All practice questions

Question 1941Question →

If $a$ , $b$ , and $c$ are non-zero integers such that $a^3 b^2 c < 0$ , $a + b$ is even, and $b + c$ is odd, which of the following expressions MUST be a negative even integer?

a c (b^2 + 1)

a b c

(a + c) b^2

a^2 c (b + 1)

a c + b

Show answer & explanation

Answer: $a c (b^2 + 1)$

Answer

The expression

a c (b^2 + 1)

must be a negative even integer.

The expression

a c (b^2 + 1)

is guaranteed to be negative because

a c < 0

(derived from

a^3 b^2 c < 0

) and

b^2 + 1 > 0

for all non-zero integers

b

. It is guaranteed to be even because

a

and

c

have opposite parity, meaning one of them must be even, rendering

a c

(and thus any integer multiple of

a c

) even.

Step-by-Step Solution

1

Analyze the sign constraint

a^3 b^2 c < 0

.

a c < 0

, which means

a

and

c

have opposite signs.

Since

b \neq 0

,

b^2

is strictly positive. Dividing

a^3 b^2 c < 0

by

b^2

yields

a^3 c < 0

. Since

a^3

has the same sign as

a

,

a c < 0

.

2

Analyze the parity constraints

a + b

is even and

b + c

is odd.

a

and

c

have opposite parity (one is even, the other is odd).

If

a + b

is even,

a

and

b

share the same parity. If

b + c

is odd,

b

and

c

have opposite parity. Substituting the parity of

a

for

b

shows that

a

and

c

must have opposite parity.

3

Determine the sign and parity of

a c (b^2 + 1)

.

a c (b^2 + 1)

is strictly negative and even.

Since

a

and

c

have opposite parity, at least one of them is even, making the product

a c

an even integer. Since

a c < 0

and

b^2 + 1 \ge 1 > 0

, the product of negative

a c

and positive

(b^2 + 1)

is negative and even.

Key Concept

Combining sign rules (

a c < 0

) with even/odd parity logic across multiple variables.

Question 1942Question →

Three water pumps, $P_1$ , $P_2$ , and $P_3$ , operate at constant individual rates to fill a large reservoir. The ratio of the pumping rate of $P_1$ to that of $P_2$ is $3 : 4$ , and the ratio of the pumping rate of $P_2$ to that of $P_3$ is $3 : 5$ .

At 8:00 AM, all three pumps begin filling an empty reservoir together. At 10:00 AM, pump $P_1$ shuts down, while $P_2$ and $P_3$ continue operating at their original rates. At 11:00 AM, the operating rate of $P_2$ is decreased by $25\%$ , and the operating rate of $P_3$ is increased by $25\%$ . The two remaining pumps continue at these adjusted rates until the reservoir is completely full at 1:00 PM.

If pump $P_3$ were to fill the empty reservoir working alone at its original constant rate, how many hours would it take?

Show answer & explanation

Answer: 9.1

Answer

It would take pump

P_3

exactly

9.1

hours (or

9.1

when entered numerically) to fill the empty reservoir alone at its original constant rate.

By unifying the given ratios

r_1 : r_2 = 3 : 4

and

r_2 : r_3 = 3 : 5

, we obtain the relative rates

r_1 = 9k

,

r_2 = 12k

, and

r_3 = 20k

. Summing the work done across the three intervals (8–10 AM at rate

41k

, 10–11 AM at rate

32k

, and 11 AM–1 PM at rate

34k

) yields a total capacity of

182k

. Dividing total work

182k

by

P_3

's original rate of

20k

gives

9.1

hours.

Step-by-Step Solution

1

Unify the individual pumping rate ratios into a single ratio

r_1 : r_2 : r_3

.

r_1 = 9k

,

r_2 = 12k

,

r_3 = 20k

Aligning the ratio of

P_1:P_2 = 3:4 = 9:12

and

P_2:P_3 = 3:5 = 12:20

establishes a common scale factor

k

.

2

Calculate the work completed in each of the three time intervals.

W_1 = 82k

,

W_2 = 32k

,

W_3 = 68k

Multiply the duration of each interval by the sum of active rates during that period.

3

Sum the total work completed to get total capacity and divide by

P_3

's original rate.

Total capacity

= 182k

; Time

= \frac{182k}{20k} = 9.1

hours

The total work equals the sum of work done across all three phases, and time is work divided by rate.

Key Concept

Multi-stage work rates, compound ratio unification, and percentage rate adjustments

Estimated Time:3m 0s

Question 1943Question →

Read the passage below and identify the contextual meaning of the highlighted word as used by the author.

Fill in the blanks below

In mid-twentieth-century urban planning, theorists frequently invoked the concept of urban fabric to describe the spatial layout of historic districts. Rather than referring to woven textiles, the term was employed to denote the physical structure and spatial continuity of buildings, streets, and public spaces that collectively defined a neighborhood. When modernizers proposed carving wide thoroughfares through old city centers, critics warned that such interventions would sever this fabric, disrupting long-established community networks and destroying architectural cohesion.

In the context of the passage, the word fabric most nearly means .

Show answer & explanation

Answer

In this context, 'fabric' refers to the structural framework or physical composition of an urban area.

The correct answer accurately reflects the passage's explicit definition of 'fabric' as the physical structure, spatial layout, and built environment of a city district.

Step-by-Step Solution

1

Analyze the passage context surrounding the target word 'fabric'.

The text explicitly contrasts 'woven textiles' with the intended definition: 'the physical structure and spatial continuity of buildings, streets, and public spaces'.

The author provides a direct clue by defining what the term denotes in urban planning versus its literal meaning.

2

Synthesize the passage clues to determine the precise contextual definition.

The word refers to the physical composition, spatial layout, or structural framework of the built environment.

The surrounding sentences emphasize architectural cohesion and spatial arrangement rather than woven cloth or figurative social harmony alone.

Key Concept

Passage-Based Word in Context Analysis

Estimated Time:1m 15s

Question 1944Question →

Let $a$ and $b$ be non-zero real numbers such that $|a + b| = 3|a - b|$ . What is the value of $\left|\frac{a^2 + b^2}{ab}\right|$ ?

\frac{3}{2}

2

\frac{5}{2}

3

\frac{10}{3}

Show answer & explanation

Answer: $\frac{5}{2}$

Answer

The value of

\left|\frac{a^2 + b^2}{ab}\right|

is

\frac{5}{2}

.

Squaring both sides of the given equation

|a + b| = 3|a - b|

eliminates the absolute values to give

(a + b)^2 = 9(a - b)^2

. Expanding and simplifying yields

2a^2 - 5ab + 2b^2 = 0

, which factors as

(2a - b)(a - 2b) = 0

. Thus,

b = 2a

or

a = 2b

. In either case, substituting into

\left|\frac{a^2 + b^2}{ab}\right|

reduces the expression to

\frac{5}{2}

.

Step-by-Step Solution

1

Square both sides of the absolute value equality

(a + b)^2 = 9(a - b)^2

Since both sides of

|a + b| = 3|a - b|

are non-negative real numbers, squaring preserves equality and eliminates the absolute value signs.

2

Expand both algebraic expressions and collect like terms

a^2 + 2ab + b^2 = 9a^2 - 18ab + 9b^2 \implies 8a^2 - 20ab + 8b^2 = 0 \implies 2a^2 - 5ab + 2b^2 = 0

Expanding the binomial squares allows combining like terms to solve for the relationship between

a

and

b

.

3

Factor the quadratic expression in terms of

a

and

b

(2a - b)(a - 2b) = 0 \implies b = 2a \text{ or } a = 2b

Factoring determines the exact proportional relationship between

a

and

b

.

4

Substitute the relation into the target expression

\left|\frac{a^2 + (2a)^2}{a(2a)}\right| = \left|\frac{5a^2}{2a^2}\right| = \frac{5}{2}

Substituting

b = 2a

allows

a^2

to cancel completely, leaving a constant numerical value.

Key Concept

Properties of Real Numbers and Absolute Value Equations

Estimated Time:2m 0s

Question 1945Question →

A non-profit foundation distributes a total grant of $\$108,000$ among three research teams: Team A, Team B, and Team C. Team B receives $\$4,000$ more than Team A. Team C receives twice as much as the combined amount received by Team A and Team B. What amount does Team B receive?

\$16,000

\$20,000

\$24,000

\$28,000

\$72,000

Show answer & explanation

Answer: $\$20,000$

Answer

Team B receives

\$20,000

.

Defining Team A's share as

x

gives Team B a share of

x + 4,000

and Team C a share of

2(x + x + 4,000) = 4x + 8,000

. Combining these yields

6x + 12,000 = 108,000

, which solves to

x = 16,000

. Therefore, Team B receives

16,000 + 4,000 = \$20,000

.

Step-by-Step Solution

1

Define variables for each team's share in terms of a single unknown variable.

Let

x

be the amount received by Team A in dollars. Then Team B receives

x + 4,000

, and Team C receives

2 \cdot [x + (x + 4,000)] = 2(2x + 4,000) = 4x + 8,000

.

Expressing all quantities in terms of

x

reduces the problem to a linear equation in one variable.

2

Set up the linear equation representing the total grant amount.

x + (x + 4,000) + (4x + 8,000) = 108,000 \implies 6x + 12,000 = 108,000

.

The sum of the individual shares must equal the total grant of

\$108,000

.

3

Solve the linear equation for

x

.

6x = 96,000 \implies x = 16,000

.

Subtract

12,000

from both sides and divide by

6

.

4

Calculate Team B's share using the value of

x

.

Team B's share

= x + 4,000 = 16,000 + 4,000 = 20,000

.

The question specifically asks for Team B's share, not Team A's.

Key Concept

Formulating and solving a linear equation in one variable from a multi-step algebraic word problem.

Estimated Time:1m 30s

Question 1946Question →

In GRE Sentence Equivalence items, test-takers must distinguish between true contextual pairs and semantic traps. Match each category of vocabulary distractor on the left with its defining structural or semantic characteristic on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

False Synonym Trap matches with words sharing a broad thematic domain whose precise definitions produce non-equivalent meanings. Topically Related Distractor matches with a single option aligning with subject matter that lacks a semantic partner. Contextually Misaligned True Pair matches with a pair of equivalent words whose shared definition contradicts stem clues. Contextually Valid Synonym Pair matches with a pair of synonymous words that satisfy sentence logic to yield identical meaning.

Each distractor category corresponds strictly to its theoretical definition in GRE Verbal Reasoning methodology. False synonyms share a broad subject area but lack precise semantic equivalence. Topically related distractors fit the narrative subject but lack a matching partner choice. Contextually misaligned true pairs are genuine synonyms whose definition conflicts with sentence clues. Contextually valid pairs meet both synonymy and sentence logic requirements.

Step-by-Step Solution

1

Identify the distinguishing characteristic of false synonyms.

Recognize that false synonyms share topical overlap but differ in exact definition.

Words in the same domain (e.g., 'frugal' and 'miserly') are often mistaken for synonyms even though their connotations and definitions differ.

2

Analyze the nature of topically related distractors.

Connect topically related distractors to single options that fit the subject matter but lack a synonym pair.

GRE Sentence Equivalence strictly requires selecting two options that create equivalent sentences.

3

Evaluate true synonym pairs that fail contextually.

Match them with the characteristic of contradicting sentence logic or tone.

A pair can be perfectly synonymous yet incorrect if it goes against contrast or causal indicators in the stem.

4

Define the target correct answer pair.

Match contextually valid synonym pairs with options that fulfill sentence logic and produce identical meanings.

This represents the fundamental criterion for solving GRE Sentence Equivalence items.

Key Concept

Distinguishing between true contextual synonym pairs and semantic distractor traps in Sentence Equivalence.

Question 1947Question →

A high-precision optical sensor measures a time interval as $T = \frac{0.00064 \times (2.5 \times 10^8)}{1.6 \times 10^{-1}}$ nanoseconds. Which of the following values are equivalent to $T$ ? Select all such values.

Select all that apply

1.0 \times 10^6

100,000

\frac{4.0 \times 10^7}{40}

(2.5 \times 10^3) \times (4.0 \times 10^2)

1.0 \times 10^{-6}

Show answer & explanation

Answer: $1.0 \times 10^6$ ; $\frac{4.0 \times 10^7}{40}$ ; $(2.5 \times 10^3) \times (4.0 \times 10^2)$

Answer

The values equivalent to

T

are

1.0 \times 10^6

,

\frac{4.0 \times 10^7}{40}

, and

(2.5 \times 10^3) \times (4.0 \times 10^2)

.

Evaluating

T

gives

\frac{(6.4 \times 10^{-4}) \times (2.5 \times 10^8)}{1.6 \times 10^{-1}} = \frac{1.6 \times 10^5}{1.6 \times 10^{-1}} = 1.0 \times 10^6

. The expression representing

1.0 \times 10^6

directly matches

T

. The quotient

\frac{4.0 \times 10^7}{40}

simplifies to

\frac{4.0 \times 10^7}{4.0 \times 10^1} = 1.0 \times 10^6

, which matches

T

. The product

(2.5 \times 10^3) \times (4.0 \times 10^2)

simplifies to

10.0 \times 10^5 = 1.0 \times 10^6

, which also matches

T

.

Step-by-Step Solution

1

Convert the decimal

0.00064

into scientific notation

0.00064 = 6.4 \times 10^{-4}

Expressing all terms in scientific notation simplifies exponent operations.

2

Simplify the numerator of the expression for

T

(6.4 \times 10^{-4}) \times (2.5 \times 10^8) = (6.4 \times 2.5) \times 10^{-4 + 8} = 16.0 \times 10^4 = 1.6 \times 10^5

Multiply coefficients directly and add exponents for product of powers with equal base.

3

Divide the numerator by the denominator

1.6 \times 10^{-1}

T = \frac{1.6 \times 10^5}{1.6 \times 10^{-1}} = \left(\frac{1.6}{1.6}\right) \times 10^{5 - (-1)} = 1.0 \times 10^6 = 1,000,000

Subtract the denominator exponent from the numerator exponent:

5 - (-1) = 6

.

4

Evaluate the choices to check equivalence to

1.0 \times 10^6

1.0 \times 10^6

,

\frac{4.0 \times 10^7}{40} = 1.0 \times 10^6

, and

(2.5 \times 10^3) \times (4.0 \times 10^2) = 10.0 \times 10^5 = 1.0 \times 10^6

are all equal to

T

.

Matching each simplified expression to the calculated value of

T

determines the correct options.

Key Concept

Operations with Decimals and Exponents in Scientific Notation

Question 1948Question →

Two positive integers $x$ and $y$ have a greatest common divisor (GCD) of $12$ and a least common multiple (LCM) of $360$ . Which of the following could be the value of $x$ ? Indicate all such values.

Select all that apply

24

48

60

90

120

Show answer & explanation

Answer: $24$ ; $60$ ; $120$

Answer

The possible values for

x

are

24

,

60

, and

120

.

Any valid value of

x

must be a multiple of

\gcd(x, y) = 12

and a divisor of

\text{lcm}(x, y) = 360

. Expressing

12

as

2^2 \cdot 3^1

and

360

as

2^3 \cdot 3^2 \cdot 5^1

,

x

must be of the form

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

with

2 \leq a \leq 3

,

1 \leq b \leq 2

, and

0 \leq c \leq 1

. The values

24

,

60

, and

120

meet all exponent constraints.

Step-by-Step Solution

1

Find the prime factorizations of the given GCD and LCM.

\gcd(x, y) = 12 = 2^2 \cdot 3^1 \cdot 5^0

and

\text{lcm}(x, y) = 360 = 2^3 \cdot 3^2 \cdot 5^1

.

Prime factorization allows analysis of exponent constraints for each prime factor.

2

Determine the constraints on any valid integer

x

.

Any valid value of

x

must be a multiple of

12

and a factor of

360

. Specifically,

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

where

2 \leq a \leq 3

,

1 \leq b \leq 2

, and

0 \leq c \leq 1

.

The GCD defines the minimum exponent for each prime factor, and the LCM defines the maximum exponent.

3

Test each provided option against the prime exponent bounds.

24 = 2^3 \cdot 3^1

satisfies the bounds (

a=3, b=1, c=0

).

48 = 2^4 \cdot 3^1

violates

a \leq 3

.

60 = 2^2 \cdot 3^1 \cdot 5^1

satisfies the bounds (

a=2, b=1, c=1

).

90 = 2^1 \cdot 3^2 \cdot 5^1

violates

a \geq 2

.

120 = 2^3 \cdot 3^1 \cdot 5^1

satisfies the bounds (

a=3, b=1, c=1

).

Options satisfying all exponent inequalities are valid possible values of

x

.

Key Concept

Prime exponent properties of GCD and LCM

Question 1949Question →

Passage:

In November 1974, paleoanthropologist Donald Johanson and graduate student Tom Gray identified a partial hominin skeleton at the site of Hadar in the Awash Valley of Ethiopia. Cataloged as AL 288-1 and popularly dubbed 'Lucy,' the specimen comprised forty percent of a single female individual's skeletal structure, making it remarkably complete for an early hominin fossil. Geological analysis of the volcanic ash layers surrounding the stratum dated the specimen to approximately 3.2 million years ago. Anatomical examination of the pelvic structure and knee joint provided definitive evidence of habitual bipedalism, resolving a long-standing debate regarding whether upright locomotion preceded cerebral expansion in hominin evolution. Despite possessing a small cranial capacity comparable to modern chimpanzees, AL 288-1 exhibited clear postcranial adaptations for bipedal walking, demonstrating that erect posture was established well prior to the marked brain enlargement observed in later species of the genus Homo.

According to the passage, which of the following was directly established by the anatomical examination of AL 288-1's pelvic structure and knee joint?

It proved that cerebral expansion occurred concurrently with postcranial adaptations for upright walking.

It established that AL 288-1 was the earliest hominin species to manufacture stone tools in the Awash Valley.

It provided definitive evidence that habitual bipedalism preceded substantial cerebral expansion in hominin evolution.

It demonstrated that AL 288-1 possessed a cranial capacity significantly larger than that of modern chimpanzees.

It served as the primary scientific method for dating the volcanic ash strata surrounding the fossil stratum.

Show answer & explanation

Answer: It provided definitive evidence that habitual bipedalism preceded substantial cerebral expansion in hominin evolution.

Answer

The anatomical examination of AL 288-1's pelvic structure and knee joint provided definitive evidence that habitual bipedalism preceded substantial cerebral expansion in hominin evolution.

The passage explicitly states that anatomical examination of AL 288-1's pelvic structure and knee joint provided definitive evidence of habitual bipedalism and resolved a debate by proving that upright locomotion preceded cerebral expansion. The correct choice directly paraphrases this fact.

Step-by-Step Solution

1

Locate the key terms in the passage stem

Identify the sentence discussing 'pelvic structure and knee joint'.

Explicit detail retrieval requires matching the exact structural context within the passage.

2

Analyze the explicit factual claim associated with those anatomical features

The text states that these features 'provided definitive evidence of habitual bipedalism, resolving a long-standing debate regarding whether upright locomotion preceded cerebral expansion'.

Understanding the direct assertion ensures precise alignment with the correct option.

3

Evaluate the choices against the passage text to select the accurate paraphrase

The option stating that habitual bipedalism occurred prior to substantial brain enlargement directly matches 'upright locomotion preceded cerebral expansion'.

Eliminate choices that misstate timing, introduce outside details, or confuse geological dating with anatomical analysis.

Key Concept

Explicit Detail Retrieval

Question 1950Question →

Let $p$ , $q$ , and $r$ be integers such that $p < 0 < q < r$ . If $p$ is an odd integer, $q$ is an even integer, and $r$ is an odd integer, which of the following expressions must be negative? Select all that apply.

Select all that apply

p^q(q - r)

p(p - q)^r

(p - r)(-p)^r

p^r(r - p)^q

(-p)^q(q - p)

Show answer & explanation

Answer: $p^q(q - r)$ ; $(p - r)(-p)^r$ ; $p^r(r - p)^q$

Answer

The expressions that must be negative are

p^q(q - r)

,

(p - r)(-p)^r

, and

p^r(r - p)^q

.

Expressions

p^q(q - r)

,

(p - r)(-p)^r

, and

p^r(r - p)^q

evaluate to the product of a positive factor and a negative factor in every case, making their values strictly negative.

Step-by-Step Solution

1

Analyze the given signs and parities of variables

p < 0

(negative, odd),

q > 0

(positive, even),

r > 0

(positive, odd), with

q < r

.

Establishing the domain and sign/parity properties of each variable is essential before evaluating exponential and subtractive terms.

2

Evaluate the sign of

p^q(q - r)

p^q > 0

because an even exponent yields a positive result for non-zero bases. Since

q < r

,

(q - r) < 0

. Thus,

\text{positive} \times \text{negative} = \text{negative}

.

Demonstrates that this expression is guaranteed to be negative.

3

Evaluate the sign of

p(p - q)^r

p < 0

.

p - q < 0

, and raising a negative number to an odd exponent

r

gives a negative result. Thus,

p(p - q)^r = \text{negative} \times \text{negative} = \text{positive}

.

Shows that this expression is positive, so it cannot be negative.

4

Evaluate the sign of

(p - r)(-p)^r

p - r < 0

because subtracting a positive number from a negative number is negative.

-p > 0

, so

(-p)^r > 0

. Thus,

\text{negative} \times \text{positive} = \text{negative}

.

Demonstrates that this expression is guaranteed to be negative.

5

Evaluate the sign of

p^r(r - p)^q

p^r < 0

because a negative number raised to an odd exponent is negative.

r - p > 0

, so

(r - p)^q > 0

. Thus,

\text{negative} \times \text{positive} = \text{negative}

.

Demonstrates that this expression is guaranteed to be negative.

6

Evaluate the sign of

(-p)^q(q - p)

-p > 0 \implies (-p)^q > 0

.

q - p > 0

. Thus,

\text{positive} \times \text{positive} = \text{positive}

.

Shows that this expression is always positive.

Key Concept

Sign rules for bases raised to even vs. odd powers, and order of operations with signed quantities.

Question 1951Question →

In Year 1, a non-profit organization received all of its funding from two sources: private donations and government grants. Private donations accounted for $60\%$ of the total funding, and government grants accounted for the remaining $40\%$ . In Year 2, private donations increased by $25\%$ compared to Year 1, while government grants decreased by $15\%$ compared to Year 1. Which of the following statements must be true? Select all such statements.

Select all that apply

The total funding received by the organization in Year 2 was

9\%

greater than the total funding received in Year 1.

In Year 2, private donations accounted for more than

68\%

of the total funding received in Year 2.

In Year 2, government grants accounted for

34\%

of the total funding received in Year 2.

In Year 2, private donations accounted for

75\%

of the total funding received in Year 2.

The ratio of private donations to government grants in Year 2 was

25\%

greater than the ratio of private donations to government grants in Year 1.

Show answer & explanation

Answer: The total funding received by the organization in Year 2 was $9\%$ greater than the total funding received in Year 1.; In Year 2, private donations accounted for more than $68\%$ of the total funding received in Year 2.

Answer

The correct statements are that the total funding in Year 2 was

9\%

greater than in Year 1, and that private donations accounted for more than

68\%

of the total funding in Year 2.

Total funding in Year 2 is

0.60(1.25) + 0.40(0.85) = 0.75 + 0.34 = 1.09

times the Year 1 total, which represents a

9\%

increase. Furthermore, private donations in Year 2 represent

\frac{0.75}{1.09} \approx 68.81\%

of the Year 2 total funding, which is greater than

68\%

.

Step-by-Step Solution

1

Define variables for Year 1 funding components relative to total Year 1 funding (

T_1

).

Private donations in Year 1:

P_1 = 0.60 T_1

. Government grants in Year 1:

G_1 = 0.40 T_1

.

Establishing Year 1 amounts in terms of total funding

T_1

provides the baseline for percentage changes.

2

Calculate Year 2 funding amounts after applying the respective percent changes.

P_2 = 1.25 \times 0.60 T_1 = 0.75 T_1

.

G_2 = 0.85 \times 0.40 T_1 = 0.34 T_1

.

Increasing private donations by

25\%

multiplies

P_1

by

1.25

; decreasing government grants by

15\%

multiplies

G_1

by

0.85

.

3

Compute total Year 2 funding (

T_2

) and compare it with

T_1

.

T_2 = P_2 + G_2 = 0.75 T_1 + 0.34 T_1 = 1.09 T_1

. Net change is a

9\%

increase.

Adding the components yields total Year 2 funding, confirming that

T_2

is

1.09

times

T_1

.

4

Calculate component shares as percentages of total Year 2 funding (

T_2

).

Share of private donations in Year 2 =

\frac{0.75 T_1}{1.09 T_1} \approx 68.81\%

. Share of government grants in Year 2 =

\frac{0.34 T_1}{1.09 T_1} \approx 31.19\%

.

Percentage share of a total requires dividing each component by the new total funding (

1.09 T_1

), not the original baseline (

T_1

).

Key Concept

Weighted percentage change and percentage base shift

Question 1952Question →

If $x$ satisfies the equation $\frac{5(x - 2)}{3} - \frac{2x - 1}{4} = \frac{x + 7}{6}$ , which of the following statements must be true? Select all that apply.

Select all that apply

x > 4

4x

is an integer

x

is an integer

2x + 1 > 10

3x - 2 = 10.75

Show answer & explanation

Answer: $x > 4$ ; $4x$ is an integer; $3x - 2 = 10.75$

Answer

The correct statements are that

x > 4

,

4x

is an integer, and

3x - 2 = 10.75

.

Solving the linear equation yields

x = 4.25

or

\frac{17}{4}

. Using this value:

4.25 > 4

is true;

4 \times \frac{17}{4} = 17

is an integer, so that statement is true; and

3(4.25) - 2 = 10.75

is also true.

Step-by-Step Solution

1

Find a common denominator to clear fractions in the given equation.

The least common multiple of denominators

3

,

4

, and

6

is

12

.

Clearing denominators simplifies multi-term fractional equations into standard linear form.

2

Multiply every term of the equation by

12

and expand numerators.

4 \cdot 5(x - 2) - 3 \cdot (2x - 1) = 2 \cdot (x + 7) \implies 20(x - 2) - 3(2x - 1) = 2(x + 7)

.

Multiplying each term by

12

eliminates all fraction bars.

3

Distribute the constants through the parentheses on both sides.

20x - 40 - 6x + 3 = 2x + 14

.

Applying the distributive property correctly accounts for negative signs across parentheses.

4

Combine like terms on the left side and solve for

x

.

14x - 37 = 2x + 14 \implies 12x = 51 \implies x = \frac{51}{12} = \frac{17}{4} = 4.25

.

Isolating

x

gives the exact rational value of the solution.

5

Evaluate each provided statement using

x = 4.25

.

The statement

x > 4

is true (

4.25 > 4

). The statement

4x

is an integer is true (

4 \times 4.25 = 17

). The statement

3x - 2 = 10.75

is true (

3 \times 4.25 - 2 = 10.75

). The other two statements are false.

Direct substitution verifies which conditions hold.

Key Concept

Solving multi-step linear equations in one variable by clearing denominators and combining variable terms.

Estimated Time:1m 30s

Question 1953Question →

At a pharmaceutical manufacturing facility, a liquid solution undergoes a three-stage purification process to remove a specific chemical compound. Stage I removes $20\%$ of the compound present at the start of Stage I. Stage II removes $25\%$ of the compound present at the start of Stage II. Stage III removes $30\%$ of the compound present at the start of Stage III. If $16.8\text{ grams}$ of the compound remain in the solution after Stage III, how many grams of the compound were present in the solution immediately before Stage I?

28.0\text{ grams}

33.6\text{ grams}

40.0\text{ grams}

56.0\text{ grams}

67.2\text{ grams}

Show answer & explanation

Answer: $40.0\text{ grams}$

Answer

40.0 grams

To find the initial quantity, we determine the compound fraction remaining after each stage. Stage I leaves

80\%

, Stage II leaves

75\%

, and Stage III leaves

70\%

. The overall fraction remaining after all three stages is

0.80 \times 0.75 \times 0.70 = 0.42

. Given that

0.42

of the original amount equals

16.8\text{ grams}

, dividing

16.8

by

0.42

gives the initial mass of

40.0\text{ grams}

.

Step-by-Step Solution

1

Determine the remaining fraction of the compound after each individual stage.

Stage I leaves

1 - 0.20 = 0.80

of its input; Stage II leaves

1 - 0.25 = 0.75

of its input; Stage III leaves

1 - 0.30 = 0.70

of its input.

Percentage decreases must be represented as multiplicative factors of the base amount entering each stage.

2

Calculate the combined multi-stage remaining multiplier.

Combined multiplier

= 0.80 \times 0.75 \times 0.70 = 0.42

(or

42\%

).

Successive percentage reductions compound multiplicatively, not additively.

3

Set up an equation relating the initial mass

X

to the final remaining mass.

0.42 X = 16.8\text{ grams}

.

The final mass is equal to the initial mass multiplied by the overall remaining decimal fraction.

4

Solve for the initial mass

X

.

X = \frac{16.8}{0.42} = 40.0\text{ grams}

.

Dividing the final amount by

0.42

yields the exact starting quantity.

Key Concept

Successive Percent Reductions and Base Shifts

Alternative Method

Work backwards from the final amount stage by stage: Before Stage III, mass

= 16.8 \div 0.70 = 24.0\text{ g}

. Before Stage II, mass

= 24.0 \div 0.75 = 32.0\text{ g}

. Before Stage I, mass

= 32.0 \div 0.80 = 40.0\text{ g}

.

Estimated Time:2m 0s

Question 1954Question →

A sports scientist records the recovery times, in minutes, for 5 subjects after an intense workout. Four of the times are $44, 16, 52,$ and $32$ minutes, and the fifth time is $x$ minutes. The 5 times are listed in no particular order. If $32 < x < 44$ and the arithmetic mean of all 5 recovery times is equal to the median of the 5 recovery times, what is the value of $x$ ?

26

29

34

36

48

Show answer & explanation

Answer: 36

Answer

The value of

x

is 36.

Because

32 < x < 44

, the 5 numbers listed in ascending order are

16, 32, x, 44, 52

. The median of these 5 numbers is the middle value,

x

. Setting the mean

\frac{16 + 32 + x + 44 + 52}{5} = \frac{144 + x}{5}

equal to the median

x

gives

\frac{144 + x}{5} = x

, which simplifies to

4x = 144

and yields

x = 36

.

Step-by-Step Solution

1

Order the dataset in ascending numerical order.

The five data values in sorted order are

16, 32, x, 44, 52

because it is given that

32 < x < 44

.

Finding the median of a dataset requires arranging all elements in ascending or descending order first.

2

Identify the median of the 5 values.

The median is the 3rd element in the sorted 5-element list, which is

x

.

For an odd number of items (

n = 5

), the median is the middle value at position

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

.

3

Express the arithmetic mean in terms of

x

.

\text{Mean} = \frac{16 + 32 + x + 44 + 52}{5} = \frac{144 + x}{5}$.

The arithmetic mean is defined as the sum of all values divided by the total count of values (

n = 5

).

4

Set the mean equal to the median and solve for

x

.

\frac{144 + x}{5} = x \implies 144 + x = 5x \implies 4x = 144 \implies x = 36$.

The problem states that the arithmetic mean equals the median.

Key Concept

Measures of Central Tendency (Mean, Median, Mode)

Estimated Time:1m 45s

Question 1955Question →

Read the passage below:

While early urban planners regarded the city's informal transit networks as chaotic and (i)________, recent spatial analysis demonstrates that these organic routes possess an underlying (ii)________, operating with an efficiency that formal municipal systems often struggle to replicate.

Which of the following combinations of words best completes the text to maintain logical coherence and structural dependency?

(i) haphazard, (ii) coherence

(i) haphazard, (ii) redundancy

(i) ubiquitous, (ii) coherence

(i) methodical, (ii) disarray

(i) labyrinthine, (ii) stagnation

Show answer & explanation

Answer: (i) haphazard, (ii) coherence

Answer

The combination of '(i) haphazard' and '(ii) coherence' best completes the text.

The sentence employs the concessionary structural signal 'While' to contrast early perceptions of urban transit networks with recent empirical findings. Blank (i) must mirror 'chaotic' in describing an apparently disordered system, making 'haphazard' appropriate. Blank (ii) must describe the true, structured nature of the system that enables high efficiency, making 'coherence' the exact logical fit. Together, '(i) haphazard, (ii) coherence' fulfills the inter-blank dependency requirements.

Step-by-Step Solution

1

Analyze the structural contrast and parallel clues in the first clause.

The word 'While' sets up a concession/contrast between early impressions and recent findings. The phrase 'chaotic and (i)________' requires Blank (i) to have a negative valence similar to 'chaotic'.

Conjunctions connecting adjectives in a paired description maintain consistent tone, while the overarching transition signals a reversal.

2

Analyze the logical requirements for the second blank.

The second clause states that recent analysis reveals an opposite, positive quality that explains how routes operate with high efficiency. Blank (ii) must denote systematic order or clarity, such as 'coherence'.

The sentence specifies an unexpected functional benefit ('efficiency') that counteracts the initial impression of chaos.

3

Verify inter-blank dependency across all candidate combinations.

Only the pairing of '(i) haphazard' and '(ii) coherence' maintains the required negative-to-positive conceptual shift across both blanks simultaneously.

Selecting choices in isolation leads to semantic mismatch with surrounding clause qualifiers.

Key Concept

Multi-Blank Dependency Tracking

Question 1956Question →

In plant neurobiology, early claims regarding electrical signaling in root apexes were initially greeted with overt hostility by mainstream botanists; nevertheless, meticulous documentation of action potentials has compelled scholars to harbor theoretical frameworks once dismissed as speculative fantasy. Far from indicating an uncritical endorsement of botanical consciousness, this shift reflects a methodical willingness to evaluate provisional models until empirical evidence dictates otherwise.

Based on the structural clues and transition signals in the passage above, which of the following best expresses the meaning of the word "harbor" as it is used in the text?

shelter

conceal

entertain

validate

suppress

Show answer & explanation

Answer: entertain

Answer

In this context, the word 'harbor' means to entertain or hold a concept or hypothesis in mind for consideration.

The contrast signal 'nevertheless' indicates a shift from the initial 'overt hostility' of botanists to a willingness to consider previously rejected ideas. The subsequent sentence reinforces this by defining the action as a 'methodical willingness to evaluate provisional models'. Therefore, 'harbor' in this context means to entertain or hold a theory in mind for evaluation.

Step-by-Step Solution

1

Analyze the structural signals surrounding the target word.

The transition signal 'nevertheless' sets up a contrast between early 'overt hostility' toward plant neurobiology claims and the later willingness of scholars to 'harbor theoretical frameworks'.

Contrast signals indicate that the second clause shifts away from initial rejection toward openness.

2

Examine continuation clues in the subsequent sentence.

The phrase 'Far from indicating an uncritical endorsement... this shift reflects a methodical willingness to evaluate provisional models' clarifies the exact nature of their action.

The elaboration explains that scholars are not confirming the theory as absolute fact, but rather giving it open consideration.

3

Match the contextual definition with the option choices.

The word 'entertain' directly captures the sense of holding a hypothesis in mind or evaluating a model provisionally.

'Entertain' fits both the structural reversal signaled by 'nevertheless' and the thematic restatement in the second sentence.

Key Concept

Deciphering Meaning via Structural Clues and Contrast/Continuation Signals

Question 1957Question →

If $A = 0.00048 \times 10^{-3}$ and $B = 1.2 \times 10^{-5}$ , what is the value of $\frac{A + B}{4 \times 10^{-8}}$ expressed in scientific notation?

3.12 \times 10^2

1.5 \times 10^3

4.2 \times 10^2

3.12 \times 10^{-4}

3.12 \times 10^1

Show answer & explanation

Answer: $3.12 \times 10^2$

Answer

3.12 \times 10^2

Converting

A = 0.00048 \times 10^{-3}

to powers of 10 gives

4.8 \times 10^{-7}

, which equals

0.048 \times 10^{-5}

. Adding

B = 1.2 \times 10^{-5}

yields

(0.048 + 1.2) \times 10^{-5} = 1.248 \times 10^{-5}

. Dividing by

4 \times 10^{-8}

gives

\frac{1.248}{4} \times 10^{-5 - (-8)} = 0.312 \times 10^3 = 3.12 \times 10^2

.

Step-by-Step Solution

1

Express

A

in standard scientific notation and then match its exponent to

B

's exponent.

A = 0.00048 \times 10^{-3} = 4.8 \times 10^{-7} = 0.048 \times 10^{-5}

To perform addition between numbers in scientific notation, their exponents must be equal.

2

Add

A

and

B

.

A + B = 0.048 \times 10^{-5} + 1.2 \times 10^{-5} = 1.248 \times 10^{-5}

Combine the coefficients once the powers of 10 match.

3

Divide

A + B

by

4 \times 10^{-8}

and express the final result in scientific notation.

\frac{1.248 \times 10^{-5}}{4 \times 10^{-8}} = \left(\frac{1.248}{4}\right) \times 10^{-5 - (-8)} = 0.312 \times 10^3 = 3.12 \times 10^2

Divide the coefficients and subtract the exponent in the denominator from the exponent in the numerator.

Key Concept

Decimals, Place Value, and Operations in Scientific Notation

Question 1958Question →

Fill in the blanks in the text below to complete the sentence logically based on inter-blank structural clues.

Fill in the blanks below

Rather than regarding the participant's initial hesitation as evidence of processing, the principal investigators asserted that such pauses were in fact strategic deliberation, serving to enhance analytical accuracy rather than reflecting cognitive fatigue.

Show answer & explanation

Answer

The sentence is completed by inserting 'deficient' (or accepted variants 'impaired' / 'flawed') into the first blank and 'integral to' (or accepted variants 'indicative of' / 'conducive to') into the second blank.

The sentence relies on a contrast framework initiated by 'Rather than regarding'. The concluding clause confirms that hesitation has a productive function ('enhance analytical accuracy'), which means the second blank must connect hesitation positively to deliberate thought ('integral to'). Consequently, the initial rejected interpretation in the first blank must represent a negative assessment of cognitive function ('deficient').

Step-by-Step Solution

1

Analyze the structural contrast pivot at the start of the sentence.

The initial phrase 'Rather than regarding X as Y' sets up an opposition between a rejected view in the first clause and the accepted view in the main clause.

Tracking contrast clues determines the required semantic polarity for both blanks.

2

Determine the orientation of the second blank using the concluding modifying clause.

The final phrase 'serving to enhance analytical accuracy rather than reflecting cognitive fatigue' indicates that the pauses play a positive, constructive role in thinking.

The second blank must express a supportive relationship to 'strategic deliberation', leading to choices like 'integral to' or 'indicative of'.

3

Derive the required word for the first blank based on the contrast established.

Since the second blank establishes hesitation as valuable, the opening clause must describe the rejected negative view (that hesitation stems from poor thinking).

The contrast requires a word with negative cognitive polarity such as 'deficient' for the first blank.

Key Concept

Multi-Blank Dependency Tracking in Text Completion

Question 1959Question →

In 2024, a municipal water treatment plant processed water using two filtration systems: System A and System B. System A processed $60\%$ of the plant's total water volume, and System B processed the remaining $40\%$ . In 2025, the volume of water processed by System A increased by $20\%$ , while the volume of water processed by System B decreased by $15\%$ . What was the net percentage change in the total volume of water processed by the plant from 2024 to 2025?

An increase of

6\%

An increase of

5\%

A decrease of

1\%

An increase of

2.5\%

An increase of

3.5\%

Show answer & explanation

Answer: An increase of $6\%$

Answer

An increase of

6\%

To find the net percentage change, calculate the weighted contribution of each component change relative to the initial total. Assuming a total volume of

100

units in 2024, System A processed

60

units and System B processed

40

units. In 2025, System A processed

60 \times 1.20 = 72

units, and System B processed

40 \times 0.85 = 34

units. The new total volume is

72 + 34 = 106

units. The net change from

100

to

106

represents an increase of

6\%

.

Step-by-Step Solution

1

Assume a convenient baseline total volume for 2024.

Let the total volume in 2024 be

100

units. System A processes

60

units and System B processes

40

units.

Choosing

100

simplifies percentage calculations.

2

Calculate the volume processed by each system in 2025.

System A:

60 \times (1 + 0.20) = 72

units.
System B:

40 \times (1 - 0.15) = 34

units.

Apply the respective percentage increase and decrease to each system's 2024 volume.

3

Calculate the total volume processed in 2025 and the net percentage change.

Total 2025 volume =

72 + 34 = 106

units.
Net change =

106 - 100 = +6

units, which is an increase of

\frac{6}{100} \times 100\% = 6\%

.

Compare the new total volume to the original baseline total volume.

Key Concept

Weighted Percent Change

Estimated Time:1m 30s

Question 1960Question →

An isosceles trapezoid has parallel base lengths of $10$ and $26$ , and an altitude of $15$ . A line segment connects the midpoints of the two non-parallel legs, dividing the figure into two smaller trapezoids. What is the area of the larger of these two smaller trapezoids?

Show answer & explanation

Answer: 165

Answer

165

The midsegment of a trapezoid connects the midpoints of the non-parallel legs, and its length is the average of the two parallel bases:

\frac{10 + 26}{2} = 18

. Because the line connects midpoints, it also bisects the altitude, making the height of each smaller trapezoid equal to

\frac{15}{2} = 7.5

. The larger of the two resulting trapezoids has bases of lengths

18

and

26

. Using the trapezoid area formula

\text{Area} = \frac{b_1 + b_2}{2} \times h

, we obtain

\frac{18 + 26}{2} \times 7.5 = 22 \times 7.5 = 165

.

Step-by-Step Solution

1

Calculate the length of the midsegment connecting the midpoints of the non-parallel legs.

The midsegment length is

18

.

The midsegment of a trapezoid is parallel to the bases and its length equals the average of the two base lengths:

\frac{10 + 26}{2} = 18

.

2

Determine the height of the smaller subtrapezoid.

The height of the subtrapezoid is

7.5

.

The segment connecting the midpoints of the legs bisects the overall altitude of

15

, giving a height of

\frac{15}{2} = 7.5

for each subtrapezoid.

3

Compute the area of the larger subtrapezoid.

The area is

165

.

The larger subtrapezoid is bounded by the midsegment (length

18

) and the bottom base (length

26

). Applying the trapezoid area formula yields

\text{Area} = \frac{18 + 26}{2} \times 7.5 = 22 \times 7.5 = 165

.

Key Concept

Trapezoid Midsegment Theorem and Subdivided Area Calculation

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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