 44

2

The correct answer is C.

7. One gallon of soft drink is made of 40% orange juice and 60% water, how many

additional gallons of orange juice must be mixed in to make the orange juice 60% of

the soft drink?

(a) 0.5

(b) 1

(d) 1.5

(e) 2

Use the average formula to solve the following equation:

1 × 40% + X × 100%

= 60%

1+ X

X=0.5 gallon. The correct answer is A.

8. A merchant gets a 5% discount on each meter of fabric he buys after the first

2,000 meters and a 7% discount on every meter after the next 1,500 meters. The

price, before discount, of one meter of fabric is $2, what is the total amount of

money the merchant spends on 4,500 meters of fabric?

(a) $8,617

(b) $8,710

(d) $8,835

(e) $8,915

The price of the first 2,000 meters of fabric is $2 = $4,000. The price of the next 1,500

meters is 95% of $2 = $1.9 = $2,850. The price of the last 1,000 meters is 93% of $2 =

$1.86 = $1,860. The total amount spent is: $4,000+$2,850+$1,860=$8,710. The correct

answer is B.

9. If X, Y and Z are positive integers, is X greater than Z – Y?

(1) X – Z – Y > 0.

(2) Z2 = X2 + Y2.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

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(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

The best answer is A.

From statement (1) we learn that X > Z + Y therefore X must be bigger than Z – Y

(positive integers).

From statement (2) we learn that X2 = Z2 – Y2 and that tells us nothing relevant.

10. Zigfield bought his car using M% of his bank savings. He also bought a house

that costs 4 times the price of the car. What is the price of the house?

(1) M = 12.

(2) The price of the car and the house was $140,000.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

With statement (1) we know that the car cost 12% of his life savings.

From statement (2) we know that $140,000 is the price of both the car and the house.

Use simple ratio to find what is the cost of the house. The correct answer is B.

11. On a summer camp, 25% of the campers paid $120 each, 35% paid $80 each and

the rest paid $65 each. What percentage of the total amount paid came from the

campers who paid $80?

(a) 18%

(b) 21%

(d) 33.3%

(e) 37.5%

Take 100 campers as an example to work with, 25 paid $120 =$3,000 total, 35 paid $80 =

$2,800 total, and 40 (the rest) paid $65 = $2,600. The total amount paid is $8,400. The

amount paid by the campers who paid $65 is

$2,800 1

= = 33.3%

$8,400 3

The correct answer is D.

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12. A snail, climbing a 20 feet high wall, climbs up 4 feet on the first day but slides

down 2 feet on the second. It climbs 4 feet on the third day and slides down again 2

feet on the fourth day. If this pattern continues, how many days will it take the snail

to reach the top of the wall?

(a) 12

(b) 16

(d) 20

(e) 21

The snail climbs 2 feet every 2 days, hence, on the 16th day he is 16 feet up the wall. The

snail needs only the 17th day to climb 4 feet more and reach the top of the wall. The

correct answer is C.

13. John spent a total of $135 on baseball tickets. The price of a ticket is either $12

for the field seats or $5 for the balcony. What is the minimum amount of dollars he

could have spent for the $12 tickets?

(a) $48

(b) $60

(d) $96

(e) $120

Since John spent all of the $135 on the tickets, their prices must sum up to $135. The

Minimum amount spent for the $12 tickets, is the smallest amount that is divisible by 12,

and still leaves an amount divisible by 5. $60 is the smallest amount both divisible by 12

and leaving $135-$60=$75, which is divisible by $5. The correct answer is B.

14. One of the solutions of the equation X2-X-k=5, is –3. What is the other solution of

the equation?

(a) 4

(b) -4

(d) 6

(e) 24

First, plug in the solution –3 to find the value of k: (-3)2 – (-3) – k =5, k=7. Then solve:

X2 – X –7 – 5 = 0 to find the second solution: X=4. The correct answer is A.

15. Is the largest of 7 consecutive numbers odd?

(1) The product of the seven numbers is zero.

(2) The sum of the seven numbers is zero.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

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(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

From statement (1) we learn that there is a 0 among the seven numbers, yet the largest

number can be odd or even. (0, 1, 2, 3, 4, 5, 6)or (-1, 0, 1, 2, 3, 4, 5).

From statement (2) we know that the numbers are located symmetrically around the zero,

therefore the largest number is even. The correct answer is B.

16. If X and Y are positive integers, what is the ratio between Y and X?

(1) XY = 150.

(2) Y is 22% of X.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

The question actually asks what is Y/X or X/Y.

Statement (1) is not sufficient because from the product of the two variables we can’t

make out the ratio. Statement (2) is sufficient by itself, Y = 22X/100 à Y/X = 11/50. The

correct answer is B.

17. What is the value of (X2 + Y2)?

(1) (X – Y)2 = 36.

(2) (X + Y)2 = 48.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) can be written as X2 – 2XY +Y2 = 36.

Statement (2) can be written as X2 + 2XY +Y2 = 48.

Adding both equations will give: 2X2 + 2Y2 = 84 à X2 + Y2 = 42.

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Therefore, both statements are needed in order to solve the question and the correct

answer is C.

18. There are X dogs in the dog hound, each dog eats Y Kg of food every day. What

percent of the total food weight does each dog eat?

(1) If there were 3 dogs less then each dog could eat 1.2 Kg more than he is does

now.

(2) If there were half the dogs, each dog could eat 3 Kg more than he is does now.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

In order to know the answer we need two equations:

From statement (1) we can write: XY = (X – 3)(Y + 1.2).

From statement (2) we can write: XY = (X/2)(Y + 3).

You don’t need to solve the equations, the answer is C, both equations are needed to

solve the question and the correct answer is C.

19. What is the sum of all even integers between 30 and 50, inclusive?

(a) 640

(b) 540

(d) 440

(e) 240

The sum of the even integers between 30 and 50 inclusive can be calculated using the

product of the number of integers by their average. The average of a fixed-difference

sequence of numbers can be calculated as the average of the largest and the smallest

numbers: (30 + 50) / 2 = 40. We remember that between 30 and 49, inclusive, there are

20 numbers of which half are even and half odd. Add 1 (the number 50) to the 10 even

numbers to receive a total of 11 even integers, multiply it by the average to get the sum:

40 x 11 = 440. The correct answer is D.

20. The price (p) of product X depends on the yearly number of units produced (n)

according to the following formula: p = (500-n) / 10. What will the estimated price

per unit be next year if an estimated number of 260 units will be sold?

(a) 24

(b) 26

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(d) 60

(e) 240

Plug the number of units into the given formula to find the price per unit: p = (500-260)

/10. P=24. The correct answer is A.

21. John must arrange 3 different physics books and 3 different mathematics books

on one shelf. If the first book on the leftmost side is a mathematics book and no

physics book is next to another physics book, how many different arrangements

exist?

(a) 6

(b) 9

(d) 120

(e) 720

The leftmost book is a mathematics book; the next is physics and so on. So, the odd

places books are the mathematics books and the even placed books are the physics books.

The mission is therefore, arranging three mathematics books in three places (3! =6

arrangements) and three physics books in three places (3! =6 arrangements). The total

number of different arrangements is the product of the two missions: 3! x 3! = 6 x 6 = 36.

The correct answer is: C.

22. An investor receives a total combined return of 7% on his two different

investments. On his $10,000 investment he receives a return of 6%. What is the

return on his $20,000 investment?

(a) 7.25%

(b) 7.5%

(d) 8.5%

(e) 9%

The combined return is 7% of $30,000 = $2,100. Subtract the 6% return on his $10,000

investment = $600. $2,100-$600=$1,500. This is the return on his second investment of

$20,000. $1,500 of $20,000 is a 7.5% return. The correct answer is B.

23. What is the probability of getting an identical result on three consecutive tosses

of a coin?

(a) 1/2

(b) 1/4

(d) 1/16

(e) 1/2 + 1/4

The first toss will be either heads or tails. The probability that the result on the second

toss is identical to the first is 1/2. The probability that the result on the third toss is

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identical to that of the second is also 1/2. The total probability is, therefore, 1/2 x 1/2 =

1/4. The correct answer is B.

24. If x and y are positive integers, is 5x(1/4)y < 1 ?

(1) Y = 3x.

(2) X = 2.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Use statement (1) to write: 5x(1/4)3x = (5/64)x. Because x is a positive integer only, the

expression will always be smaller than 1. This statement alone provides us the answer.

Use statement (2) to write: 52(1/4)y à the answer here is dependent on y, a different

combinations of the variable y will give different results. Therefore the correct answer is

25. If A is a prime number, what is the value of A?

(1) 0 < A < 10.

(2) (A – 2) is divisible by 3.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) narrows down A to be one of: 1, 2, 3, 5 and 7. This statement is

insufficient.

Statement (2) is also insufficient; there are a lot of numbers that fulfill this statement, like

17, 23 and many more. Even after you combine both statements, there are still two

options: 5 and 2. In both, when subtracting 2 we get a number that is divisible by 3. The

correct answer is E.

26. What is the value of (A/5 + B/5)?

(1) A + B = 100.

A+ B

(2)= 10 .

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(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

Statement (1) is sufficient enough because all you need to do is divide both sides by to

get the answer, (20).

Statement (2) is also sufficient by it self, because all you need to do is to multiply both

sides by 2 to get the correct answer. The correct answer is D.

27. A salesperson receives a base salary of $1000 per month and a commission of

7.5% of the selling price of each item sold above the first 50 items. If this month she

sold 210 items for $150 each, what will her monthly salary be?

(a) $1,500

(b) $1,800

(d) $2,500

(e) $2,800

The salesperson receives a commission on 210-50=160 sales. The commission on every

sale is 7.5% of $150 = $11.25. And 160 x 11.25 = $1800. The correct answer is: B.

28. What is the probability that the sum of the results when two dice are rolled

simultaneously will be 9?

(a) 1/3

(b) 7/36

(d) 10/36

(e) 1/9

When two dice are rolled together, there are 4 different possibilities to receive a sum of 9:

[3,6], [4,5], [5,4], [6,3], out of a total of 36 different possibilities.

36 9

The correct answer is E.

29. What is the units’ digit of the following expression (13) 5(15) 4(17) 5?

(a) 0

(b) 1

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(d) 5

(e) 9

When multiplying any number by a number with a units’ digit of 5, the units’ digit of the

product will be either 5 (when all numbers are odd) or 0 (when there is at least one even

number). Since in this question there are no even numbers, the units’ digit is 5. The

correct answer is D.

30. Is B 3 ≥ B 2 ?

(1) B is an integer.

(2) B is positive.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

There are two cases in which the inequality will not be true, when B is a fraction or when

B is negative. Statements (1) and (2) cover both of these cases and therefore both

statements, taken together, are sufficient. The correct answer is C.

31. If Peter spends $460 on three pairs of shoes, how much did the least expansive

shoes cost?

(1) The ratio between the most expansive shoes to the least expansive shoes is 3 to 1.

(2) The ratio between the least expansive shoes to all the other ones is 1 to 5.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

We need to find what is the price of the least expansive among three shoes.

Statement (1) gives us the ratio between the most expansive shoes and the least expansive

one but that’s not sufficient because we have no information about the middle priced

shoes.

Statement (2) is sufficient. If the ratio between the least expansive to all the shoes is 1 to

5 than we can calculate the price of the least expansive shoes, (460/6). The correct

answer is B.

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32. What percent of 20 is Y?

(1) 50 percent of Y is 5.

(2) Y percent of 200 is 20.

(a) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by

itself is not.

(b) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by

itself is not.

even though NEITHER statement BY ITSELF is sufficient.

(d) Either statement BY ITSELF is sufficient to answer the question.

(e) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the

question, requiring more data pertaining to the problem.

The question, in other words, is asking what is the value of (Y/20 x 100). So all we need

to find out is the value of Y. Statement (1) gives us Y, it is equal to 10.

Statement (2) is also sufficient, it also tells us that Y is equal to 10. The correct answer is

33. What is the number of different ways to choose a chairman, two deputies and

two assistants for the class committee out of 7 students up for election?

(a) 25

(b) 210

(d) 840

(e) 2520

There are 7 different choices for chairman,

 66!6×5

 == 2  2!×4! 2 × 1 = 15



choices for the two deputies (choosing 2 out of 6), and

 44!4× 3

 == 2  2!×2! 2 × 1 = 6



choices for the two assistants. The total is 7 x 15 x 6 = 630 choices. The correct answer

is: C.

34. A factory has a fixed cost of $45,000 a month, and a cost of $2.5 for every item

produced. If the selling price of a single item is $5, what is the number of items must

be sold monthly for the factory to cover its cost exactly?

(a)

(c)

(d)

9,000

14,000

18,000

22,500

27,000

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To find the number of items sold that makes the revenues equal to the cost, solve the

following equation:

$45,000 + $2.5 × X = $5 × X

X=18,000. The correct answer is C.

35. John traveled 80% of the way from Yellow-town to Green-fields by train at an

average speed of 80 miles per hour. The rest of the way John traveled by car at an

average speed of v miles per hour. If the average speed for the entire trip was 60

miles per hour, what is v in miles per hour?

(a) 30

(b) 40

(d) 55

(e) 70

Take 100 miles as the total mileage traveled, then calculate the total distance divided by

the total time to receive the average speed for the whole trip:

100

= 60

80 20

80 v

And v=30 miles per hour. The correct answer is B.

36. The ratio between the number of sheep and the number of horses at the Stewart

farm is 4 to 7, If each horse is fed 230 ounces of horse food per day and the farm

needs a total 12,880 ounces of horse food per day, what is the number of sheep in the

farm?

(a) 18

(b) 28

(d) 56

(e) 60

The number of horses can be calculated using the total weight of daily horse food divided

by the weight each horse is fed daily:

12,880

= 56

230

There are 56 horses at the farm. Since the ratio is 4 to7, between the sheep and horses, the

number of sheep is:

= 8 ⇒ 8 × 4 = 32

There are 32 sheep. The correct answer is C.

37. The volume of water inside a swimming pool doubles every hour. If the pool is

filled to its full capacity within 8 hours, in how many hours was it filled to one

quarter of its capacity?

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(a) 2

(b) 4

(d) 6

(e) 7

Solve this question backwards. If the pool was full on the 8th hour, it was half filled on

the 7th and one quarter filled on the 6th. The correct answer is D.

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