Arithmetic Practice Questions — GRE General Test — Page 2

Question 21Question →

If $x$ and $y$ are positive integers such that $5^x \cdot 2^y = 10^{2x-1} \cdot 4^{x+1}$ , what is the value of $y - x$ ?

2

3

4

5

6

Show answer & explanation

Answer: 4

Answer

The value of

y - x

is 4.

By prime-factorizing the bases on the right-hand side,

10^{2x-1} \cdot 4^{x+1}

becomes

(2 \cdot 5)^{2x-1} \cdot (2^2)^{x+1} = 5^{2x-1} \cdot 2^{4x+1}

. Matching the powers of 5 gives

x = 2x - 1

, which yields

x = 1

. Matching the powers of 2 gives

y = 4x + 1

, which yields

y = 5

. Subtracting

x

from

y

gives

5 - 1 = 4

.

Step-by-Step Solution

1

Rewrite composite bases into prime factor bases on the right side of the equation.

10^{2x-1} = (2 \cdot 5)^{2x-1} = 2^{2x-1} \cdot 5^{2x-1}

and

4^{x+1} = (2^2)^{x+1} = 2^{2(x+1)} = 2^{2x+2}

.

Converting all terms to prime bases (2 and 5) allows equating corresponding exponents.

2

Combine terms with identical bases on the right side.

10^{2x-1} \cdot 4^{x+1} = 5^{2x-1} \cdot 2^{(2x-1) + (2x+2)} = 5^{2x-1} \cdot 2^{4x+1}

.

Applying the product rule of exponents

a^m \cdot a^n = a^{m+n}

simplifies the right side.

3

Equate exponents of corresponding prime bases from both sides of

5^x \cdot 2^y = 5^{2x-1} \cdot 2^{4x+1}

.

Equating powers of 5 yields

x = 2x - 1 \implies x = 1

. Equating powers of 2 yields

y = 4x + 1

.

Since 2 and 5 are distinct prime numbers, their corresponding exponents must be equal.

4

Calculate

y

and evaluate

y - x

.

y = 4(1) + 1 = 5

, so

y - x = 5 - 1 = 4

.

Substituting

x = 1

gives

y = 5

, satisfying the final question requirement.

Key Concept

Decomposing exponential bases into prime factors and applying exponent rules (

a^m \cdot a^n = a^{m+n}

and

(a^m)^n = a^{mn}

) to solve system equations of powers.

Question 22Question →

Consider the expression $K = \frac{0.000072 \times (1.5 \times 10^{-4})}{3.6 \times 10^{-11}}$ . Which of the following values or expressions are equivalent to $K$ ? Select all that apply.

Select all that apply

3.0 \times 10^2

0.3 \times 10^3

30,000 \times 10^{-2}

3.0 \times 10^{-2}

3.0 \times 10^4

Show answer & explanation

Answer: $3.0 \times 10^2$ ; $0.3 \times 10^3$ ; $30,000 \times 10^{-2}$

Answer

The expressions equivalent to

K

are

3.0 \times 10^2

,

0.3 \times 10^3

, and

30,000 \times 10^{-2}

.

Evaluating the expression

K

gives

300

. First, rewrite

0.000072

as

7.2 \times 10^{-5}

. Multiplying by

1.5 \times 10^{-4}

gives

(7.2 \times 1.5) \times 10^{-9} = 10.8 \times 10^{-9}

. Next, dividing by

3.6 \times 10^{-11}

yields

(10.8 / 3.6) \times 10^{-9 - (-11)} = 3.0 \times 10^2 = 300

. Testing the choices shows that

3.0 \times 10^2 = 300

,

0.3 \times 10^3 = 300

, and

30,000 \times 10^{-2} = 300

are all equal to

300

.

Step-by-Step Solution

1

Convert decimal numbers in the numerator to scientific notation.

0.000072 = 7.2 \times 10^{-5}

.

Converting all terms to powers of 10 simplifies multiplication.

2

Multiply the terms in the numerator.

(7.2 \times 10^{-5}) \times (1.5 \times 10^{-4}) = (7.2 \times 1.5) \times 10^{-5 + (-4)} = 10.8 \times 10^{-9}

.

Coefficients multiply together and exponents add during multiplication of powers with the same base.

3

Divide the numerator by the denominator.

\frac{10.8 \times 10^{-9}}{3.6 \times 10^{-11}} = \left(\frac{10.8}{3.6}\right) \times 10^{-9 - (-11)} = 3.0 \times 10^2 = 300

.

Dividing coefficients gives

3.0

, and subtracting the exponent of the denominator

-11

from

-9

yields

-9 + 11 = 2

.

4

Verify each option against the value

300

.

3.0 \times 10^2 = 300

,

0.3 \times 10^3 = 300

, and

30,000 \times 10^{-2} = 300

are all equivalent to

300

.

Matching each option's evaluated value ensures all correct representations are selected.

Key Concept

Simplifying numerical expressions involving decimals and scientific notation rules.

Question 23Question →

If $(5.0 \times 10^{-4}) \times (4.0 \times 10^{7}) = 2.0 \times 10^n$ , what is the value of $n$ ?

Show answer & explanation

Answer: 4

Answer

The value of

n

is 4.

Multiplying the coefficients yields

5.0 \times 4.0 = 20.0

, and multiplying the powers of ten yields

10^{-4} \times 10^7 = 10^3

. Combining these gives

20.0 \times 10^3

. Rewriting

20.0 \times 10^3

into standard scientific notation gives

2.0 \times 10^4

. Therefore,

n = 4

.

Step-by-Step Solution

1

Multiply the numerical coefficients

5.0 \times 4.0 = 20.0

When multiplying numbers in scientific notation, separate the coefficients from the exponential terms.

2

Add the exponents of the base 10 terms

10^{-4} \times 10^{7} = 10^{3}

By exponent rules,

10^a \times 10^b = 10^{a+b}

.

3

Adjust the product to standard scientific notation form

20.0 \times 10^3 = 2.0 \times 10^4

Shift the decimal point one place to the left to obtain a coefficient

2.0

(

1 \le a < 10

), which increases the power of 10 by 1.

4

Determine the exponent value

n

n = 4

Comparing

2.0 \times 10^4

to

2.0 \times 10^n

gives

n = 4

.

Key Concept

Scientific notation multiplication and place value rules

Question 24Question →

If

x

is a positive integer such that

\frac{\sqrt{54 \cdot 4^{2x+1} + 9 \cdot 4^{2x}}}{\sqrt{2^{x+3} + 2^x}} = 2560

what is the value of

x

?

Show answer & explanation

Answer: 6

Answer

The value of

x

is

6

.

Factoring out common exponential terms inside both radicals yields

\sqrt{225 \cdot 2^{4x}} = 15 \cdot 2^{2x}

for the numerator and

\sqrt{9 \cdot 2^x} = 3 \cdot 2^{x/2}

for the denominator. Dividing these gives

5 \cdot 2^{3x/2}

. Equating this to

2560

results in

2^{3x/2} = 512 = 2^9

, which simplifies to

\frac{3x}{2} = 9

, or

x = 6

.

Step-by-Step Solution

1

Simplify the numerator inside the radical expression.

\sqrt{54 \cdot 4^{2x+1} + 9 \cdot 4^{2x}} = 15 \cdot 2^{2x}

Rewrite

4^{2x+1}

as

4 \cdot 4^{2x}

. Then factor out

4^{2x}

:

54(4 \cdot 4^{2x}) + 9 \cdot 4^{2x} = (216 + 9)4^{2x} = 225 \cdot 4^{2x}

. Taking the square root gives

\sqrt{225} \cdot \sqrt{(2^2)^{2x}} = 15 \cdot 2^{2x}

.

2

Simplify the denominator inside the radical expression.

\sqrt{2^{x+3} + 2^x} = 3 \cdot 2^{x/2}

Rewrite

2^{x+3}

as

2^3 \cdot 2^x = 8 \cdot 2^x

. Factoring out

2^x

gives

(8 + 1)2^x = 9 \cdot 2^x

. Taking the square root gives

\sqrt{9} \cdot \sqrt{2^x} = 3 \cdot 2^{x/2}

.

3

Simplify the quotient of the two radical expressions.

\frac{15 \cdot 2^{2x}}{3 \cdot 2^{x/2}} = 5 \cdot 2^{3x/2}

Divide the constants

\frac{15}{3} = 5

and subtract exponents with the same base:

2x - \frac{x}{2} = \frac{3x}{2}

.

4

Equate to 2560 and solve for

x

.

x = 6

Divide both sides by 5:

2^{3x/2} = \frac{2560}{5} = 512

. Express 512 as a power of 2:

512 = 2^9

. Therefore,

\frac{3x}{2} = 9 \implies 3x = 18 \implies x = 6

.

Key Concept

Exponent and Radical Simplification using Base Prime Factorization

Estimated Time:2m 0s

Question 25Question →

On the real number line, the set of all real numbers $x$ that satisfy the inequality $|3x - 7| \le 11$ forms a closed interval $[a, b]$ . What is the value of $|a + b|$ ?

\frac{14}{3}

\frac{22}{3}

6

\frac{10}{3}

\frac{26}{3}

Show answer & explanation

Answer: $\frac{14}{3}$

Answer

The value of

|a + b|

is

\frac{14}{3}

.

Rewriting the inequality

|3x - 7| \le 11

as

-11 \le 3x - 7 \le 11

and solving yields

-\frac{4}{3} \le x \le 6

. Thus, the endpoints are

a = -\frac{4}{3}

and

b = 6

. Summing these values gives

a + b = \frac{14}{3}

, and taking the absolute value yields

\frac{14}{3}

.

Step-by-Step Solution

1

Express the absolute value inequality as a compound inequality.

-11 \le 3x - 7 \le 11

An inequality of the form

|u| \le k

for

k \ge 0

is equivalent to

-k \le u \le k

.

2

Isolate

3x

by adding

7

to all parts of the inequality.

-11 + 7 \le 3x \le 11 + 7 \implies -4 \le 3x \le 18

Adding a constant to all parts preserves the direction of the inequality.

3

Divide all parts by

3

to solve for

x

.

-\frac{4}{3} \le x \le 6

Dividing by a positive number isolates

x

without flipping inequality signs.

4

Identify interval bounds

a

and

b

, then calculate

|a + b|

.

a = -\frac{4}{3}

,

b = 6 \implies a + b = -\frac{4}{3} + \frac{18}{3} = \frac{14}{3} \implies \left|\frac{14}{3}\right| = \frac{14}{3}

The question asks for the absolute value of the sum of the endpoints of interval

[a, b]

.

Key Concept

Absolute value inequalities on the number line and interval endpoints

Question 26Question →

If

k

is a positive integer, which of the following expressions is equivalent to

\frac{2^{k+2} \cdot 3^{2k-1} + 6^k \cdot 3^{k-1}}{12^k \cdot 2^{-1}}

for all values of

k

?

\frac{5}{3} \left(\frac{3}{2}\right)^k

\frac{10}{3} \left(\frac{3}{2}\right)^k

\frac{5}{3} \cdot 6^k

5 \left(\frac{3}{2}\right)^k

\frac{10}{3} \cdot 3^k

Show answer & explanation

Answer: $\frac{10}{3} \left(\frac{3}{2}\right)^k$

Answer

The expression simplifies to

\frac{10}{3} \left(\frac{3}{2}\right)^k

.

Factoring the terms in the numerator into base

18^k

gives

\frac{4}{3} \cdot 18^k + \frac{1}{3} \cdot 18^k = \frac{5}{3} \cdot 18^k

. The denominator equals

\frac{1}{2} \cdot 12^k

. Dividing numerator by denominator yields

\frac{5/3}{1/2} \cdot \left(\frac{18}{12}\right)^k = \frac{10}{3} \left(\frac{3}{2}\right)^k

.

Step-by-Step Solution

1

Rewrite each term in the numerator using prime base factorization

2^{k+2} \cdot 3^{2k-1} = 2^k \cdot 2^2 \cdot (3^2)^k \cdot 3^{-1} = 4 \cdot 2^k \cdot 9^k \cdot \frac{1}{3} = \frac{4}{3} \cdot 18^k

, and

6^k \cdot 3^{k-1} = (2 \cdot 3)^k \cdot 3^k \cdot 3^{-1} = 2^k \cdot 9^k \cdot \frac{1}{3} = \frac{1}{3} \cdot 18^k

.

Converting all powers to base 18 allows terms with identical exponential factors to be combined.

2

Combine the terms in the numerator

\frac{4}{3} \cdot 18^k + \frac{1}{3} \cdot 18^k = \left(\frac{4}{3} + \frac{1}{3}\right) \cdot 18^k = \frac{5}{3} \cdot 18^k

.

Adding coefficients of like exponential terms.

3

Simplify the denominator expression

12^k \cdot 2^{-1} = \frac{1}{2} \cdot 12^k

.

Applying the negative exponent rule

a^{-n} = \frac{1}{a^n}

.

4

Divide the numerator by the denominator

\frac{\frac{5}{3} \cdot 18^k}{\frac{1}{2} \cdot 12^k} = \frac{5/3}{1/2} \cdot \left(\frac{18}{12}\right)^k = \left(\frac{5}{3} \cdot 2\right) \cdot \left(\frac{3}{2}\right)^k = \frac{10}{3} \left(\frac{3}{2}\right)^k

Dividing fractions by multiplying by the reciprocal and applying quotient rule for powers with the same exponent.

Key Concept

Prime base factorization and laws of exponents

Question 27Question →

An index value $I_0 = 1{}000$ increases over a 6-month period. For each month $n$ from 1 to 6, the index value $I_n$ is calculated by increasing the previous month's value $I_{n-1}$ by $5\%$ and then rounding the result to the nearest integer (with half-integers rounded up). Let $U_6 = 1{}000 \times (1.05)^6$ represent the exact unrounded compounded value at month 6.

Which of the following statements must be true? Select all such statements.

Select all that apply

I_2 = 1{}103

The exact unrounded value

U_6 = 1{}000 \times (1.05)^6

is strictly greater than the rounded index value

I_6

.

The sequence of monthly increments

I_n - I_{n-1}

for

n = 1, 2, \dots, 6

is strictly increasing.

The sum of the first two monthly fractional growth rates,

\frac{I_1 - I_0}{I_0} + \frac{I_2 - I_1}{I_1}

, is equal to

\frac{I_2 - I_0}{I_0 + I_1}

.

The exponential growth factor

(1 + 0.05)^6

is equivalent to

1^6 + (0.05)^6

.

Show answer & explanation

Answer: $I_2 = 1{}103$ ; The sequence of monthly increments $I_n - I_{n-1}$ for $n = 1, 2, \dots, 6$ is strictly increasing.

Answer

The statement specifying that

I_2 = 1{}103

and the statement asserting that the sequence of monthly increments

I_n - I_{n-1}

is strictly increasing are both correct.

The statement giving

I_2 = 1{}103

is correct because

1{}050 \times 1.05 = 1{}102.5

, which rounds up to

1{}103

. The statement regarding the sequence of monthly increments is correct because the increments

50, 53, 55, 58, 61, 64

strictly increase.

Step-by-Step Solution

1

Calculate each term of the sequence

I_n

by applying a 5% increase and rounding to the nearest integer.

I_0 = 1{}000

;

I_1 = \text{round}(1{}000 \times 1.05) = 1{}050

;

I_2 = \text{round}(1{}050 \times 1.05) = \text{round}(1{}102.5) = 1{}103

;

I_3 = \text{round}(1{}103 \times 1.05) = \text{round}(1{}158.15) = 1{}158

;

I_4 = \text{round}(1{}158 \times 1.05) = \text{round}(1{}215.9) = 1{}216

;

I_5 = \text{round}(1{}216 \times 1.05) = \text{round}(1{}276.8) = 1{}277

;

I_6 = \text{round}(1{}277 \times 1.05) = \text{round}(1{}340.85) = 1{}341

.

This establishes the exact sequence of rounded monthly values.

2

Evaluate the statement that

I_2 = 1{}103

.

From Step 1,

I_2 = 1{}103

.

This directly confirms the validity of the first statement.

3

Compute the sequence of monthly increments

I_n - I_{n-1}

for

n = 1, 2, \dots, 6

.

Increments:

I_1 - I_0 = 50

,

I_2 - I_1 = 53

,

I_3 - I_2 = 55

,

I_4 - I_3 = 58

,

I_5 - I_4 = 61

,

I_6 - I_5 = 64

.

Since

50 < 53 < 55 < 58 < 61 < 64

, the sequence of increments is strictly increasing.

4

Compare

I_6

with

U_6 = 1{}000 \times (1.05)^6

.

U_6 = 1{}000 \times 1.3400956... \approx 1{}340.10

. Since

I_6 = 1{}341

,

I_6 > U_6

.

The statement claiming

U_6 > I_6

is false.

Key Concept

Error propagation in recursive sequence rounding and non-distributivity of exponents over sums

Estimated Time:2m 30s

Question 28Question →

If $x$ is a negative real number such that $\sqrt{(-x)^3 \cdot x^2} = 32$ , what is the value of $x$ ?

-16

-4

-2

2

4

Show answer & explanation

Answer: $-4$

Answer

-4

Simplifying the expression inside the radical gives

(-x)^3 \cdot x^2 = -x^5

. Setting

\sqrt{-x^5} = 32

and squaring both sides gives

-x^5 = 32^2 = 1024

, which means

x^5 = -1024

. The fifth root of

-1024

is

-4

. Since

-4

is a negative real number, it satisfies all conditions of the problem.

Step-by-Step Solution

1

Simplify the expression under the square root

Since

(-x)^3 = -x^3

, we have

(-x)^3 \cdot x^2 = (-x^3) \cdot x^2 = -x^5

.

Applying exponent addition rules

x^a \cdot x^b = x^{a+b}

and odd power rules for negative quantities.

2

Square both sides of the equation to eliminate the square root

-x^5 = 32^2 = 1024

.

Squaring both sides of

\sqrt{-x^5} = 32

isolates the radicand.

3

Solve for x

x^5 = -1024 \implies x = \sqrt[5]{-1024} = -4

.

Taking the 5th root of

-1024 = (-2)^{10} = (-4)^5

yields

x = -4

, which satisfies the given condition

x < 0

.

Key Concept

Simplifying expressions with powers and square roots involving negative variables.

Estimated Time:1m 15s

Question 29Question →

A bookstore received a shipment of $N$ identical books, each with an original full price of $P$ dollars. The sales over three consecutive months proceeded as follows:

- In May, the store sold $40\%$ of the total initial shipment at full price $P$ .
- In June, the store discounted the price of the unsold books by $25\%$ off the original full price and sold $50\%$ of the remaining unsold books.
- In July, the store discounted the June price of the remaining unsold books by an additional $20\%$ and sold all remaining books.

Which of the following statements regarding the bookstore's sales must be true? Select all such statements.

Select all that apply

The number of books sold in July accounted for exactly

30\%

of the total initial shipment.

The total revenue generated from sales in June was greater than the total revenue generated from sales in July.

The total revenue from selling the entire shipment was equal to

80.5\%

of the revenue that would have been earned if all books were sold at full price.

The discounted price per book in July represented a

45\%

decrease from the original full price

P

.

The number of books sold in June was equal to the number of books sold in May.

Show answer & explanation

Answer: The number of books sold in July accounted for exactly $30\%$ of the total initial shipment.; The total revenue generated from sales in June was greater than the total revenue generated from sales in July.; The total revenue from selling the entire shipment was equal to $80.5\%$ of the revenue that would have been earned if all books were sold at full price.

Answer

The statements asserting that July sales comprised 30% of the initial shipment, that June revenue exceeded July revenue, and that overall revenue equaled 80.5% of full-price potential revenue are all correct.

The statement regarding July sales accounting for 30% of the initial shipment is correct because 50% of the 60% remaining after May leaves 30% unsold, all of which were sold in July. The statement comparing June and July revenue is correct because June revenue (0.225NP) is greater than July revenue (0.180NP). The statement regarding total revenue is correct because 0.400NP + 0.225NP + 0.180NP = 0.805NP, which is 80.5% of the total potential revenue.

Step-by-Step Solution

1

Calculate the quantity of books sold and remaining in each month as a fraction of total shipment N

May: Sold

0.40N

, Remaining

0.60N

. June: Sold

0.50 \times 0.60N = 0.30N

, Remaining

0.30N

. July: Sold all remaining

0.30N

.

Tracking the base shift after each monthly sale ensures accurate inventory proportions.

2

Calculate the selling price per book for each month relative to original price P

May price =

1.00P

. June price =

P \times (1 - 0.25) = 0.75P

. July price =

0.75P \times (1 - 0.20) = 0.60P

.

Successive percent discounts compound on the previous month's price, not the original price.

3

Calculate revenue for each month and total revenue

May:

0.40N \times 1.00P = 0.400NP

. June:

0.30N \times 0.75P = 0.225NP

. July:

0.30N \times 0.60P = 0.180NP

. Total revenue =

(0.400 + 0.225 + 0.180)NP = 0.805NP

.

Revenue equals quantity sold multiplied by price per unit for each individual phase.

4

Evaluate each given statement against calculated values

July quantity =

30\%

of N (Statement 1 is True). June revenue

0.225NP > 0.180NP

July revenue (Statement 2 is True). Total revenue

0.805NP = 80.5\%

of full potential

1.000NP

(Statement 3 is True). July price discount is

1 - 0.60 = 40\% \neq 45\%

(Statement 4 is False). June sales (

30\%

)

\neq

May sales (

40\%

) (Statement 5 is False).

Determines which set of statements must be selected.

Key Concept

Successive Percent Change and Base Shift in Multi-Step Scenarios

Estimated Time:1m 45s

Question 30Question →

If $n$ is a positive integer such that $n^2$ is divisible by 72, what is the smallest possible value of $n$ ?

6

12

18

24

36

Show answer & explanation

Answer: 12

Answer

12

The prime factorization of 72 is

2^3 \times 3^2

. For

n^2

to be a multiple of 72, the prime factorization of

n^2

must contain at least three factors of 2 and two factors of 3. Since

n^2

is a perfect square, the exponents of all prime factors in

n^2

must be even. Therefore, the exponent of 2 in

n^2

must be at least 4, making the minimum value of

n^2

equal to

2^4 \times 3^2 = 144

. Taking the square root yields

n = 12

, which is the smallest positive integer.

Step-by-Step Solution

1

Find the prime factorization of 72.

72 = 2^3 \times 3^2

.

Decomposing 72 into prime factors reveals the minimum prime factor requirements for

n^2

.

2

Determine the prime factorization required for

n^2

.

For

n^2

to be divisible by

2^3 \times 3^2

, its prime factorization must contain at least three factors of 2 and two factors of 3. Because

n^2

is a perfect square, all exponents in its prime factorization must be even numbers. Thus,

n^2

must contain at least

2^4 \times 3^2 = 144

.

Exponents of prime factors in any perfect square must be even integers.

3

Take the square root to find the minimum value of

n

.

n = \sqrt{2^4 \times 3^2} = 2^2 \times 3^1 = 12

.

The smallest positive integer

n

must equal the square root of the minimal perfect square multiple of 72.

Key Concept

Prime Factorization and Divisibility of Perfect Squares

Question 31Question →

A budget planner allocates $\frac{1}{3}$ of a monthly stipend to housing expenses and $\frac{2}{5}$ of the stipend to food. If no other expenses are incurred from these two categories, what fraction of the total monthly stipend remains unallocated?

\frac{3}{8}

\frac{4}{15}

\frac{5}{8}

\frac{7}{15}

\frac{11}{15}

Show answer & explanation

Answer: $\frac{4}{15}$

Answer

\frac{4}{15}

of the total monthly stipend remains unallocated.

To find the remaining unallocated fraction, first determine the total fraction spent by adding

\frac{1}{3}

and

\frac{2}{5}

using the common denominator

15

, yielding

\frac{5}{15} + \frac{6}{15} = \frac{11}{15}

. Subtracting this total from

1

gives

1 - \frac{11}{15} = \frac{4}{15}

. Thus, the option equal to

\frac{4}{15}

is correct.

Step-by-Step Solution

1

Calculate the total fraction allocated to housing and food.

\frac{1}{3} + \frac{2}{5} = \frac{5}{15} + \frac{6}{15} = \frac{11}{15}

Find a common denominator (15) to combine the fractions.

2

Subtract the allocated fraction from 1 to determine the unallocated portion.

1 - \frac{11}{15} = \frac{15}{15} - \frac{11}{15} = \frac{4}{15}

The total stipend is represented by 1 whole.

Key Concept

Adding and subtracting rational numbers using common denominators

Estimated Time:1m 0s

Question 32Question →

An artist sold $\frac{1}{3}$ of her paintings on the first day of an exhibition. On the second day, she sold $\frac{2}{5}$ of the paintings that remained after the first day. On the third day, she sold $\frac{1}{4}$ of the paintings that remained after the second day. If $18$ paintings remained unsold at the end of the third day, how many paintings did the artist have originally?

45

60

72

90

108

Show answer & explanation

Answer: 60

Answer

The artist originally had 60 paintings.

To find the original total, determine the remaining fraction of paintings after each day sequentially. After day one,

\frac{2}{3}

remain. After day two,

\frac{3}{5}

of

\frac{2}{3}

remain, which is

\frac{2}{5}

. After day three,

\frac{3}{4}

of

\frac{2}{5}

remain, which simplifies to

\frac{3}{10}

of the original total. Given that

\frac{3}{10}

of the total equals

18

, dividing

18

by

\frac{3}{10}

gives

60

.

Step-by-Step Solution

1

Calculate the fraction of paintings remaining after the first day.

Remaining fraction after Day 1 =

1 - \frac{1}{3} = \frac{2}{3}

.

Selling

\frac{1}{3}

of the total leaves

\frac{2}{3}

of the original total.

2

Calculate the fraction of paintings remaining after the second day.

Remaining fraction after Day 2 =

\frac{2}{3} \times \left(1 - \frac{2}{5}\right) = \frac{2}{3} \times \frac{3}{5} = \frac{2}{5}

.

Selling

\frac{2}{5}

of the remaining paintings leaves

\frac{3}{5}

of that remaining portion.

3

Calculate the fraction of paintings remaining after the third day.

Remaining fraction after Day 3 =

\frac{2}{5} \times \left(1 - \frac{1}{4}\right) = \frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10}

.

Selling

\frac{1}{4}

of the remaining paintings leaves

\frac{3}{4}

of the second day's remaining portion.

4

Set up the equation with the given remaining quantity and solve for the total number of paintings

N

.

\frac{3}{10} N = 18 \implies N = 18 \times \frac{10}{3} = 60

.

The final remaining fraction represents

18

paintings.

Key Concept

Sequential Fraction of Remainder Problems

Question 33Question →

At a certain technology company, $\frac{4}{9}$ of the employees work in the sales department and $\frac{1}{3}$ of the employees work in the engineering department. If all of the remaining employees work in the administration department, what fraction of the total employees works in the administration department?

\frac{2}{9}

\frac{7}{9}

\frac{5}{12}

\frac{4}{9}

\frac{1}{9}

Show answer & explanation

Answer: $\frac{2}{9}$

Answer

\frac{2}{9}

Expressing

\frac{1}{3}

as

\frac{3}{9}

allows direct addition with

\frac{4}{9}

, giving

\frac{7}{9}

for sales and engineering combined. Subtracting

\frac{7}{9}

from 1 leaves

\frac{2}{9}

for administration.

Step-by-Step Solution

1

Find the total fraction of employees working in sales and engineering.

\frac{4}{9} + \frac{1}{3} = \frac{4}{9} + \frac{3}{9} = \frac{7}{9}

Convert

\frac{1}{3}

to an equivalent fraction with a common denominator of 9 before adding.

2

Subtract the combined fraction from 1 to find the remaining fraction.

1 - \frac{7}{9} = \frac{9}{9} - \frac{7}{9} = \frac{2}{9}

The sum of all departmental fractions must equal 1 whole.

Key Concept

Adding fractions with unlike denominators and determining the remaining part of a whole.

Question 34Question →

Let $p$ and $q$ be positive integers such that $18$ is a factor of $p$ and $45$ is a factor of $q$ . Which of the following statements MUST be true? Select all such statements.

Select all that apply

The sum

p + q

is divisible by

9

.

The product

p \cdot q

is divisible by

270

.

The sum

p + q

is divisible by

18

.

The product

p \cdot q

is divisible by

1620

.

The sum

p^2 + q^2

is divisible by

81

.

Show answer & explanation

Answer: The sum $p + q$ is divisible by $9$ .; The product $p \cdot q$ is divisible by $270$ .; The sum $p^2 + q^2$ is divisible by $81$ .

Answer

The statements asserting that the sum

p + q

is divisible by

9

, the product

p \cdot q

is divisible by

270

, and the sum

p^2 + q^2

is divisible by

81

MUST be true.

The correct options are those stating that the sum

p + q

is divisible by

9

, the product

p \cdot q

is divisible by

270

, and the sum

p^2 + q^2

is divisible by

81

. Each of these statements can be proven algebraically by substituting

p = 18a

and

q = 45b

into the expressions and factoring out the required divisor.

Step-by-Step Solution

1

Express

p

and

q

in terms of their prime factorizations and integer multipliers.

p = 18a = (2 \cdot 3^2)a

and

q = 45b = (3^2 \cdot 5)b

for positive integers

a

and

b

.

Establishing explicit algebraic representations allows verification of divisibility for any combination of

p

and

q

.

2

Evaluate the divisibility of the sum

p + q

.

p + q = 18a + 45b = 9(2a + 5b)

.

Factoring out

9

proves that

p + q

is always divisible by

9

. However,

2a + 5b

is not necessarily even (e.g., if

a=1, b=1

,

2a+5b=7

), so

p+q

is not guaranteed to be divisible by

18

.

3

Evaluate the divisibility of the product

p \cdot q

.

p \cdot q = (18a)(45b) = 810ab = 270(3ab)

.

Since

810ab

is a multiple of

270

, the product is always divisible by

270

. However, it is not necessarily a multiple of

1620

unless

ab

contains an additional factor of

2

.

4

Evaluate the divisibility of the sum of squares

p^2 + q^2

.

p^2 + q^2 = (18a)^2 + (45b)^2 = 324a^2 + 2025b^2 = 81(4a^2 + 25b^2)

.

Factoring out

81

proves that

p^2 + q^2

is always divisible by

81

.

Key Concept

Divisibility rules and algebraic factoring of linear and quadratic integer expressions

Question 35Question →

What is the smallest positive integer $n$ such that $n$ is a multiple of $180$ , the only prime factors of $n$ are $2$ , $3$ , and $5$ , and $n$ has exactly $36$ positive integer divisors?

Show answer & explanation

Answer: 1440

Answer

1440

To minimize

n = 2^a \cdot 3^b \cdot 5^c

subject to

(a+1)(b+1)(c+1) = 36

with

a \ge 2

,

b \ge 2

, and

c \ge 1

, we evaluate all valid factor partitions of

36

. The partition

(a+1, b+1, c+1) = (6, 3, 2)

yields

(a, b, c) = (5, 2, 1)

, giving

n = 2^5 \cdot 3^2 \cdot 5^1 = 1440

, which is the smallest possible integer satisfying all criteria.

Step-by-Step Solution

1

Express

n

in terms of prime factorization and establish exponent inequalities.

n = 2^a \cdot 3^b \cdot 5^c

with

a \ge 2

,

b \ge 2

, and

c \ge 1

.

Since

n

is a multiple of

180 = 2^2 \cdot 3^2 \cdot 5^1

and contains no other prime factors, its prime powers must at least match those of

180

.

2

Set up the divisor count equation.

(a+1)(b+1)(c+1) = 36

, where

a+1 \ge 3

,

b+1 \ge 3

, and

c+1 \ge 2

.

The total number of positive divisors of

2^a \cdot 3^b \cdot 5^c

is given by

(a+1)(b+1)(c+1)

.

3

Determine all valid factorizations of

36

into three factors

(x, y, z) = (a+1, b+1, c+1)

.

The valid factor sets

\{x, y, z\}

satisfying

x, y \ge 3

and

z \ge 2

are

\{4, 3, 3\}

and

\{6, 3, 2\}

.

Factor sets containing a factor of

2

for

x

or

y

(such as

\{9, 2, 2\}

) are invalid because

a+1 \ge 3

and

b+1 \ge 3

.

4

Calculate the value of

n

for all valid assignments of exponents.

From

\{4, 3, 3\}

:

(3, 2, 2) \implies 1800

,

(2, 3, 2) \implies 2700

,

(2, 2, 3) \implies 4500

.
From

\{6, 3, 2\}

:

(5, 2, 1) \implies 1440

,

(2, 5, 1) \implies 4860

.

Assigning larger exponents to smaller prime bases minimizes the overall product.

5

Select the minimum integer value among all candidates.

The smallest value is

1440

.

Comparing all valid candidates

1440, 1800, 2700, 4500, 4860

, the minimum is

1440

.

Key Concept

Prime Factorization and Divisor Count Constraints

Estimated Time:2m 30s

Question 36Question →

When the positive integer $n$ is divided by $12$ , the remainder is $7$ . Which of the following could be the remainder when $n^2 + 5n + 2$ is divided by $24$ ?

1

2

5

7

11

Show answer & explanation

Answer: 2

Answer

2

Because

n \equiv 7 \pmod{12}

,

n

can be expressed as

12k + 7

. Testing the two cases for the integer

k

(even vs. odd) shows that

n \equiv 7 \pmod{24}

or

n \equiv 19 \pmod{24}

. Substituting

n = 19

into

n^2 + 5n + 2

yields

458

, and

458 = 19 \times 24 + 2

, which leaves a remainder of

2

. Therefore, the value

2

is a valid possible remainder.

Step-by-Step Solution

1

Express

n

algebraically based on the given remainder condition.

Since

n

leaves a remainder of

7

when divided by

12

,

n

can be written as

n = 12k + 7

for some non-negative integer

k

.

By the division algorithm, any integer leaving remainder

r

when divided by

m

is of the form

mk + r

.

2

Analyze

n

modulo

24

by considering the parity of

k

.

If

k

is even (

k = 2m

), then

n = 24m + 7 \equiv 7 \pmod{24}

. If

k

is odd (

k = 2m + 1

), then

n = 24m + 19 \equiv 19 \pmod{24}

.

To evaluate an expression modulo

24

when given information modulo

12

, split the integer into even and odd multiplier cases.

3

Evaluate

n^2 + 5n + 2 \pmod{24}

for both possible residue classes of

n \pmod{24}

.

Case 1 (

n \equiv 7 \pmod{24}

):

7^2 + 5(7) + 2 = 49 + 35 + 2 = 86

. Since

86 = 3 \times 24 + 14

, the remainder is

14

. Case 2 (

n \equiv 19 \pmod{24}

):

19^2 + 5(19) + 2 = 361 + 95 + 2 = 458

. Since

458 = 19 \times 24 + 2

, the remainder is

2

.

Substituting representative values of

n

into

n^2 + 5n + 2

determines all possible remainder outputs modulo

24

.

4

Identify which of the options matches a valid remainder.

The only possible remainders are

14

and

2

. Among the given choices,

2

is present.

Matching the derived valid remainder set

\{2, 14\}

against the options identifies the correct answer.

Key Concept

Modular Arithmetic and Remainder Properties under Quadratic Transformations

Estimated Time:2m 0s

Question 37Question →

At the start of the year, a community library owned $8,000$ books. During the course of the year, the total number of books in the collection increased by $15\%$ . How many books were in the library's collection at the end of the year?

1,200

6,800

8,015

9,200

9,400

Show answer & explanation

Answer: $9,200$

Answer

The total number of books in the library's collection at the end of the year is

9,200

.

To determine the final quantity after a percent increase, first compute

15\%

of the initial amount:

0.15 \times 8,000 = 1,200

. Adding this increase to the original

8,000

books yields

8,000 + 1,200 = 9,200

. Alternatively, multiply the initial value directly by

1.15

:

8,000 \times 1.15 = 9,200

.

Step-by-Step Solution

1

Calculate the number of additional books added during the year.

15\% \text{ of } 8,000 = 0.15 \times 8,000 = 1,200

To find the amount of increase, multiply the original total by the percentage increase converted to a decimal.

2

Add the increase to the original number of books to find the final total.

8,000 + 1,200 = 9,200

The final total equals the initial quantity plus the percentage increase.

Key Concept

Percent Increase

Estimated Time:45s

Question 38Question →

If $x$ and $y$ are non-zero rational numbers such that $\frac{x}{y} < -\frac{1}{2}$ and $x + y > 0$ , which of the following statements must be true? Select all that apply.

Select all that apply

xy < 0

\frac{1}{x} + \frac{1}{y} < 0

x > y

|x| > |y|

x^2 + y^2 > (x+y)^2

Show answer & explanation

Answer: $xy < 0$ ; $\frac{1}{x} + \frac{1}{y} < 0$ ; $x^2 + y^2 > (x+y)^2$

Answer

The statements

xy < 0

,

\frac{1}{x} + \frac{1}{y} < 0

, and

x^2 + y^2 > (x+y)^2

must be true.

Because

\frac{x}{y} < -\frac{1}{2}

,

x

and

y

must carry opposite algebraic signs, establishing that

xy < 0

. Combining the reciprocals into

\frac{x+y}{xy}

places a positive numerator over a negative denominator, ensuring the sum of reciprocals is strictly negative. Expanding

(x+y)^2 = x^2 + y^2 + 2xy

shows that adding the negative term

2xy

makes

(x+y)^2 < x^2 + y^2

.

Step-by-Step Solution

1

Determine the sign of the product

xy

xy < 0

The quotient of two non-zero real numbers is negative if and only if they have opposite signs. Since

\frac{x}{y} < -\frac{1}{2} < 0

,

x

and

y

must have opposite signs, so their product is negative.

2

Evaluate the sum of reciprocals

\frac{1}{x} + \frac{1}{y}

\frac{1}{x} + \frac{1}{y} < 0

Finding a common denominator gives

\frac{1}{x} + \frac{1}{y} = \frac{x+y}{xy}

. Given

x+y > 0

(positive) and

xy < 0

(negative), dividing a positive number by a negative number yields a negative result.

3

Compare

x^2 + y^2

with

(x+y)^2

x^2 + y^2 > (x+y)^2

Using algebraic expansion,

(x+y)^2 = x^2 + y^2 + 2xy

. Because

xy < 0

,

2xy

is negative. Subtracting a positive value (or adding a negative value) to

x^2 + y^2

results in a smaller quantity.

4

Test counterexamples for

x > y

and

|x| > |y|

Neither statement is required to be true.

Let

x = -2

and

y = 3

. Both are rational numbers. Check conditions:

\frac{-2}{3} < -\frac{1}{2}

holds, and

-2 + 3 = 1 > 0

holds. For this counterexample,

x = -2 < 3 = y

and

|-2| = 2 < 3 = |y|

.

Key Concept

Signs and inequalities of rational numbers and reciprocal operations

Question 39Question →

A chemical storage vessel contains a mixture composed of three liquid components: Component $X$ , Component $Y$ , and Component $Z$ . Initially, Component $X$ accounts for $\frac{3}{8}$ of the total mixture volume, and Component $Y$ accounts for $\frac{5}{12}$ of the total mixture volume, with Component $Z$ occupying the remainder of the volume.

During a processing stage, $\frac{1}{3}$ of Component $X$ is extracted and $\frac{2}{5}$ of Component $Y$ is extracted, while Component $Z$ remains completely unchanged. If the total volume of the mixture remaining in the vessel after processing is 170 milliliters, what was the total volume, in milliliters, of the mixture in the vessel before processing?

Show answer & explanation

Answer: 240

Answer

The initial total volume of the mixture was 240 milliliters.

To find the initial total volume, first find the fraction of the initial mixture that is Component Z:

1 - (3/8 + 5/12) = 5/24

. Next, compute the remaining amounts of each component relative to the initial total volume

V

: Component X has

(1 - 1/3) \times 3/8 = 1/4 = 6/24

remaining; Component Y has

(1 - 2/5) \times 5/12 = 1/4 = 6/24

remaining; Component Z retains its

5/24

. Summing these yields

6/24 + 6/24 + 5/24 = 17/24

of the original volume. Setting

(17/24)V = 170

mL gives

V = 170 \times (24/17) = 240

mL.

Step-by-Step Solution

1

Find the initial fraction of the mixture representing Component

Z

.

Component

Z = 1 - \left(\frac{3}{8} + \frac{5}{12}\right) = 1 - \left(\frac{9}{24} + \frac{10}{24}\right) = 1 - \frac{19}{24} = \frac{5}{24}

.

The sum of all component fractions must equal 1.

2

Calculate the remaining fraction for each component relative to the initial total volume

V

.

Remaining

X = \left(1 - \frac{1}{3}\right) \times \frac{3}{8}V = \frac{2}{3} \times \frac{3}{8}V = \frac{1}{4}V = \frac{6}{24}V

.
Remaining

Y = \left(1 - \frac{2}{5}\right) \times \frac{5}{12}V = \frac{3}{5} \times \frac{5}{12}V = \frac{1}{4}V = \frac{6}{24}V

.
Remaining

Z = \frac{5}{24}V

.

When a fraction of a component is removed, the remaining fraction of that component is multiplied by its original portion of the total mixture.

3

Sum the remaining component fractions to find the total remaining volume as a fraction of

V

.

Total Remaining Fraction

= \frac{6}{24}V + \frac{6}{24}V + \frac{5}{24}V = \frac{17}{24}V

.

Combining the remaining amounts gives the fraction of the initial mixture left.

4

Solve for the initial total volume

V

using the given final volume of 170 mL.

\frac{17}{24}V = 170 \implies V = 170 \times \frac{24}{17} = 10 \times 24 = 240 \text{ mL}.

Multiplying the final volume by the reciprocal of the remaining fraction yields the original volume.

Key Concept

Multi-step fraction operations and solving for original whole amounts

Question 40Question →

When the positive integer $n$ is divided by $9$ , the remainder is $5$ . What is the remainder when $n^2 + 4n + 7$ is divided by $9$ ?

Show answer & explanation

Answer: 7

Answer

The remainder when

n^2 + 4n + 7

is divided by

9

is

7

.

Any positive integer

n

that leaves a remainder of

5

when divided by

9

can be represented as

n = 9k + 5

for some non-negative integer

k

. Substituting this into

n^2 + 4n + 7

yields

(9k + 5)^2 + 4(9k + 5) + 7 = 81k^2 + 90k + 25 + 36k + 20 + 7 = 81k^2 + 126k + 52

. Since

81k^2

and

126k

are both divisible by

9

, the remainder of the entire expression when divided by

9

is determined entirely by

52

. Dividing

52

by

9

gives

52 = 9 \times 5 + 7

, so the final remainder is

7

.

Step-by-Step Solution

1

Express the integer

n

in terms of its remainder upon division by

9

.

n = 9k + 5

for some non-negative integer

k

, or equivalently

n \equiv 5 \pmod{9}

.

By the division algorithm, any integer divided by

9

can be written as a multiple of

9

plus the remainder.

2

Substitute

n \equiv 5 \pmod{9}

into the polynomial expression

n^2 + 4n + 7

.

n^2 + 4n + 7 \equiv 5^2 + 4(5) + 7 = 25 + 20 + 7 = 52 \pmod{9}

.

Properties of modular arithmetic allow substitution of remainder values into polynomial expressions.

3

Find the remainder of

52

when divided by

9

.

52 = 9 \times 5 + 7

, which gives a remainder of

7

.

Dividing

52

by

9

yields a quotient of

5

and a remainder of

7

, which is strictly between

0

and

8

.

Key Concept

Properties of Integer Remainders and Modular Substitution

Estimated Time:1m 15s

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