GMAT Math Problems with Solutions

Solution:

If we recognize this question as a difference of squares, we can solve it in less than 30 seconds, doing all the calculations in our heads.

We can recall the fact that, for any numbers a and b, a^2 – b^2 = (a – b)(a + b), we can apply this equation to the expression in the question stem, with a = 998 and b = 2. Thus, we can re-express 998^2 – 2^2 as follows:

998^2 – 2^2 = (998 – 2)(998 + 2) = (996)(1,000) = 996,000

Note that we could have instead solved this question by performing the unwieldy and time-consuming calculation of 998^2, and then subtracting 4. But, instead, by knowing and applying the difference of squares concept, we have avoided the risk of making an arithmetic mistake, and we’ve gained valuable extra time for solving later questions in the quant section.

Answer: D

Solution:

When we raise the base 7 to successive integer powers, a pattern of the units (ones) digit is revealed. Let’s determine that pattern now by calculating the first few powers of 7.

7^1 = 7 units digit is 7

7^2 = 49 units digit is 9

7^3 = 343 units digit is 3

7^4 = 2,401 units digit is 1

7^5 = 16,807 units digit is 7

So, we see that the units digits of powers of 7 have a repeating pattern of 4 numbers: 7-9-3-1. Thus, every 4th exponent has the same units digit. For example, we see that 7^4, 7^8, 7^12, … etc., will all have the same units digit of 1.

Therefore the most straightforward way of determining the units digit of 7^15 is to find an exponent that is a multiple of 4 close to 15. We know from our pattern that 7^16 must have a units digit of 1. Now, using the pattern 7-9-3-1, we know that 7^15 must have a units digit of 3.

Answer: B

Solution:

First, notice that we can’t combine any of the expressions in the equation until we re-express the terms with like bases. In this case, the current bases, 16, 8, and 32, can all be expressed as powers of 2. So, we will use the facts that 16 = 2^4, 8 = 2^3, and 32 = 2^5 to rewrite the equation.

16^{x + 2} * 8⁶ = 32⁶

(2⁴)^{x + 2} * ( 2³)⁶ = ( 2⁵)⁶

2^{4x + 8} * 2¹⁸ = 2³⁰

So now that all terms in the equation now have the same base, we can combine them, using the fact that when we multiply two terms with like bases, we add the exponents:

2^{4x + 8 + 18} = 2³⁰

2^{4x + 26} = 2³⁰

We can now use the fact that when we have two expressions like bases on either side of an equation, we can equate the exponents.

^{4x + 26 = 30}

^{4x = 4}

^{x = 1}

Answer: C

Solution:

First, let’s define our two variables:

M = Marla’s age today

A = Angelina’s age today

Next, we can write two equations from the information presented in the problem stem.

Since Marla is 20 years older than Angelina, we have:

M = A + 20 (equation 1)

Since in 5 years, Marla will be 3 times as old as Angelina, we have:

M + 5 = 3(A + 5)

M + 5 = 3A + 15

M = 3A + 10 (equation 2)

Next, from equation 1, we can substitute A + 20 for M in equation 2, and then solve for A:

A + 20 = 3A + 10

10 = 2A

5 = A

Finally, we see that Angelina is 5 years old. Thus, Marla is currently 5 + 20 = 25 years old. So, in 3 years, Marla will be 28 years old.

Answer: C

Solution:

Since we have an average rate question we can use the following formula:

average rate = total distance / total time

Since the distance is the same in both directions, we can use a smart number to represent the one-way distance. A good number to use would be one that is divisible by both 5 and 12, so we can let the distance each way = 60.

So, the time going to work is 60/5 = 12, and the time going home from work is 60/12 = 5.

Finally we can determine the average rate:

average rate = total distance / total time

average rate = (60 + 60)/(12 + 5)

average rate = 120/17

Answer: C

Solution:

First, we can note that 500% of a number is equivalent to 5 times that number, and 400% of a number is equivalent to 4 times that number. Next, with these facts in mind, we can create two equations:

p = 5q (equation 1)

q = 4z (equation 2)

We can substitute 4z for q in equation 1:

p = 5(4z)

p = 20z (equation 3)

Now, to find out what percent z is of p, we can divide these two quantities and multiply the result by 100:

z/p ✕ 100

Finally, we can simplify this expression if we use equation 3, plugging in 20z for p:

z/p ✕ 100 = z/20z ✕ 100 = 1/20 ✕ 100 = 100/20 = 5

Thus, z is 5% of p.

Answer: B

Solution:

First, he chooses 3 rubies out of 5. Since the order doesn’t matter, we use combinations to count the number of possibilities. Recall the formula for a combination choosing k objects out of n objects is nCk = n! / k!(n-k)!. Thus, the number of ways to choose just the 3 rubies is:

5C3 = 5! / 3!(5-3)! = 5! / 3!(2)! = 5x4x3x2x1 / 3x2x1x2x1 = 20 / 2 = 10

Now, there are 2 remaining choices to make, and there are 7 non-rubies remaining in the chest. Thus, the number of ways to choose the 2 jewels out of 7 is:

7C2 = 7! / 2!(7-2)! = 7! / 2!5! = 7x6x5x4x3x2x1 / 2x1x5x4x3x2x1 = 42 / 2 = 21

There are 10 ways to choose the rubies and 21 ways to choose the remaining jewels.

Therefore, the total number of ways to choose the 5 jewels is 10 ✕ 21 = 210.

Answer: D

Solution:

First, we can define our two variables:

L = length

W = width

We know that the length of the rectangle is twice the width. This can be expressed as:

L = 2W

Next, we are told that the area of the rectangle is 72. Recall that the area of any rectangle is the product of its length and its width:

A = L ✕ W

72 = L ✕ W

Lastly, we can substitute 2W for L to solve for W:

72 = 2W ✕ W

72 = 2W^2

36 = W^2

6 = W

Note that we chose only the positive square root of 36 because the width cannot be negative.

Since L = 2W, we see that L = 2 ✕ 6 = 12, and so the length of the rectangle is 12.

Answer: C

Solution:

Statement (1) Alone:

It might be tempting to divide both sides of the given equation by n. But recall that we cannot divide both sides of an equation by a variable unless we are sure that the variable can’t be equal to 0. In statement (1), we are told nothing about whether n is equal to 0. So, we have to do some other algebra to solve for n:

n² = 24n

n² – 24n = 0

n(n – 24) = 0

We see that there are two solutions to the equation: n = 0 or n = 24. Thus, we cannot determine a unique value of n. Therefore, statement (1) alone is not sufficient to answer the question, “What is the value of n?” So, we can eliminate answer choices A and D.

Statement (2) Alone:

Knowing only that n is greater than 0 is not sufficient to determine a single value for n. Under this constraint, n could be any positive number. Therefore, statement (2) alone is also not sufficient to answer the question, and we can eliminate answer choice B.

Both Statements Together:

If we look at our algebra above, we can see that statement (1) tells us that n = 0 or n = 24. Statement (2), meanwhile, tells us that n > 0. Therefore, n cannot equal 0, meaning n must equal 24. Both statements, used together, give us sufficient information to answer the question.

Answer: C

Solution:

First, we can solve this absolute value equation for two cases: when (3y – 2) is positive and when (3y – 2) is negative, as follows:

Case 1: 3y – 2 is positive.

3y – 2 = 1

3y = 3

y = 1

Case 2: 3y – 2 is negative.

-(3y – 2) = 1

-3y + 2 = 1

-3y = -1

y = 1/3

The absolute value equation has two solutions: y = 1 and y = 1/3.

Statement (1) Alone:

Statement (1) says that y is a positive number. Both possible values of y are positive, so statement (1) by itself is not sufficient to determine a unique value for y. We can eliminate answer choices A and D.

Statement (2) Alone:

We know that the only way that y² can be less than y is if y is a positive proper fraction — that is, if y is between 0 and 1.

To illustrate this point, consider some possible values for y, as follows:

1. If y is a negative number, then y² will always be positive, and thus y² will always be greater than y. Thus, in order to satisfy the inequality y² < y, we see that y cannot be negative.
2. If y is a positive number greater than 1, then y² will always be greater than y. Thus, in order to satisfy the inequality, y also cannot be a positive number greater than 1.
3. If y is a positive proper fraction, when a positive proper fraction is squared, this squared value is less than the value of the original fraction. Thus, we see that y² < y is true only if y is a positive proper fraction.

From the question stem, we determined that the two possible values of y are y = 1 and y = 1/3. Statement (2) tells us that y must be a positive proper fraction, and only one possible value of y meets this criterion: y = 1/3. Therefore, tatement (2) by itself is sufficient to answer the question. We can eliminate answer choices C and E.

Answer: B

Solution:

Many DS questions can be solved using math tricks, and this is one such question! Let’s evaluate each statement.

Statement (1) Alone:

The mean of Set T is 10.

Just knowing the mean of Set T does not provide enough information to determine the set’s standard deviation. We can eliminate answers A and D.

Statement (2) Alone:

The largest data point in Set T is equal to the mean.

When the largest data point in a set is equal to the mean, all the numbers in the set must be the same.

When all numbers in a data set are the same, the standard deviation of the set is equal to 0. Thus, the standard deviation of Set T is zero, and statement (2) alone was sufficient to answer the question.

Answer: B

Solution:

Question Stem Analysis:

First, let’s rewrite the inequality in the question stem by adding m to both of its sides in order to put the variables on the same side. Thus, the question becomes: Is n + m > 17?

Statement (1) Alone:

We know only that n < 9. So, without any information about the value of m, statement (1) is insufficient. Eliminate A and D.

Statement (2) Alone:

We know only that m < 8. So, without any information about the value of n, statement (2) is insufficient. Eliminate B.

Both Statements Together:

If n is less than 9 and m is less than 8, then the sum of n and m must be less than the sum of 9 and 8, which is 17:

n + m < 17

Thus, we can definitely say that the answer to the question “Is n + m > 17?” is no. The sum of n and m must be less than 17, not greater than it. Therefore, statements (1) and (2) together are sufficient to answer the yes/no question.

Answer: C

Solution:

Let’s let T represent the total number of volunteers. If the volunteers can be evenly placed into 3 groups, then T must be divisible by 3.

Statement (1) Alone:

If the researcher reduces the number of volunteers by 16 percent, then he still has 84% of T, or (84/100)T volunteers remaining. We are told that this number of volunteers is evenly divisible by 9, so we can say that the expression (84/100)T / 9 yields an integer. Let’s simplify this expression:

(21/25)T / 9

(21 x T) / (25 x 9)

7T / 25×3

Thus, the expression 7T / 25×3 must also equal an integer, but 7 is not divisible by either 25 or 3. Thus, T must be divisible by both 25 and 3. Divisibility by 3 indicates that the researcher can divide the T participants into 3 equal groups.

Statement (1) is sufficient. We can eliminate answer choices B, C, and E.

Statement (2) Alone:

If the researcher reduces the number of volunteers by 6 percent, then he still has 94% of T, or (94/100)T volunteers remaining. We are told that this number of volunteers is evenly divisible by 3, so we can say that (94/100)T / 3 is an integer. We can simplify this expression as we did in the analysis of statement (1):

(47/50)T / 3

(47 x T) / (50 x 3)

Thus, this simplified form is also an integer, but 47 is not divisible by either 50 or 3. Thus, T must be divisible by both 50 and 3. Because T is divisible by 3, the researcher again can divide those volunteers into 3 equal groups.

Statement (2) is sufficient. We can eliminate answer choice A.

Therefore, the correct answer is D. Each statement by itself allows us to answer the question.

Answer: D

Solution:

Question Stem Analysis:

In order to answer the question, we need to know more about line k.

Statement (1) Alone:

If the y-intercept of line k is -2, then we know it passes through the point (0, -2), but it may or may not pass through the point (2, 5), depending on the slope of line k. Statement (1) is not sufficient. We can eliminate answer choices A and C.

Statement (2) Alone:

Just knowing only that the slope of line k is positive is not sufficient for answering the question. There are many, many lines with positive slopes. We can eliminate answer choice B.

Both Statements Together:

Even knowing both that the y-intercept of line k is -2 and that its slope is positive will not tell us whether the line passes through (2, 5). For instance, the equation y = x – 2 represents a line with a y-intercept equal to -2 and a positive slope equal to 1 that does not pass through the point (2, 5). The equation y = 3.5x – 2, meanwhile, represents a line that has a y-intercept equal to -2 and a positive slope equal to 3.5, and it does pass through the point (2, 5). Therefore, both statements together are not sufficient to answer the question.

Answer: E