Free GMAT Math Test

99 Percentile GMAT Math

Here is a Full Length Quantative Reasoning Math Test.
A majority of the questions are at the 700+ level to test your understanding of the difficult concepts tested on the GMAT.
Answers and Complete Solutions (alongwith Shortcuts) are provided to help you Excel!

 

  1. The sum of the even numbers between 1 and n is 79 × 80, where n is an odd number, then n = ?

A) 77

B) 78

C) 79

D) 80

E) 87

  1. The price of a bushel of corn is currently $3.20, and the price of a peck of wheat is $5.80. The price of corn is increasing at a constant rate of \( 5x \) cents per day while the price of wheat is decreasing at a constant rate of \( \sqrt{2}x – x \) cents per day.

What is the approximate price when a bushel of corn costs the same amount as a peck of wheat?


A) $4.50

B) $5.10

C) $5.30

D) $5.50

E) $5.60

  1. How many randomly assembled people do you need to have a better than 50% probability that at least 1 of them was born in a leap year?

A) 2

B) 3

C) 4

D) 5

E) 7

  1. In a basketball contest, players must make 10 free throws. Assuming a player has a 90% chance of making each of his shots, how likely is it that he will make all of his first 10 shots?

A) 20%

B) 35%

C) 50%

D) 81%

E) 85%

  1. AB + CD = AAA, where AB and CD are two-digit numbers and AAA is a three-digit number. Where A, B, C, and D are distinct positive integers. What is the value of C?

A) 1

B) 3

C) 7

D) 9

E) 11

  1. A certain quantity of 40% solution is replaced with 25% solution such that the new concentration is 35%. What fraction of the solution was replaced?

A) 1/4

B) 1/3

C) 1/2

D) 2/3

E) 3/4

  1. A bag contains 3 red, 4 black, and 2 white balls. What is the probability of drawing a red and a white ball in two successive draws, each ball being put back after it is drawn?

A) 2/27

B) 1/9

C) 1/3

D) 4/27

E) 2/9

  1. What is the least possible distance between a point on the circle \( x^2 + y^2 = 1 \) and a point on the line \( y = \frac{3}{4}x – 3 \)?

A) 1.4

B) \( \sqrt{2} \)

C) 1.7

D) \( \sqrt{3} \)

E) 2.0

  1. The average of temperatures at noontime from Monday to Friday is 50; the lowest one is 45. What is the possible maximum range of the temperatures?

A) 20

B) 25

C) 40

D) 45

E) 75

  1. If n is an integer from 1 to 96, what is the probability that \( n(n+1)(n+2) \) is divisible by 8?

A) 25%

B) 50%

C) 62.5%

D) 72.5%

E) 75%


  1. Kurt, a painter, has 9 jars of paint:
    4 are yellow, 2 are red, and the rest are brown.
    Kurt will combine 3 jars of paint into a new container to make a new color, which he will name according to the following conditions:
  • Brun Y if the paint contains 2 jars of brown paint and no yellow.
  • Brun X if the paint contains 3 jars of brown paint.
  • Jaune X if the paint contains at least 2 jars of yellow.
  • Jaune Y if the paint contains exactly 1 jar of yellow.

What is the probability that the new color will be Jaune?


A) 5/42

B) 37/42

C) 1/21

D) 4/9

E) 5/9

  1. TWO couples and a single person are to be seated on 5 chairs such that no couple is seated next to each other. What is the probability of the above?

A) 1/5

B) 3/5

C) 2/5

D) 7/5

E) 9/5

  1. An express train traveled at an average speed of 100 km per hour, stopping for 3 minutes after every 75 km. A local train traveled at an average speed of 50 km, stopping for 1 minute after every 25 km. If the trains began traveling at the same time, how many kilometers did the local train travel in the time it took the express train to travel 600 km?

A) 300

B) 305

C) 307.5

D) 1200

E) 1236

  1. Matt starts a new job, with the goal of doubling his old average commission of $400. He has a 10% commission, making commissions of $100.00, $200.00, $250.00, $700.00, and $1,000 on his first 5 sales. If Matt made two sales on the last day of the week, how much would Matt have had to sell in order to meet his goal?

A) $20,000

B) $25,000

C) $33,500

D) $40,000

E) $35,000

  1. How many ways can the letters of the word “COMPUTER” be arranged if all the vowels are together?

A) 360

B) 720

C) 1,440

D) 4,320

E) 5,390

  1. In how many ways can one choose 6 cards from a normal deck of cards so as to have all suits present?

A) \( 13^4 \times 48 \times 47 \)

B) \( 13^4 \times 27 \times 47 \)

C) 6

D) 134

E) \( 13^4 \times {}^{28}C_7 \)

  1. Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into a hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10?

A) 3

B) 4

C) 5

D) 6

E) 7

  1. If \( \frac{(13!)^{16} – (13!)^8}{(13!)^8 + (13!)^4} = a \), what is the unit’s digit of \( \frac{a}{(13!)^4} \)?

A) 0

B) 1

C) 3

D) 5

E) 9

  1. How many different subsets of the set {10,14,17,24} are there that contain an odd number of elements?

A) 3

B) 6

C) 8

D) 10

E) 12

  1. Seven men and seven women have to sit around a circular table so that no two women are together. In how many ways can this be done?

A) 120

B) 3,628,800

C) 400

D) 240

E) 1,230

  1. If the sum of five consecutive positive integers is A, then the sum of the next five consecutive integers in terms of A is:

A) A+1

B) A+5

C) A+24

D) 2A

E) 5A

Test 1 – Answers with Explanations

1. Answer: C (79)

Explanation:
First term \( a = 2 \), common difference \( d = 2 \)

Since even number therefore sum to first n numbers of Arithmetic progression would be

\( S_n = \frac{n}{2} \left[ 2a + (n-1)d \right] \)
\( S_n = \frac{n}{2} \left[ 2(2) + (n-1)2 \right] \)

\( S_n = n(n+1) \)

Since it is given that sum is equal to \( 79 \times 80 \)
[i.e. \( n(n+1) = 79 \times 80 \) ], therefore \( n=79 \), which is odd.

2. Answer: E ($5.60$)

Explanation:
\( 320 + 5x = 580 – ( \sqrt{2} x – x) \)
Solving for \( x \), we get approximately 48 days.
The required price = \( 320 + 5 \times 48 = 560 \) cents = $5.60

3. Answer: B (3 people)

Explanation:
Probability of a randomly selected person not being born in a leap year: \( \frac{3}{4} \)
For 2 people:
\( \frac{3}{4} \times \frac{3}{4} = \frac{9}{16} \) (probability of none being born in a leap year)
Probability that at least one was born in a leap year:

\( 1 – \frac{9}{16} = \frac{7}{16} < 0.5 \)

For 3 people:
\( \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} = \frac{27}{64} \)
Probability that at least one was born in a leap year:
\( 1 – \frac{27}{64} = \frac{37}{64} > 0.5 \)
Thus, a minimum of 3 people is required.

4. Answer: B (35%)

Explanation:
The probability of making all 10 shots is:
\( (0.9)^{10} = 0.348 \)
Final probability: 35%

5. Answer: D (9)

Explanation:
Since \( AB + CD = AAA \), and \( AAA = 111 \):

  • If \( B = 3 \), then \( CD = 111 – 13 = 98 \) and \( C = 9 \)
  • If \( B = 9 \), then \( CD = 111 – 19 = 92 \) and \( C = 9 \)
    For all values of B between 3 & 9, C = 9.

6. Answer: B (1/3)

Explanation:
Let \( Y \) be the fraction of solution that is replaced.

\( Y \times 25\% + (1-Y) \times 40\% = 35\% \)

Solving for \( Y \), we get

\( Y = \frac{1}{3} \)

7. Answer: D (4/27)

Explanation:
Case 1: Red ball first, then white ball

\( P_1 = \frac{3}{9} \times \frac{2}{9} = \frac{2}{27} \)

Case 2: White ball first, then red ball

\( P_2 = \frac{2}{9} \times \frac{3}{9} = \frac{2}{27} \)

Total probability:

\( P_1 + P_2 = \frac{4}{27} \)

8. Answer: A (1.4 units)

Explanation:
The equation of the line is:
\( 3x – 4y – 12 = 0 \)
The shortest distance from the origin to the line is:
\( \frac{| -12 |}{\sqrt{3^2 + 4^2}} = 2.4 \)
Since the radius of the circle is 1 unit, the shortest distance is:
\( 2.4 – 1 = 1.4 \) units

9. Answer: B (25)

Explanation:
The average of the 5 temperatures is:
\( \frac{a + b + c + d + e}{5} = 50 \)
One of these temperatures is 45:
\( \frac{a + b + c + d + 45}{5} = 50 \)
Solving for the variables:
\( a + b + c + d = 205 \)
To maximize the range, minimize all temperatures but one:
\( 45 + 45 + 45 + d = 205 \)
Solving for \( d \):
\( d = 70 \)
Range: \( 70 – 45 = 25 \)

10. Answer: C (62.5%)

Explanation:

\( E = n(n+1)(n+2) \)

E is divisible by 8 if ( n ) is even.
Number of even numbers from 1 to 96: 48

E is also divisible by 8 when ( n = 8k – 1 ), totaling 12 cases.
Total favorable cases: 60
Total cases: 96
Probability:

\( \frac{60}{96} = 62.5\% \)

11. Answer: B (37/42)

Explanation

  1. This has at least 2 yellow, meaning: a) There can be all three yellow:
\( {}^4C_3 = \frac{4!}{3!(4-3)!} \)

OR

b) 2 yellow and 1 out of 2 red and 3 brown:

\( {}^4C_2 \times {}^5C_1 = \frac{4!}{2!(4-2)!} \times \frac{5!}{1!(5-1)!} \)

Total = 34

This has exactly 1 yellow, with the remaining 2 out of 5:

\( {}^4C_1 \times {}^5C_2 = \frac{4!}{1!(4-1)!} \times \frac{5!}{2!(5-2)!} \)

Total = 40

Total possibilities:

\( \frac{9!}{3!6!} = 84 \)

Adding both probabilities:

\( \frac{74}{84} = \frac{37}{42} \)

12. Answer: C (2/5)

Explanation:
Ways for one couple to sit together: \( 2 \times 4! \)
Ways for both couples to sit together: \( 2 \times 2 \times 3! \)
Ways for at least one couple to sit together: \( 3 \times 4! \)
Total ways: \( 5! \)
Probability: \( \frac{3 \times 4!}{5!} = \frac{3}{5} \)
Thus, probability of no couples sitting together: \( 1 – \frac{3}{5} = \frac{2}{5} \)

13. Answer: C (307.5 km)

Explanation:
Express train time: 6 hours + 7 stops × 3 minutes = 6 hours 21 minutes
Local train distance in 6 hours: 300 km
Extra 9 minutes distance: 7.5 km
Total distance: 307.5 km

14. Answer: C ($33,500$)

Explanation:
Average commission goal: $800
Total commission required: $5,600
Sales on last day: $33,500

15. Answer: D (4,320)

Explanation:
Vowels grouped as one unit → CMPTR(AUE)
Ways to arrange: \( 6! \times 3! \)
Total: 4,320

16. Answer: A ( \( 13^4 \times 48 \times 47 \) )

52 cards in a deck – 13 cards per suit.
First card – let us say from suit hearts: \( {}^{13}C_1 = 13 \).
Second card – let us say from suit diamonds: \( {}^{13}C_1 = 13 \).
Third card – let us say from suit spades: \( {}^{13}C_1 = 13 \).
Fourth card – let us say from suit clubs: \( {}^{13}C_1 = 13 \).

Remaining cards in the deck: \( 52 – 4 = 48 \).
Fifth card – any card in the deck: \( {}^{48}C_1 \).
Sixth card – any card in the deck: \( {}^{47}C_1 \).

Total number of ways: \( 13 \times 13 \times 13 \times 13 \times 48 \times 47 = \left(13^4\right) \times 48 \times 47 \).

17. Answer: E (7 draws)

Explanation:
Worst case: drawing numbers 0, 1, 2, 3, 4, 5
The first valid sum of 10 occurs when 6 is drawn
Thus, minimum draws required: 7

18. Answer: E (9)

Explanation:
Given:
\( \frac{(13!)^{16} – (13!)^8}{(13!)^8 + (13!)^4} = a \)
Unit digit of:
\( \frac{a}{(13!)^4} \)
Final unit digit: 9

19. Answer: C (8 subsets)

Explanation:
Odd subsets of {10,14,17,24}:

  • 1-element subsets: {10}, {14}, {17}, {24}
  • 3-element subsets: {10,14,17}, {14,17,24}, {17,24,10}, {24,10,14}
    Total: 8

20. Answer: B (3,628,800)

Explanation:
Ways to arrange men: 6!
Ways to seat women: 7!
Total: 6! × 7! = 3,628,800

21. Answer: C (A + 25)

Explanation:
Sum of first 5 numbers: A
Each new number is 5 greater than the previous:
A + 5 + 5 + 5 + 5 + 5 = A + 25