All practice questions

Question 1821Question →

At a research institute, an annual equipment budget is distributed among three departments. First, $\frac{2}{5}$ of the total budget is allocated to the Biology department. Next, $\frac{3}{8}$ of the remaining budget is allocated to the Chemistry department. The Physics department receives the remaining balance of $\$45,000$ . What was the institute's total annual equipment budget?

$100,000

$112,500

$120,000

$180,000

$200,000

Show answer & explanation

Answer: $120,000

Answer

$120,000

Subtracting the Biology department's share of

\frac{2}{5}

leaves

\frac{3}{5}

of the total budget. The Chemistry department receives

\frac{3}{8}

of this remainder, leaving

\frac{5}{8}

of the remainder for the Physics department. Multiplying these fractions yields

\frac{5}{8} \times \frac{3}{5} = \frac{3}{8}

of the total budget for Physics. Equating

\frac{3}{8}

of the total budget to

\$45,000

gives a total budget of

\$120,000

.

Step-by-Step Solution

1

Determine the fraction of the budget remaining after the Biology allocation.

The remaining fraction is

1 - \frac{2}{5} = \frac{3}{5}

of the total budget.

The Biology department receives

\frac{2}{5}

of the overall budget.

2

Calculate the fraction of the total budget allocated to Physics.

The Physics department receives

1 - \frac{3}{8} = \frac{5}{8}

of the remaining budget, which equals

\frac{5}{8} \times \frac{3}{5} = \frac{3}{8}

of the total budget.

The Chemistry department receives

\frac{3}{8}

of the remaining budget, leaving

\frac{5}{8}

of that remainder for Physics.

3

Solve for the total budget

B

.

\frac{3}{8}B = 45,000 \implies B = 45,000 \times \frac{8}{3} = 120,000.

The Physics allocation of

\$45,000

represents

\frac{3}{8}

of the total budget

B

.

Key Concept

Sequential Fraction of Remainder Calculations

Estimated Time:1m 30s

Question 1822Question →

A water reservoir initially contains $450$ liters of water. Water drains out of the reservoir at a constant rate of $r$ liters per hour, while an inlet pipe supplies water at a constant rate of $18$ liters per hour. If the reservoir contains $390$ liters of water after $6$ hours, what is the value of $r$ ?

Show answer & explanation

Answer: 28

Answer

The value of

r

is

28

.

The reservoir starts with

450

liters. Over

6

hours, water enters at

18

liters/hour and leaves at

r

liters/hour, giving a net volume equation of

450 + 6(18 - r) = 390

. Simplifying yields

558 - 6r = 390

, which subtracts to

-6r = -168

, giving

r = 28

.

Step-by-Step Solution

1

Set up the linear equation representing the net change in water volume over time.

450 + 6(18 - r) = 390

The final volume equals the initial volume plus the net water added (inflow rate minus outflow rate multiplied by hours).

2

Expand and simplify the linear expression.

558 - 6r = 390

Distribute

6

across

(18 - r)

to obtain

108 - 6r

, then add to

450

.

3

Isolate the variable term

-6r

.

-6r = -168

Subtract

558

from both sides of the equation.

4

Solve for the rate

r

.

r = 28

Divide both sides by

-6

.

Key Concept

Formulating and solving a linear equation in one variable from a rate problem context.

Estimated Time:1m 30s

Question 1823Question →

A commercial trucking fleet monitored its vehicle fuel efficiency (in miles per gallon) and average fuel price (in dollars per gallon) over a two-year period. In Year 1, vehicle fuel efficiency increased by $25\%$ , while the price of fuel increased by $15\%$ . In Year 2, vehicle fuel efficiency increased by an additional $20\%$ relative to Year 1, while the price of fuel decreased by $10\%$ relative to Year 1. Assuming the total annual miles driven per vehicle remained constant throughout the two-year period, by what percent did the total annual fuel cost per vehicle change from the beginning of Year 1 to the end of Year 2?

31\%

decrease

69\%

decrease

40\%

decrease

28\%

decrease

45\%

increase

Show answer & explanation

Answer: $31\%$ decrease

Answer

A

31\%

decrease

The total fuel cost is given by

\text{Cost} = \text{Distance} \times \frac{\text{Price}}{\text{Efficiency}}

. After Year 1, efficiency becomes

1.25 E_0

and price becomes

1.15 P_0

. After Year 2, efficiency increases to

1.25 E_0 \times 1.20 = 1.50 E_0

, while price changes to

1.15 P_0 \times 0.90 = 1.035 P_0

. The new cost ratio is

\frac{1.035}{1.50} = 0.69

, meaning the new cost is

69\%

of the original cost. This represents a net decrease of

100\% - 69\% = 31\%

.

Step-by-Step Solution

1

Formulate the fuel cost relationship in terms of distance, efficiency, and price.

Initial cost

C_0 = D \times \frac{P_0}{E_0}

, where

D

is distance driven,

P_0

is initial price per gallon, and

E_0

is initial fuel efficiency in miles per gallon.

Total fuel used is

\frac{D}{E_0}

, so multiplying by price per gallon gives total cost.

2

Calculate the updated fuel efficiency

E_2

and price

P_2

at the end of Year 2 using successive percentage changes.

E_2 = E_0 \times 1.25 \times 1.20 = 1.50 E_0

and

P_2 = P_0 \times 1.15 \times 0.90 = 1.035 P_0

.

Compounding changes sequentially: Year 1 efficiency factor is

1.25

, Year 2 factor is

1.20

; Year 1 price factor is

1.15

, Year 2 factor is

0.90

.

3

Compute the final fuel cost

C_2

as a fraction of the initial cost

C_0

.

C_2 = D \times \frac{1.035 P_0}{1.50 E_0} = \frac{1.035}{1.50} \times C_0 = 0.69 C_0

.

Substituting the expressions for

P_2

and

E_2

into the fuel cost formula.

4

Determine the net percentage change from

C_0

to

C_2

.

\text{Percent Change} = \frac{0.69 C_0 - C_0}{C_0} \times 100\% = -31\%

, representing a

31\%$ decrease.

Subtracting

1.00

from

0.69

yields

-0.31

, which corresponds to a

31\%

reduction.

Key Concept

Successive Percentage Changes and Rate Relationships

Question 1824Question →

A positive integer $n$ has a prime factorization of the form $2^x \times 3^y \times 7^z$ , where $x$ , $y$ , and $z$ are positive integers. The greatest common divisor of $n$ and $420$ is $84$ , and the least common multiple of $n$ and $420$ is $8,820$ . What is the total number of positive integer factors of $n$ ?

8

12

18

27

36

Show answer & explanation

Answer: $27$

Answer

The total number of positive integer factors of

n

is

27

.

By prime factorizing

420

,

84

, and

8,820

, we find that

420 = 2^2 \times 3^1 \times 5^1 \times 7^1

,

\gcd(n, 420) = 2^2 \times 3^1 \times 7^1

, and

\text{lcm}(n, 420) = 2^2 \times 3^2 \times 5^1 \times 7^2

. Comparing the minimum and maximum powers for each prime factor shows that

n = 2^2 \times 3^2 \times 7^2

. Applying the divisor counting formula gives

(2+1)(2+1)(2+1) = 27

.

Step-by-Step Solution

1

Express all given values in their prime factorized forms.

420 = 2^2 \times 3^1 \times 5^1 \times 7^1

,

\gcd(n, 420) = 84 = 2^2 \times 3^1 \times 7^1

, and

\text{lcm}(n, 420) = 8,820 = 2^2 \times 3^2 \times 5^1 \times 7^2

.

Finding the prime factorizations allows comparison of prime exponent bounds for GCD and LCM.

2

Determine the exponents

x

,

y

, and

z

for

n = 2^x \times 3^y \times 7^z

.

For prime

2

:

\max(x, 2) = 2

and

\min(x, 2) = 2 \implies x = 2

.
For prime

3

:

\max(y, 1) = 2

and

\min(y, 1) = 1 \implies y = 2

.
For prime

7

:

\max(z, 1) = 2

and

\min(z, 1) = 1 \implies z = 2

.

GCD takes the minimum exponent of each prime factor, while LCM takes the maximum exponent.

3

Calculate the total number of positive factors of

n = 2^2 \times 3^2 \times 7^2

.

Total factors

= (x + 1)(y + 1)(z + 1) = (2 + 1)(2 + 1)(2 + 1) = 3 \times 3 \times 3 = 27

.

The number of positive divisors of

p_1^{a} p_2^{b} \dots p_k^{k}

is

(a+1)(b+1)\dots(k+1)

.

Key Concept

Relationship between prime factorization, greatest common divisor (GCD), least common multiple (LCM), and the number of positive divisors.

Question 1825Question →

Consider the expression $N = \frac{0.00048 \times (2.5 \times 10^7)}{1.2 \times 10^{-2}}$ . Which of the following expressions are equal to $N$ ? Select all such values.

Select all that apply

1.0 \times 10^6

(4.0 \times 10^4) \times 25

1.0 \times 10^2

\frac{5.0 \times 10^{-3}}{5.0 \times 10^{-9}}

0.00012 \times 10^9

Show answer & explanation

Answer: $1.0 \times 10^6$ ; $(4.0 \times 10^4) \times 25$ ; $\frac{5.0 \times 10^{-3}}{5.0 \times 10^{-9}}$

Answer

The expressions equal to

N

are

1.0 \times 10^6

,

(4.0 \times 10^4) \times 25

, and

\frac{5.0 \times 10^{-3}}{5.0 \times 10^{-9}}

.

Evaluating

N

simplifies to

1.0 \times 10^6

. The expression stating

1.0 \times 10^6

is an exact scientific notation match. The expression

(4.0 \times 10^4) \times 25

computes to

40,000 \times 25 = 1,000,000

. The fraction

\frac{5.0 \times 10^{-3}}{5.0 \times 10^{-9}}

evaluates to

1.0 \times 10^{-3 - (-9)} = 1.0 \times 10^6

.

Step-by-Step Solution

1

Express all numbers in the stem's numerator in standard scientific notation

0.00048 = 4.8 \times 10^{-4}

and the numerator becomes

(4.8 \times 10^{-4}) \times (2.5 \times 10^7)

Standardizing numbers into scientific notation simplifies operations involving powers of 10

2

Multiply the coefficients and combine the powers of 10 in the numerator

(4.8 \times 2.5) \times 10^{-4 + 7} = 12 \times 10^3 = 1.2 \times 10^4

Coefficients are multiplied directly while exponents are added according to index laws

3

Divide the simplified numerator by the denominator

\frac{1.2 \times 10^4}{1.2 \times 10^{-2}} = \left(\frac{1.2}{1.2}\right) \times 10^{4 - (-2)} = 1.0 \times 10^6 = 1,000,000

Subtracting a negative exponent in the denominator is equivalent to adding its positive value

4

Evaluate each given option to determine equivalence to

1.0 \times 10^6

The expressions representing

1.0 \times 10^6

are

1.0 \times 10^6

,

(4.0 \times 10^4) \times 25

, and

\frac{5.0 \times 10^{-3}}{5.0 \times 10^{-9}}

Direct comparison verifies which options match the calculated value of

N

Key Concept

Operations with Decimals and Scientific Notation

Question 1826Question →

If $x$ is a real number such that $x^4 = 16$ , which of the following values could be equal to $x^3$ ? Select all that apply.

Select all that apply

-8

-4

4

8

16

Show answer & explanation

Answer: $-8$ ; $8$

Answer

The values

-8

and

8

are the possible values of

x^3

.

Solving

x^4 = 16

gives two real solutions,

x = 2

and

x = -2

, because raising any real number to an even power yields a non-negative result. Cubing each solution gives

2^3 = 8

and

(-2)^3 = -8

. Thus, both

-8

and

8

are correct values for

x^3

.

Step-by-Step Solution

1

Find all real solutions for

x

in the equation

x^4 = 16

.

x = 2

or

x = -2

.

Taking the fourth root of both sides gives

|x| = \sqrt[4]{16} = 2

, so

x

can be positive or negative.

2

Calculate

x^3

for the positive root

x = 2

.

2^3 = 8

.

Cubing a positive number yields a positive result.

3

Calculate

x^3

for the negative root

x = -2

.

(-2)^3 = -8

.

Cubing a negative number yields a negative result because an odd exponent preserves the sign.

Key Concept

Even and odd power rules for positive and negative real bases

Question 1827Question →

If $N = 2^4 \times 3^3 \times 5^2$ , how many positive integer factors of $N$ are divisible by $6$ but not divisible by $12$ ?

Show answer & explanation

Answer: 9

Answer

The number of positive integer factors of

N

divisible by 6 but not by 12 is 9.

Any factor of

N = 2^4 \times 3^3 \times 5^2

is of the form

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

. For the factor to be divisible by

6

, we must have

a \ge 1

and

b \ge 1

. For it to not be divisible by

12 = 2^2 \times 3^1

, we must have

a < 2

. Therefore,

a

must equal

1

. The possible choices for

a

are

1

value (

1

), for

b

are

3

values (

1, 2, 3

), and for

c

are

3

values (

0, 1, 2

). Multiplying these options gives

1 \times 3 \times 3 = 9

.

Step-by-Step Solution

1

Express the prime factorization structure of a factor of

N

.

Any factor of

N

takes the form

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

with bounds

0 \le a \le 4

,

0 \le b \le 3

, and

0 \le c \le 2

.

Divisors of a number are formed by taking prime factors with exponents between zero and their maximum powers in the original number.

2

Determine exponent constraints for divisibility by 6 and non-divisibility by 12.

Divisibility by

6

requires

a \ge 1

and

b \ge 1

. Non-divisibility by

12

requires

a < 2

. Thus,

a = 1

exactly.

A factor must contain at least one factor of 2 and one factor of 3 to be a multiple of 6, but containing two or more factors of 2 makes it a multiple of 12.

3

Count combinations of choices for the exponents.

Exponent

a

has

1

option (

a = 1

),

b

has

3

options (

b \in \{1, 2, 3\}

), and

c

has

3

options (

c \in \{0, 1, 2\}

). Total

= 1 \times 3 \times 3 = 9

.

Applying the fundamental counting principle by multiplying the number of choices for independent prime factor exponents.

Key Concept

Counting Divisors with Prime Factorization and Divisibility Constraints

Question 1828Question →

At the beginning of Year 1, an investor divided a sum of money between two portfolio accounts, Account X and Account Y, such that the initial balance of Account X was $25\%$ greater than the initial balance of Account Y.

Over a two-year period:
- Account X earned compound interest at a constant annual rate of $20\%$ per year.
- Account Y decreased in value by $10\%$ during Year 1, and then increased in value by $r\%$ during Year 2.

If the total combined value of both accounts at the end of Year 2 was $32\%$ greater than the total combined initial balance at the beginning of Year 1, what is the value of $r$ ?

22

25

27

30

36

Show answer & explanation

Answer: $30$

Answer

The value of

r

is

30

.

The correct value is

30

. Account X starts at

125\%

of Account Y's initial value (

125

vs

100

, totaling

225

). Compounding Account X at

20\%

per year for 2 years yields

125 \times 1.20^2 = 180

. Account Y drops

10\%

to

90

in Year 1. For the overall total to reach

225 \times 1.32 = 297

, Account Y must reach

297 - 180 = 117

at the end of Year 2. The percentage increase from

90

to

117

is

\frac{117 - 90}{90} \times 100\% = \frac{27}{90} \times 100\% = 30\%

.

Step-by-Step Solution

1

Define initial account balances using a suitable variable base

Let initial balance of Account Y be

Y_0 = 100

units. Initial balance of Account X is

X_0 = 1.25 \times 100 = 125

units. Combined initial balance is

T_0 = 125 + 100 = 225

units.

Establishing a standard numerical base simplifies tracking multi-account percentage changes.

2

Calculate the value of Account X at the end of Year 2

X_2 = 125 \times (1 + 0.20)^2 = 125 \times 1.44 = 180

units.

Account X compounds annually at

20\%

for 2 full years.

3

Express the value of Account Y at the end of Year 2 in terms of

r

After Year 1:

Y_1 = 100 \times (1 - 0.10) = 90

units. After Year 2:

Y_2 = 90 \times \left(1 + \frac{r}{100}\right)

units.

Account Y first loses

10\%

of its initial value, creating a new base of

90

units for the Year 2 growth rate.

4

Determine the required total combined final value

T_2 = 225 \times (1 + 0.32) = 225 \times 1.32 = 297

units.

The overall portfolio value grew by

32\%

relative to the initial combined balance of

225

units.

5

Set up and solve the combined equation for

r

180 + 90 \left(1 + \frac{r}{100}\right) = 297 \implies 90 \left(1 + \frac{r}{100}\right) = 117 \implies 1 + \frac{r}{100} = 1.30 \implies r = 30

.

Equating the sum of individual final account balances to the target portfolio balance yields the exact value of

r

.

Key Concept

Successive percent changes on shifting bases combined with compound interest calculations across multiple accounts.

Question 1829Question →

Passage:
In 1821, Swiss engineer Ignaz Venetz presented a paper asserting that alpine glaciers had once extended far beyond their current boundaries, citing the widespread distribution of massive, polished rocks known as erratic boulders across the Swiss plain. While contemporary naturalists largely attributed the movement of these boulders to catastrophic ancient deluges, Venetz argued that only slow-moving ice sheets possessed sufficient density to transport such immense stone masses without pulverizing them. To support his claim, Venetz pointed to distinct striations—parallel linear grooves carved into bedrock—which he observed exclusively beneath active glaciers and along valley walls where erratic boulders were concentrated. Although his hypothesis was initially dismissed by the scientific establishment as overly speculative, it eventually provided the empirical foundation for Louis Agassiz’s comprehensive theory of a global Ice Age published two decades later.

According to the passage, which of the following specifies the primary physical reason Venetz gave for why glaciers, rather than water deluges, relocated erratic boulders?

Ice sheets possessed adequate structural density to convey massive stones without shattering them.

Glaciers moved at rapid speeds capable of incising parallel grooves directly into valley bedrock.

Glaciers deposited unique mineral trace signatures along valley walls as they expanded across the plain.

Ice sheets were observed expanding across the Swiss plain by contemporary naturalists during the early nineteenth century.

Catastrophic water deluges lacked the velocity required to deposit boulders along elevated valley walls.

Show answer & explanation

Answer: Ice sheets possessed adequate structural density to convey massive stones without shattering them.

Answer

Venetz maintained that glaciers relocated erratic boulders because ice sheets possessed adequate structural density to convey massive stones without shattering them.

The passage explicitly states that 'Venetz argued that only slow-moving ice sheets possessed sufficient density to transport such immense stone masses without pulverizing them.' The option stating that ice sheets possessed adequate structural density to convey massive stones without shattering them is an exact, accurate paraphrase of this explicit fact.

Step-by-Step Solution

1

Identify the key terms in the question prompt.

The prompt asks for the explicit physical reason Venetz gave for why glaciers, rather than deluges, relocated erratic boulders.

Scanning the passage for 'Venetz argued', 'deluges', and boulder transport mechanisms isolates the target sentence.

2

Locate and extract the exact sentence in the passage matching the target concept.

The text explicitly states: 'Venetz argued that only slow-moving ice sheets possessed sufficient density to transport such immense stone masses without pulverizing them.'

This direct statement establishes density as the physical property enabling transport without pulverization (shattering).

3

Match the extracted fact with the accurate paraphrase among the options.

The choice stating that ice sheets possessed adequate structural density to convey massive stones without shattering them perfectly matches the passage detail.

'Sufficient density' is paraphrased as 'adequate structural density', 'transport' as 'convey', and 'without pulverizing' as 'without shattering'.

Key Concept

Explicit Detail Retrieval

Estimated Time:1m 30s

Question 1830Question →

Although early twentieth-century astrophysicists initially regarded cosmic microwave background predictions as mere theoretical ______, subsequent observational evidence firmly established these calculations as foundational principles of modern cosmology. Which TWO of the following answer choices, when inserted into the sentence, produce completed sentences that are most similar in meaning?

Select all that apply

speculations

conjectures

dogmas

tenets

curiosities

truisms

Show answer & explanation

Answer: speculations; conjectures

Answer

The two words that complete the sentence to produce sentences with equivalent meaning are 'speculations' and 'conjectures'.

The contrast transition 'Although' signals an opposition between the initial status of cosmic microwave background predictions and their ultimate recognition as 'foundational principles.' The blank therefore requires words signifying tentative, unverified theoretical suppositions. Both 'speculations' and 'conjectures' fit this precise contextual demand and are functional near-synonyms, yielding sentences with identical meaning.

Step-by-Step Solution

1

Analyze structural contrast clues in the sentence stem.

The concessive transition 'Although' sets up a contrast between how astrophysicists 'initially regarded' predictions and how subsequent evidence 'firmly established' them as 'foundational principles.'

The blank requires noun choices that describe tentative, unproven ideas to contrast with established principles.

2

Identify synonymous word choices that fit the required meaning.

'Speculations' and 'conjectures' both mean tentative conclusions or hypotheses formed without firm proof.

Inserting either word produces a sentence expressing that predictions were viewed as mere unproven theoretical ideas.

3

Evaluate distractor options against error patterns.

'Dogmas' and 'tenets' form a valid synonym pair but conflict with the contrast signal. 'Curiosities' fits contextually as a single word but has no synonym partner. 'Truisms' contradicts the unverified nature of the initial predictions.

GRE Sentence Equivalence requires both contextual appropriateness and semantic equivalence between the chosen pair.

Key Concept

Identifying Synonym and Near-Synonym Pairs in Sentence Equivalence Contexts

Question 1831Question →

An electronics retailer evaluated its annual revenue over a four-year period. In Year 2, the revenue increased by $25\%$ compared to Year 1. In Year 3, the revenue decreased by $20\%$ compared to Year 2. In Year 4, the revenue increased by $15\%$ compared to Year 3.

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

Select all that apply

The revenue in Year 3 is equal to the revenue in Year 1.

The overall percentage change in revenue from Year 1 to Year 4 is a

15\%

increase.

The revenue in Year 2 is

25\%

greater than the revenue in Year 3.

The overall percentage change in revenue from Year 1 to Year 3 is a

5\%

increase.

The percent increase from Year 3 to Year 4 is greater than the percent decrease from Year 2 to Year 3.

Show answer & explanation

Answer: The revenue in Year 3 is equal to the revenue in Year 1.; The overall percentage change in revenue from Year 1 to Year 4 is a $15\%$ increase.; The revenue in Year 2 is $25\%$ greater than the revenue in Year 3.

Answer

The statements asserting that Year 3 revenue equals Year 1 revenue, that the overall revenue change from Year 1 to Year 4 is a 15% increase, and that Year 2 revenue is 25% greater than Year 3 revenue are all true.

Let the revenue in Year 1 be

R

. Revenue in Year 2 is

1.25R

. Revenue in Year 3 is

1.25R \times 0.80 = 1.00R

, which equals the revenue in Year 1. Revenue in Year 4 is

1.00R \times 1.15 = 1.15R

, representing a net

15\%

increase from Year 1. Furthermore, Year 2 revenue (

1.25R

) compared to Year 3 revenue (

1.00R

) is calculated as

\frac{1.25R - 1.00R}{1.00R} = 0.25

, or a

25\%

increase. Therefore, the first three statements are mathematically true.

Step-by-Step Solution

1

Define variables and calculate Year 2 revenue relative to Year 1

Let Year 1 revenue be

R

. Year 2 revenue =

R \times (1 + 0.25) = 1.25R

.

A 25% increase corresponds to a multiplier of 1.25.

2

Calculate Year 3 revenue relative to Year 2 and Year 1

Year 3 revenue =

1.25R \times (1 - 0.20) = 1.25R \times 0.80 = 1.00R

.

A 20% decrease applies to the new base value of

1.25R

.

3

Calculate Year 4 revenue relative to Year 3 and Year 1

Year 4 revenue =

1.00R \times (1 + 0.15) = 1.15R

.

A 15% increase applies to Year 3 revenue, which equals

R

.

4

Evaluate each statement against the calculated values

Year 3 revenue (

1.00R

) equals Year 1 revenue (

R

). Year 4 revenue (

1.15R

) is 15% greater than Year 1 revenue (

R

). Year 2 revenue (

1.25R

) is 25% greater than Year 3 revenue (

1.00R

).

Determines which statements hold true mathematically.

Key Concept

Successive Percent Changes and Base Value Shifting

Question 1832Question →

Let $r$ and $s$ be rational numbers such that $0 < r < \frac{1}{2}$ and $1 < s < 2$ . Which of the following inequalities MUST be true? Select all that apply.

Select all that apply

r \cdot s < 1

\frac{s}{r} > 2

s - r > 1

\frac{r}{s} > \frac{1}{4}

r + s > \frac{3}{2}

Show answer & explanation

Answer: $r \cdot s < 1$ ; \frac{s}{r} > 2

Answer

The correct inequalities that must be true are

r \cdot s < 1

and

\frac{s}{r} > 2

.

The inequality stating that the product of

r

and

s

is less than

1

is guaranteed because multiplying the maximum possible bounds (

\frac{1}{2}

and

2

) gives

1

. The inequality stating that the ratio of

s

to

r

is greater than

2

is guaranteed because the numerator is strictly greater than

1

and the denominator is strictly less than

\frac{1}{2}

.

Step-by-Step Solution

1

Analyze

r \cdot s < 1

Since

r < \frac{1}{2}

and

s < 2

, and both are positive rational numbers, multiplying their upper limits gives

r \cdot s < \left(\frac{1}{2}\right)(2) = 1

. This statement MUST be true.

Properties of inequality multiplication for positive rational numbers.

2

Analyze

\frac{s}{r} > 2

Since

s > 1

and

r < \frac{1}{2}

, taking the ratio gives

\frac{s}{r} > \frac{1}{1/2} = 2

. This statement MUST be true.

Dividing a larger positive number by a fraction less than

1/2

amplifies the quotient.

3

Test counterexamples for remaining options

For

s - r > 1

, pick

s = 1.2

and

r = 0.4 \Rightarrow s - r = 0.8 \le 1

(False). For

\frac{r}{s} > \frac{1}{4}

, pick

r = 0.1

and

s = 1.6 \Rightarrow \frac{r}{s} = \frac{1}{16} < \frac{1}{4}

(False). For

r + s > \frac{3}{2}

, pick

r = 0.1

and

s = 1.1 \Rightarrow r + s = 1.2 < 1.5

(False).

A single valid counterexample disproves a 'must be true' statement.

Key Concept

Properties of Rational Numbers and Inequalities

Question 1833Question →

If

x

satisfies the linear equation

4(x - 2) - 15 = 2x + 3

which of the following statements must be true? Select all that apply.

Select all that apply

x

is a prime number

2x - 5

is equal to

21

x + 7

is a multiple of

5

x

is an even integer

\frac{x + 1}{2} = 6

Show answer & explanation

Answer: $x$ is a prime number; $2x - 5$ is equal to $21$ ; $x + 7$ is a multiple of $5$

Answer

The correct statements are that

x

is a prime number,

2x - 5

is equal to

21

, and

x + 7

is a multiple of

5

.

Solving the equation

4(x - 2) - 15 = 2x + 3

yields

x = 13

. Evaluating the choices with

x = 13

confirms that

13

is a prime number,

2(13) - 5 = 21

, and

13 + 7 = 20

(a multiple of

5

).

Step-by-Step Solution

1

Expand the left side of the linear equation by distributing the constant term.

4x - 8 - 15 = 2x + 3

Distribution removes parentheses so like terms can be combined.

2

Combine constant terms on the left side of the equation.

4x - 23 = 2x + 3

Simplifying

-8 - 15

yields

-23

.

3

Isolate variable terms on the left side and constant terms on the right side by subtracting

2x

and adding

23

to both sides.

2x = 26

Subtracting

2x

gives

2x

on the left, and adding

23

gives

26

on the right.

4

Divide both sides of the equation by

2

to solve for

x

.

x = 13

Dividing

26

by

2

determines the unique value of

x

.

5

Test each statement using

x = 13

.

13

is prime (True);

2(13) - 5 = 21

(True);

13 + 7 = 20

which is a multiple of

5

(True);

13

is even (False);

\frac{13 + 1}{2} = 7 \neq 6

(False).

Direct evaluation identifies which statements hold true.

Key Concept

Solving linear equations in one variable and evaluating numerical properties of the solution.

Estimated Time:1m 0s

Question 1834Question →

Let $n$ be an integer such that $-15 \le n \le 15$ . How many integer values of $n$ satisfy both of the following conditions?

1. $(-1)^{n^2 + n + 1} < 0$
2. $(n - 3)(n + 4)^2 < 0$

Show answer & explanation

Answer: 17

Answer

17

Condition 1 is satisfied by every integer because

n^2 + n = n(n+1)

is always even (as the product of two consecutive integers), which makes

n^2 + n + 1

always odd and

(-1)^{\text{odd}} = -1 < 0

. Condition 2 requires

(n - 3)(n + 4)^2 < 0

. Since

(n+4)^2 > 0

for all

n \neq -4

, this reduces to

n < 3

while excluding

n = -4

. Within the range

-15 \le n \le 15

, there are 18 integers strictly less than 3, and removing

n = -4

leaves 17 valid values.

Step-by-Step Solution

1

Determine the parity of the exponent

n^2 + n + 1

n^2 + n + 1

is always odd for any integer

n

, making

(-1)^{n^2 + n + 1} = -1 < 0

unconditionally true.

The product of consecutive integers

n(n+1)

is always even, so adding 1 results in an odd number.

2

Solve the sign inequality

(n - 3)(n + 4)^2 < 0

n < 3

with

n \neq -4

.

A squared expression is strictly positive except when its base is zero. At

n = -4

, the product becomes 0, violating the strict inequality.

3

Count the integer solutions in the range

-15 \le n \le 15

17 integers satisfy both conditions.

There are 18 integers less than 3 in the interval

[-15, 15]

, and excluding

n = -4

gives

18 - 1 = 17

.

Key Concept

Parity of consecutive integer products and sign rules for squared terms in inequalities

Estimated Time:2m 0s

Question 1835Question →

In a given fiscal quarter, a retail company increased the unit price of its primary product by $p\%$ while the total quantity of the product sold decreased by $p\%$ , where $p > 0$ . As a result of these two concurrent changes, the total revenue generated from the product experienced a net decrease of exactly $4\%$ . Which of the following statements must be true? Select all that apply.

Select all that apply

The value of

p

is

20

.

The final revenue generated from the product is equal to

96\%

of the initial revenue.

If the unit price had instead decreased by

p\%

and the quantity sold had increased by

p\%

, total revenue would still have decreased by

4\%

.

The net percentage change in total revenue depends directly on the initial unit price of the product.

The sequential percentage changes of

+p\%

and

-p\%

offset each other, resulting in zero net change in total revenue.

Show answer & explanation

Answer: The value of $p$ is $20$ .; The final revenue generated from the product is equal to $96\%$ of the initial revenue.; If the unit price had instead decreased by $p\%$ and the quantity sold had increased by $p\%$ , total revenue would still have decreased by $4\%$ .

Answer

The correct statements are that the value of p is 20, the final revenue is 96% of the initial revenue, and reversing the direction of the percentage changes would still result in a 4% decrease.

The correct options accurately identify that

p = 20

, that the final revenue is

96\%

of the initial value, and that reversing the roles of increase and decrease produces the exact same net multiplier of

0.96

. Using the difference of squares identity

(1 + x)(1 - x) = 1 - x^2

, the combined multiplier for price and quantity shifts is

1 - (p/100)^2 = 0.96

. Solving gives

p^2 = 400

, so

p = 20

. Additionally,

(1 - 0.20)(1 + 0.20) = 0.96

, showing symmetry in net outcome.

Step-by-Step Solution

1

Express the final revenue as a function of initial revenue, price change, and quantity change.

Final Revenue = Initial Revenue

\times (1 + \frac{p}{100}) \times (1 - \frac{p}{100})

Total revenue is Price

\times

Quantity, so independent percentage changes multiply.

2

Apply the difference of squares formula and set equal to the given net revenue decrease.

(1 + \frac{p}{100})(1 - \frac{p}{100}) = 1 - \frac{p^2}{10000} = 1 - 0.04 = 0.96

A

4\%

decrease leaves

96\%

or

0.96

of the original value.

3

Solve the algebraic equation for

p

.

\frac{p^2}{10000} = 0.04 \implies p^2 = 400 \implies p = 20

Isolating

p

demonstrates that

p = 20

.

4

Evaluate alternative scenario with price decrease and quantity increase.

(1 - \frac{20}{100})(1 + \frac{20}{100}) = (0.80)(1.20) = 0.96

Multiplication is commutative, yielding the exact same

4\%

net decrease.

Key Concept

Successive Percent Changes and Base Shift Multipliers

Question 1836Question →

Although nineteenth-century archivists assumed that the newly uncovered royal decree was completely ________, subsequent paleographic analysis revealed a web of subtle linguistic contradictions, proving the document's meaning to be thoroughly ambiguous. Which word, if inserted into the blank, best completes the sentence in accordance with the structural contrast clues?

enigmatic

unequivocal

spurious

tendentious

pedantic

Show answer & explanation

Answer: unequivocal

Answer

unequivocal

The sentence relies on the structural transition word 'Although', which establishes a contrast between the initial assumption of the archivists and the later findings. Because the later findings prove that the decree's meaning is 'thoroughly ambiguous', the word in the blank must express the opposite quality—namely, being clear, direct, and unambiguous. The word 'unequivocal' precisely captures this meaning of absolute clarity and lack of ambiguity.

Step-by-Step Solution

1

Identify the structural contrast signal in the sentence.

The sentence begins with the concessionary conjunction 'Although', indicating that the initial assumption about the decree contrasts directly with later findings.

Contrast pivots establish an opposite relationship between the clause containing the blank and the rest of the sentence.

2

Analyze the context clues after the structural pivot.

The main clause states that paleographic analysis 'revealed a web of subtle linguistic contradictions, proving the document's meaning to be thoroughly ambiguous.'

The target word must be an antonym of 'ambiguous' (or 'contradictory') to satisfy the reversal.

3

Evaluate the vocabulary options to find the semantic opposite of 'ambiguous'.

The word 'unequivocal' means clear, unambiguous, and unambiguous in intent, which directly contrasts with 'ambiguous'.

Inserting 'unequivocal' creates a coherent statement: although archivists thought the decree was clear, it turned out to be ambiguous.

Key Concept

Identifying Contrast and Reversal Structural Clues

Estimated Time:1m 0s

Question 1837Question →

A laboratory technician prepares a cleaning solution by mixing a concentrated formula and distilled water in a ratio of $2 : 7$ by volume. If the technician uses $14$ liters of the concentrated formula, how many liters of distilled water are required?

4

35

49

63

98

Show answer & explanation

Answer: $49$

Answer

49

liters of distilled water

The ratio of concentrated formula to distilled water is given as

2 : 7

. Since

14

liters of concentrated formula represents

2

ratio parts (

14 = 2 \times 7

), each ratio part equals

7

liters. Multiplying the

7

ratio parts of distilled water by

7

liters gives

7 \times 7 = 49

liters.

Step-by-Step Solution

1

Set up the proportion relating concentrated formula to distilled water

\frac{\text{Concentrated Formula}}{\text{Distilled Water}} = \frac{2}{7} = \frac{14}{x}

The given ratio is part-to-part (

2

parts formula for every

7

parts water).

2

Solve for the unknown volume

x

using cross-multiplication

2 \cdot x = 7 \cdot 14 \implies 2x = 98

Cross-multiplying equates the products of the diagonal terms in a proportion.

3

Divide by

2

to isolate

x

x = 49

Isolating

x

provides the required amount of distilled water in liters.

Key Concept

Direct Part-to-Part Proportions

Question 1838Question →

Passage:
In 1952, English architect and self-taught linguist Michael Ventris deciphered Linear B, an ancient script found on clay tablets at Knossos and Pylos. Prior to Ventris’s breakthrough, prevailing scholarly opinion, spearheaded by archaeologist Sir Arthur Evans, held that Linear B encoded an unknown, non-Greek Minoan language unrelated to any known Indo-European tongue. Ventris initially shared this belief, attempting to analyze the syllabic signs under the assumption that the language was related to Etruscan. However, by constructing a grid to chart the contextual distribution and phonetic associations of recurring signs, Ventris discovered that certain sign combinations corresponded to geographical place names in Crete. When he substituted syllabic values derived from these place names into other tablet inscriptions, the resulting words closely matched an archaic form of Mycenaean Greek. Contrary to Evans's long-standing hypothesis, Ventris demonstrated that the Linear B tablets were administrative inventory records written in Greek, establishing that Greek literacy existed centuries earlier than previously believed.

According to the passage, which of the following statements about Michael Ventris's decipherment of Linear B are explicitly supported?

Select all that apply.

Select all that apply

Ventris initially conducted his analysis of Linear B under the assumption that its underlying language was related to Etruscan.

Sir Arthur Evans assisted Ventris in constructing the grid that mapped the phonetic associations of recurring signs.

The Linear B tablets analyzed by Ventris functioned as administrative inventory records.

Show answer & explanation

Answer: Ventris initially conducted his analysis of Linear B under the assumption that its underlying language was related to Etruscan.; The Linear B tablets analyzed by Ventris functioned as administrative inventory records.

Answer

The statements confirming that Ventris initially assumed a relationship between Linear B and Etruscan, and that the Linear B tablets functioned as administrative inventory records, are both explicitly supported by the text.

The passage explicitly supports two statements. First, it explicitly confirms that Ventris initially analyzed Linear B under the assumption that it was related to Etruscan. Second, it explicitly confirms that Ventris demonstrated the tablets were administrative inventory records.

Step-by-Step Solution

1

Scan the passage for details regarding Ventris's initial working assumptions.

The text states that Ventris 'attempted to analyze the syllabic signs under the assumption that the language was related to Etruscan.'

This confirms the factual accuracy of the statement regarding Ventris's initial assumption of an Etruscan connection.

2

Evaluate the passage details concerning Sir Arthur Evans's involvement in constructing the grid.

The text mentions Evans spearheaded the earlier prevailing opinion, but notes that 'by constructing a grid... Ventris discovered that certain sign combinations corresponded to geographical place names.' Evans did not participate in creating the grid.

This refutes the statement claiming Evans assisted in constructing the grid.

3

Scan the passage for explicit statements regarding the function of the Linear B tablets.

The text explicitly asserts that 'Ventris demonstrated that the Linear B tablets were administrative inventory records written in Greek.'

This confirms the factual accuracy of the statement regarding the administrative nature of the tablets.

Key Concept

Explicit Detail Retrieval

Question 1839Question →

An investment fund managed two separate accounts, Account X and Account Y, over a two-year period.

- At the start of Year 1, Account X held $\$100,000$ and Account Y held $\$150,000$ .
- During Year 1, Account X increased in value by $20\%$ , while Account Y decreased in value by $x\%$ .
- During Year 2, Account X decreased in value by $x\%$ , while Account Y increased in value by $20\%$ .
- At the end of Year 2, the combined total value of both accounts was $\$243,000$ .

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

Select all that apply

The value of

x

is equal to

19

.

At the end of Year 2, Account X experienced a net decrease of

2.8\%

relative to its initial value.

At the end of Year 1, Account Y had a greater dollar value than Account X.

At the end of Year 2, the overall percentage decrease in the value of Account Y from its initial value was greater than that of Account X.

The overall percent change of the combined portfolio over the two-year period was a decrease of

7.0\%

.

Show answer & explanation

Answer: The value of $x$ is equal to $19$ .; At the end of Year 2, Account X experienced a net decrease of $2.8\%$ relative to its initial value.; At the end of Year 1, Account Y had a greater dollar value than Account X.

Answer

The correct statements are those asserting that

x = 19

, that Account X experienced a net decrease of

2.8\%

relative to its initial value, and that at the end of Year 1 Account Y had a greater dollar value than Account X.

The statement specifying that

x = 19

is correct because solving the combined account equation

\$300,000(1 - x/100) = \$243,000

yields

1 - x/100 = 0.81

, giving

x = 19

. The statement regarding Account X experiencing a net decrease of

2.8\%

is correct because

1.20 \times 0.81 = 0.972

, which corresponds to a

2.8\%

decrease from initial value. The statement asserting that Account Y had a greater dollar value at the end of Year 1 than Account X is correct because Account Y was worth

\$121,500

compared to

\$120,000

for Account X.

Step-by-Step Solution

1

Express the value of each account at the end of Year 1 in terms of

x

.

Account X value =

\$100,000 \times (1 + 0.20) = \$120,000

. Account Y value =

\$150,000 \times (1 - x/100)

.

Calculate the intermediate values after Year 1 percentage changes.

2

Express the value of each account at the end of Year 2 and sum them to form an equation for the combined total.

Account X end of Year 2 =

\$120,000(1 - x/100)

. Account Y end of Year 2 =

\$150,000(1 - x/100)(1 + 0.20) = \$180,000(1 - x/100)

. Combined total =

(\$120,000 + \$180,000)(1 - x/100) = \$300,000(1 - x/100) = \$243,000

.

Use the given final total value of

\$243,000

to solve for

x

.

3

Solve for

x

.

1 - x/100 = 243,000 / 300,000 = 0.81 \implies x/100 = 0.19 \implies x = 19

.

Find the numerical value of

x

.

4

Evaluate each statement against the calculated values.

1.

x = 19

is true.
2. Net change multiplier for Account X =

1.20 \times 0.81 = 0.972

, representing a

2.8\%

decrease (true).
3. End of Year 1 values: Account X =

\$120,000

, Account Y =

\$150,000 \times 0.81 = \$121,500

. Since

\$121,500 > \$120,000

, Account Y was greater (true).
4. Account Y net change multiplier =

0.81 \times 1.20 = 0.972

, identical to Account X, so percentage decreases are equal (false).
5. Overall combined percent change =

(243,000 - 250,000)/250,000 = -7,000/250,000 = -2.8\%

(false).

Determine which statements are mathematically true.

Key Concept

Successive Percentage Changes and Base Shifts in Multi-Asset Portfolios

Question 1840Question →

Based on the contextual clues provided across the passage, which terms best complete each corresponding blank to maintain logical coherence?

Fill in the blanks below

Far from being a purely endeavor, early modern cartography was deeply entwined with political ambition and imperial expansion. Mapmakers did not merely record geographic realities; rather, they frequently territorial boundaries to project sovereign authority over contested lands. Consequently, historians who treat these vintage charts as records of objective spatial truth misinterpret their primary rhetorical function.

Show answer & explanation

Answer

The passage is correctly completed by 'dispassionate' for the first blank, 'manipulated' for the second blank, and 'unvarnished' for the third blank.

The sentence structure relies on clear contrastive and causal markers. In sentence one, 'Far from being...' sets up a contrast with political ambition, making 'dispassionate' (neutral/unbiased) the precise fit for blank 1. In sentence two, the contrast between merely recording reality and projecting authority requires 'manipulated' for blank 2. In sentence three, 'Consequently' links the misinterpretation of maps to historians treating distorted maps as 'unvarnished' (plain, accurate) records, making 'unvarnished' the only logically coherent fit for blank 3.

Step-by-Step Solution

1

Analyze the structural contrast clue in the first sentence to determine the meaning of blank 1.

The phrase 'Far from being a purely...' signals a contrast with 'deeply entwined with political ambition'. The first blank requires a word meaning neutral or objective.

Choosing 'dispassionate' establishes the required contrast between perceived scientific neutrality and political bias.

2

Evaluate the semi-colon and restatement shift in the second sentence for blank 2.

The clause states that mapmakers 'did not merely record... rather, they frequently... to project sovereign authority'. This indicates an active alteration or distortion of maps for political gain.

'Manipulated' fits the context of altering boundaries to project power rather than objectively surveying them.

3

Examine the cause-and-effect relationship set up by 'Consequently' in the third sentence for blank 3.

Because mapmakers distorted maps for political purposes, historians who view these maps as plain, truthful depictions are misinterpreting them.

'Unvarnished' (meaning straightforward or untruthfully modified) correctly describes the objective spatial records that historians mistakenly assume these maps to be.

Key Concept

Multi-Blank Text Completion: Cross-Sentence Contextual Coherence

Estimated Time:1m 30s

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