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 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 exponential expressions with 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. 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 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.

Now, we don’t know which value of y to choose as the answer. We use the given fact that y^2 < y. Let’s test each possible answer, either y = 1 or y = 1/3.

If y = 1, then y^2 = 1. Thus, y cannot be equal to 1.

If y = 1/3, then y^2 = (1/3)^2 = 1/9, and we see that since 1/9 < 1/3 , then y^2 < y.

Thus, the correct answer is 1/3.

We could have skipped these last steps by recalling a fact from Number Properties: 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.

Answer: C