All practice questions

Question 1321Question →

An agricultural research station divides an experimental plot of land among three crops: Wheat, Rye, and Barley. Initially, Wheat occupies $\frac{4}{9}$ of the total plot area, Rye occupies $\frac{1}{3}$ of the total area, and Barley occupies the remaining area. First, $\frac{1}{3}$ of the Wheat area is converted to Rye. Next, $\frac{2}{5}$ of the new total Rye area is converted to Barley. Finally, $\frac{1}{4}$ of the new total Barley area is converted back to Wheat. If the final area occupied by Wheat is $126$ hectares, what was the total area, in hectares, of the experimental plot?

210

315

270

378

540

Show answer & explanation

Answer: 315

Answer

315 hectares

By tracking the dynamic changes to each crop's area step-by-step, the final Wheat area is

\frac{54}{135} = \frac{2}{5}

of the total land area. Setting

\frac{2}{5} T = 126

yields

T = 315

hectares.

Step-by-Step Solution

1

Determine initial fraction allocations for each crop

Wheat:

\frac{4}{9}

, Rye:

\frac{1}{3} = \frac{3}{9}

, Barley:

1 - \left(\frac{4}{9} + \frac{3}{9}\right) = \frac{2}{9}

The sum of all three crop areas must equal the total area (

1

).

2

Apply the first conversion (

\frac{1}{3}

of Wheat area converted to Rye)

Wheat transferred:

\frac{1}{3} \times \frac{4}{9} = \frac{4}{27}

. New Wheat:

\frac{4}{9} - \frac{4}{27} = \frac{8}{27}

. New Rye:

\frac{1}{3} + \frac{4}{27} = \frac{13}{27}

. Barley:

\frac{2}{9} = \frac{6}{27}

.

Wheat loses

\frac{1}{3}

of its current area, and Rye gains that exact amount.

3

Apply the second conversion (

\frac{2}{5}

of new Rye area converted to Barley)

Rye transferred:

\frac{2}{5} \times \frac{13}{27} = \frac{26}{135}

. New Rye:

\frac{13}{27} - \frac{26}{135} = \frac{39}{135}

. New Barley:

\frac{6}{27} + \frac{26}{135} = \frac{56}{135}

. Wheat remains

\frac{8}{27} = \frac{40}{135}

.

Rye loses

\frac{2}{5}

of its updated area, and Barley gains that exact amount.

4

Apply the third conversion (

\frac{1}{4}

of new Barley area converted to Wheat)

Barley transferred:

\frac{1}{4} \times \frac{56}{135} = \frac{14}{135}

. New Wheat:

\frac{40}{135} + \frac{14}{135} = \frac{54}{135} = \frac{2}{5}

.

Barley loses

\frac{1}{4}

of its updated area, and Wheat gains that exact amount.

5

Solve for the total plot area

T

\frac{2}{5} T = 126 \implies T = \frac{126 \times 5}{2} = 315

hectares.

Set the final calculated fraction of Wheat equal to the given area of

126

hectares.

Key Concept

Sequential fractional updates of non-disjoint sub-amounts

Estimated Time:2m 30s

Question 1322Question →

Fill in the blank in the passage below with the single word that best completes the sentence according to its secondary meaning.

Fill in the blanks below

Faced with severe supply shortages during the prolonged siege, the logistics officer realized that the garrison could endure the harsh winter only if he learned to their dwindling reserves of grain and fuel with rigorous frugality.

Show answer & explanation

Answer

husband

The correct response is 'husband'. In academic and literary English, 'husband' functions as a verb meaning to manage prudently or conserve resources economically. Context clues such as 'supply shortages', 'dwindling reserves', and 'rigorous frugality' directly mandate a word meaning to economize or save.

Step-by-Step Solution

1

Analyze contextual clues in the sentence

Identified clues such as 'severe supply shortages', 'dwindling reserves', and 'rigorous frugality'.

The surrounding context indicates a need to carefully manage, conserve, or use resources sparingly.

2

Identify the word whose secondary meaning fits this exact context

Selected 'husband' in its secondary verb form.

While its primary definition is a noun referring to a married man, as a transitive verb 'husband' means to manage economically or conserve resources prudently.

Key Concept

Secondary Meanings of Common Words

Question 1323Question →

In 1928, British bacteriologist Frederick Griffith conducted a landmark experiment investigating Streptococcus pneumoniae, the bacterium responsible for pneumonia. Griffith utilized two distinct strains: the virulent 'S' (smooth) strain, enclosed in a protective polysaccharide capsule that shielded it from the host's immune system, and the non-virulent 'R' (rough) strain, which lacked this capsule. When Griffith injected mice with heat-killed S-strain bacteria, the mice survived, confirming that dead bacteria alone were harmless. However, when heat-killed S-strain bacteria were combined with living R-strain bacteria and injected into mice, the animals succumbed to pneumonia, and living S-strain bacteria were subsequently isolated from their blood. Griffith concluded that a non-living 'transforming principle' had passed from the dead S-strain bacteria to the living R-strain bacteria, permanently endowing them with the ability to synthesize the protective capsule. Although Griffith did not identify the chemical identity of this transforming principle, his findings established that genetic traits could be horizontally transferred between bacterial strains.

According to the passage, which of the following statements regarding Griffith's 1928 experiment are explicitly supported by the text?

Select all that apply

Mice injected exclusively with heat-killed S-strain bacteria survived the procedure.

The protective polysaccharide capsule of the R-strain bacterium was chemically destroyed by heat prior to injection.

Griffith succeeded in isolating living S-strain bacteria from mice exposed to a mixture of live R-strain and heat-killed S-strain bacteria.

Show answer & explanation

Answer: Mice injected exclusively with heat-killed S-strain bacteria survived the procedure.; Griffith succeeded in isolating living S-strain bacteria from mice exposed to a mixture of live R-strain and heat-killed S-strain bacteria.

Answer

The statements confirming that mice injected exclusively with heat-killed S-strain bacteria survived and that living S-strain bacteria were isolated from mice injected with the mixture of live R-strain and heat-killed S-strain bacteria are both explicitly supported.

The passage provides direct factual support for two statements. First, it explicitly confirms that mice injected with heat-killed S-strain bacteria survived. Second, it explicitly states that after injecting the mixture of live R-strain and heat-killed S-strain bacteria, living S-strain bacteria were isolated from the blood of the affected mice.

Step-by-Step Solution

1

Locate explicit passage facts regarding the heat-killed S-strain injection in isolation.

The text explicitly states: 'When Griffith injected mice with heat-killed S-strain bacteria, the mice survived.' This directly validates the statement about mice surviving exclusive injection with heat-killed S-strain bacteria.

Explicit detail retrieval requires matching statement claims directly against textual evidence.

2

Evaluate the statement concerning the capsule of the R-strain bacterium.

The text states that the R-strain 'lacked this capsule.' It does not claim that the R-strain possessed a capsule that was destroyed by heat.

Recognize misread details where traits of one strain are misattributed to another.

3

Locate explicit passage evidence regarding bacterial isolation following the combination injection.

The text states that when heat-killed S-strain and living R-strain were combined, 'living S-strain bacteria were subsequently isolated from their blood.' This confirms the statement regarding the recovery of live S-strain bacteria.

Direct paraphrase matching confirms explicit support.

Key Concept

Explicit detail verification requires confirming that each chosen statement reflects explicitly stated facts without misattributing modifiers or characteristics.

Estimated Time:1m 30s

Question 1324Question →

Read the passage below:

Far from representing an (i)________ departure from the prevailing scientific orthodoxy of its era, the monograph actually (ii)________ the very tenets it ostensibly sought to overturn, revealing under critical analysis that its most provocative claims remained firmly tethered to traditional doctrine.

Which of the following pairs of terms best completes the passage in a logically coherent manner?

(i) iconoclastic, (ii) buttressed

(i) iconoclastic, (ii) dismantled

(i) orthodox, (ii) buttressed

(i) anomalous, (ii) subverted

(i) derivative, (ii) expanded

Show answer & explanation

Answer: (i) iconoclastic, (ii) buttressed

Answer

The pair '(i) iconoclastic, (ii) buttressed' logically completes both blanks by respecting the contrast structural pivot 'Far from' and maintaining alignment with the explanatory clause at the end of the sentence.

The correct option pair '(i) iconoclastic, (ii) buttressed' creates complete contextual coherence. 'Far from representing an iconoclastic departure' signals that although the monograph seemed rebellious, it actually buttressed (supported) traditional scientific tenets, which directly matches the elaboration that its claims 'remained firmly tethered to traditional doctrine.'

Step-by-Step Solution

1

Analyze structural transition signals and overall sentence architecture.

The initial clause 'Far from representing an (i)________ departure...' sets up a contrast with the main clause. The final clause 'revealing under critical analysis that its most provocative claims remained firmly tethered to traditional doctrine' provides the definitive clue for the text's true relationship with traditional doctrine.

Tracking cross-clause dependencies allows us to determine the semantic polarity required for both blanks.

2

Determine the required valence for Blank 2 using the explanatory elaboration clue.

Since the monograph's claims remained 'firmly tethered to traditional doctrine,' Blank 2 must mean supported, reinforced, or sustained. 'Buttressed' perfectly fits this meaning.

Blank 2 describes what the monograph actually did to the tenets of scientific orthodoxy.

3

Determine the required valence for Blank 1 based on the contrast established by 'Far from' and Blank 2.

Because the monograph actually supported orthodoxy, what it 'far from' represented was a radical or rebellious departure. Thus, Blank 1 requires a word meaning revolutionary or attacking traditions, making 'iconoclastic' the correct fit.

The phrase 'Far from [Blank 1] departure' indicates that the departure was merely apparent, not real.

Key Concept

Multi-Blank Dependency Tracking

Question 1325Question →

An investor divides a total initial investment of $\$10,000$ between Account $X$ and Account $Y$ . Account $X$ earns simple annual interest at a rate of $6\%$ , while Account $Y$ earns annual interest at a rate of $10\%$ compounded annually. If no additional deposits or withdrawals are made and the total interest earned from both accounts combined at the end of $2$ years is $\$1,560$ , how much money was originally invested in Account $Y$ ?

\$3,500

\$4,000

\$4,500

\$6,000

\$6,400

Show answer & explanation

Answer: $\$4,000$

Answer

\$4,000

The correct answer of

\$4,000

is determined by calculating the interest earned by each account over two years. Account X earns simple interest of

6\% \times 2 = 12\%

, producing interest equal to

0.12(10,000 - P_Y)

. Account Y earns compound interest at

10\%

annually, yielding an interest factor of

1.10^2 - 1 = 0.21

, or

0.21 P_Y

. Adding the interest expressions yields

1,200 + 0.09 P_Y = 1,560

, which simplifies to

0.09 P_Y = 360

and gives

P_Y = \$4,000

.

Step-by-Step Solution

1

Define variables for the initial principal in each account

Let

P_Y

be the principal in Account

Y

. Then the principal in Account

X

is

10,000 - P_Y

.

The total investment across both accounts is

\$10,000

.

2

Calculate the interest earned by Account

X

over

2

years

\text{Interest}_X = (10,000 - P_Y) \times 0.06 \times 2 = 0.12(10,000 - P_Y) = 1,200 - 0.12 P_Y.

Account

X

earns simple annual interest of

6\%

per year for

2

years.

3

Calculate the interest earned by Account

Y

over

2

years

\text{Interest}_Y = P_Y \times ((1 + 0.10)^2 - 1) = P_Y \times (1.21 - 1) = 0.21 P_Y.

Account

Y

earns interest compounded annually at

10\%

per year for

2

years.

4

Formulate and solve the equation for total interest

(1,200 - 0.12 P_Y) + 0.21 P_Y = 1,560 \implies 1,200 + 0.09 P_Y = 1,560 \implies 0.09 P_Y = 360 \implies P_Y = 4,000.

The combined interest from both accounts equals

\$1,560

.

Key Concept

Simple vs Compound Interest over multiple time periods

Question 1326Question →

Let $m$ and $n$ be positive integers whose prime factorizations consist only of the prime factors $2$ , $3$ , and $5$ . The greatest common divisor of $m$ and $n$ is $360$ , and their least common multiple is $108,000$ . If the power of $2$ in the prime factorization of $m$ is strictly greater than the power of $2$ in the prime factorization of $n$ , and $m$ has exactly $36$ positive divisors, what is the total number of positive divisors of $n$ ?

32

48

64

72

96

Show answer & explanation

Answer: $64$

Answer

The total number of positive divisors of

n

is

64

.

By writing the GCD and LCM in prime factorized form,

\text{GCD}(m,n) = 2^3 \times 3^2 \times 5^1

and

\text{LCM}(m,n) = 2^5 \times 3^3 \times 5^3

. Given

a > x

for the exponent of

2

, we obtain

a=5

and

x=3

. Using the divisor count formula for

m

,

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

, which requires

(b+1)(c+1) = 6

. Testing possible exponent values

b \in \{2,3\}

reveals that only

b=2

yields an integer exponent

c=1

. This uniquely determines

n = 2^3 \times 3^3 \times 5^3

. Calculating

(3+1)(3+1)(3+1)

gives

64

positive divisors.

Step-by-Step Solution

1

Express the greatest common divisor (GCD) and least common multiple (LCM) in prime factorization form.

\text{GCD}(m, n) = 360 = 2^3 \times 3^2 \times 5^1

and

\text{LCM}(m, n) = 108,000 = 2^5 \times 3^3 \times 5^3

.

Prime factorizations allow direct comparison of the minimum and maximum exponents for each prime factor.

2

Determine the exponent of

2

for both integers.

For prime factor

2

,

\min(a, x) = 3

and

\max(a, x) = 5

. Since the exponent of

2

in

m

is strictly greater than in

n

(

a > x

),

a = 5

and

x = 3

.

The GCD gives the lower exponent bound while the LCM gives the upper exponent bound.

3

Use the number of divisors of

m

to solve for its remaining exponents.

Let

m = 2^5 \times 3^b \times 5^c

. The number of positive divisors of

m

is

(5+1)(b+1)(c+1) = 6(b+1)(c+1) = 36

, so

(b+1)(c+1) = 6

. Since

b \in \{2, 3\}

, if

b = 3

then

b+1=4

which does not divide

6

. Thus,

b = 2

and

c = 1

.

The divisor count formula requires integer exponents, uniquely identifying

b = 2

and

c = 1

for

m

.

4

Determine the exponents for

n

and calculate its divisor count.

Since

b = 2

and

\max(b, y) = 3

,

y = 3

. Since

c = 1

and

\max(c, z) = 3

,

z = 3

. Thus

n = 2^3 \times 3^3 \times 5^3

, and its divisor count is

(3+1)(3+1)(3+1) = 64

.

Combining the max/min constraints from LCM/GCD yields all exponents of

n

.

Key Concept

GCD and LCM Prime Exponent Relations & Divisor Counting Formula

Estimated Time:3m 0s

Question 1327Question →

The market value of a rare piece of artwork increased by $x\%$ during its first decade in a gallery and then decreased by $x\%$ during its second decade. If the artwork's market value at the end of the second decade was $\$19,200$ , which represented a net decrease of $\$800$ from its initial market value prior to the two decades, what is the value of $x$ ?

4

10

16

20

25

Show answer & explanation

Answer: $20$

Answer

The value of

x

is

20

.

The initial value of the artwork is

\$19,200 + \$800 = \$20,000

. Applying an increase of

x\%

followed by a decrease of

x\%

gives a net value of

\$20,000 \times (1 + \frac{x}{100})(1 - \frac{x}{100}) = \$20,000 \left(1 - \frac{x^2}{10,000}\right)

. Setting this equal to

\$19,200

yields

1 - \frac{x^2}{10,000} = 0.96

, which simplifies to

\frac{x^2}{10,000} = 0.04

, so

x^2 = 400

and

x = 20

.

Step-by-Step Solution

1

Determine the initial market value before the two decades.

Initial Value

= \$19,200 + \$800 = \$20,000

.

The final value of

\$19,200

reflects a net decrease of

\$800

from the original starting value.

2

Express the successive percentage changes in terms of

x

.

Final Value

= \text{Initial Value} \times \left(1 + \frac{x}{100}\right) \times \left(1 - \frac{x}{100}\right) = \text{Initial Value} \times \left(1 - \frac{x^2}{10,000}\right)

.

An increase of

x\%

followed by a decrease of

x\%

on the new base yields the difference of squares formula for percent multipliers.

3

Set up the equation with known numerical values and solve for

x^2

.

20,000 \left(1 - \frac{x^2}{10,000}\right) = 19,200 \implies 1 - \frac{x^2}{10,000} = \frac{19,200}{20,000} = 0.96 \implies \frac{x^2}{10,000} = 0.04 \implies x^2 = 400

.

Dividing the final value by the initial value gives the net multiplier of

0.96

, meaning a

4\%

overall decrease.

4

Take the positive square root to find

x

.

x = \sqrt{400} = 20

.

Since

x

represents a positive percentage rate,

x = 20

.

Key Concept

Successive percentage changes and base shifts using the difference of squares

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

.

Estimated Time:2m 0s

Question 1328Question →

The policy analyst observed that the sudden removal of price controls did not merely fail to curb inflation; consequently, the unmitigated market volatility ________ existing socioeconomic disparities across the region.

Select the two answer choices that, when inserted into the sentence, produce completed sentences that are equivalent in meaning and logically fit the sentence structure. Which two choices best complete the sentence?

Select all that apply

exacerbated

aggravated

alleviated

mitigated

highlighted

underscored

Show answer & explanation

Answer: exacerbated; aggravated

Answer

The correct answer choices are 'exacerbated' and 'aggravated'.

The sentence establishes a cause-and-effect relationship using the causal signal 'consequently'. The cause—'unmitigated market volatility' resulting from uncurbed inflation—leads to a negative outcome regarding 'existing socioeconomic disparities'. The terms 'exacerbated' and 'aggravated' both mean to intensify or make worse a bad situation, creating coherent and semantically equivalent sentences.

Step-by-Step Solution

1

Analyze the structural signals and sentence context

The clause opens with 'did not merely fail to curb inflation' and transitions with the causal structural signal 'consequently'.

The word 'consequently' indicates that the second clause must express a direct, logical result of the unmitigated market volatility stemming from the failed policy.

2

Determine the required tone and direction for the blank

Since 'unmitigated market volatility' is a negative cause, its direct result on 'existing socioeconomic disparities' must be negative (making them worse).

The blank requires a verb meaning to worsen, intensify, or make more severe.

3

Evaluate candidate words for contextual fit and synonymy

'Exacerbated' and 'aggravated' both mean to make a problem or bad situation worse, providing identical logical and stylistic fit.

Selecting both words maintains semantic equivalence and aligns precisely with the causal continuation signal.

Key Concept

Interpreting Causal Structural Signals in Sentence Equivalence

Question 1329Question →

At the beginning of 2025, Company X's allocation for Marketing was $40\%$ greater than its allocation for Research. Over the course of the year, the Marketing allocation was decreased by $25\%$ , while the Research allocation was increased by $p\%$ . At the end of 2025, the revised Research allocation was $20\%$ greater than the revised Marketing allocation. What is the value of $p$ ?

20

25

26

38

50

Show answer & explanation

Answer: 26

Answer

The value of

p

is 26.

The correct option correctly applies successive percentage changes to each variable. Starting with Research as

R

, Marketing is

1.40R

. Reducing Marketing by

25\%

yields

1.40R \times 0.75 = 1.05R

. Increasing this result by

20\%

gives the final Research amount of

1.20 \times 1.05R = 1.26R

, corresponding to a

26\%

increase over the initial Research allocation

R

.

Step-by-Step Solution

1

Define variables for the initial allocations of both departments.

Let

R

represent the initial Research allocation. Then the initial Marketing allocation is

1.40R

.

Marketing was

40\%

greater than Research at the start of the year.

2

Calculate the revised Marketing allocation at the end of the year.

Revised Marketing allocation

= 1.40R \times (1 - 0.25) = 1.40R \times 0.75 = 1.05R

.

The Marketing allocation decreased by

25\%

from its initial value.

3

Determine the revised Research allocation using the end-of-year relationship.

Revised Research allocation

= 1.20 \times 1.05R = 1.26R

.

The revised Research allocation was

20\%

greater than the revised Marketing allocation.

4

Find the percentage increase

p

for the Research allocation.

R \left(1 + \frac{p}{100}\right) = 1.26R \implies 1 + \frac{p}{100} = 1.26 \implies p = 26

.

The change from

R

to

1.26R

represents a

26\%

increase.

Key Concept

Successive percent changes and base shifts in multi-variable algebraic scenarios

Estimated Time:2m 30s

Question 1330Question →

On a number line, point $P$ corresponds to $-3$ and point $Q$ corresponds to $5$ . If point $R$ is the midpoint of line segment $PQ$ , what is the value of $|R - P|$ ?

2

4

8

1

15

Show answer & explanation

Answer: $4$

Answer

The correct distance between point P and midpoint R is 4.

The total distance between

P (-3)

and

Q (5)

on the real number line is

|5 - (-3)| = 8

. Since

R

is the midpoint of segment

PQ

, it splits the total length into two equal halves of length

4

. Therefore, the distance from

P

to

R

, represented by

|R - P|

, is

4

.

Step-by-Step Solution

1

Calculate the total distance between points P and Q on the number line.

Distance

PQ = |5 - (-3)| = |5 + 3| = 8

.

The distance between any two points

a

and

b

on a number line is given by

|b - a|

.

2

Find the coordinate of the midpoint R.

Coordinate of

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

.

The midpoint coordinate is the average of the coordinates of the two endpoints.

3

Calculate the absolute distance between R and P.

|R - P| = |1 - (-3)| = |1 + 3| = 4

.

The absolute value gives the non-negative distance between the midpoint and point P.

Key Concept

Distance on a Number Line and Midpoint Formula

Question 1331Question →

Let $M = 3.2 \times 10^{-5}$ and $N = 8.0 \times 10^{-4}$ . Which of the following statements must be true? Select all that apply.

Select all that apply

N - M = 7.68 \times 10^{-4}

The ratio

\frac{N}{M}

is equal to

25

\sqrt{M \cdot N} = 1.6 \times 10^{-5}

M + N = 1.12 \times 10^{-3}

\frac{1}{M} = 3.125 \times 10^4

Show answer & explanation

Answer: $N - M = 7.68 \times 10^{-4}$ ; The ratio $\frac{N}{M}$ is equal to $25$ ; $\frac{1}{M} = 3.125 \times 10^4$

Answer

The correct statements are

N - M = 7.68 \times 10^{-4}

, the ratio

\frac{N}{M}

is equal to

25

, and

\frac{1}{M} = 3.125 \times 10^4

.

The statement

N - M = 7.68 \times 10^{-4}

is true because expressing both numbers with the exponent

10^{-4}

gives

(8.0 - 0.32) \times 10^{-4} = 7.68 \times 10^{-4}

. The statement that the ratio

\frac{N}{M}

is equal to

25

is true because

\frac{8.0}{3.2} \times 10^{-4 - (-5)} = 2.5 \times 10 = 25

. The statement

\frac{1}{M} = 3.125 \times 10^4

is true because

\frac{1}{3.2} \times 10^5 = 0.3125 \times 10^5 = 3.125 \times 10^4

.

Step-by-Step Solution

1

Evaluate

N - M

N - M = 7.68 \times 10^{-4}

Convert

M

to the exponent

10^{-4}

:

M = 0.32 \times 10^{-4}

. Then

N - M = (8.0 - 0.32) \times 10^{-4} = 7.68 \times 10^{-4}

.

2

Evaluate

\frac{N}{M}

\frac{N}{M} = 25

Divide coefficients and subtract exponents:

\frac{8.0}{3.2} \times 10^{-4 - (-5)} = 2.5 \times 10^1 = 25

.

3

Evaluate

\sqrt{M \cdot N}

\sqrt{M \cdot N} = 1.6 \times 10^{-4}

Compute

M \cdot N = 25.6 \times 10^{-9} = 2.56 \times 10^{-8}

. Take the square root:

\sqrt{2.56} \times \sqrt{10^{-8}} = 1.6 \times 10^{-4}

.

4

Evaluate

M + N

M + N = 8.32 \times 10^{-4}

Express with common powers of ten:

0.32 \times 10^{-4} + 8.0 \times 10^{-4} = 8.32 \times 10^{-4}

.

5

Evaluate

\frac{1}{M}

\frac{1}{M} = 3.125 \times 10^4

Compute

\frac{1}{3.2 \times 10^{-5}} = 0.3125 \times 10^5 = 3.125 \times 10^4

.

Key Concept

Operations with numbers in scientific notation and place value adjustments

Question 1332Question →

At a financial trading firm, incoming trade orders pass through a three-stage automated verification pipeline:

1. In Stage 1, $\frac{1}{6}$ of all incoming orders are rejected as invalid, and $\frac{1}{5}$ of the remaining valid orders are flagged for manual review.
2. In Stage 2, of the orders that were cleared directly without manual review, $\frac{3}{8}$ are executed immediately, while the rest are placed in a waiting queue.
3. In Stage 3, $\frac{1}{3}$ of the queued orders are processed, leaving the remaining queued orders unprocessed.

If exactly $250$ queued orders remain unprocessed at the end of Stage 3, how many incoming orders were flagged for manual review in Stage 1?

100

150

180

200

225

Show answer & explanation

Answer: 150

Answer

150 orders were flagged for manual review in Stage 1.

Let

N

represent the total number of incoming trade orders. In Stage 1, after

\frac{1}{6}N

orders are rejected,

\frac{5}{6}N

remain. Flagging

\frac{1}{5}

of these remaining valid orders means

\frac{1}{5} \times \frac{5}{6}N = \frac{1}{6}N

orders are flagged for manual review, leaving

\frac{5}{6}N - \frac{1}{6}N = \frac{2}{3}N

orders cleared directly. In Stage 2,

\frac{5}{8}

of the cleared orders enter the queue, which equals

\frac{5}{8} \times \frac{2}{3}N = \frac{5}{12}N

. In Stage 3,

\frac{2}{3}

of the queued orders remain unprocessed, giving

\frac{2}{3} \times \frac{5}{12}N = \frac{5}{18}N

. Setting

\frac{5}{18}N = 250

yields

N = 900

. The number of orders flagged for manual review is

\frac{1}{6} \times 900 = 150

.

Step-by-Step Solution

1

Determine the fraction of orders cleared directly in Stage 1.

The fraction of valid orders remaining after rejection is

1 - \frac{1}{6} = \frac{5}{6}

. The fraction flagged for manual review is

\frac{1}{5} \times \frac{5}{6} = \frac{1}{6}

of the total. Thus, the fraction cleared directly is

\frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}

of total incoming orders.

Sequential fractional deductions must be applied to the remaining amounts at each sub-stage.

2

Determine the fraction of orders placed in the queue during Stage 2.

Of the

\frac{2}{3}

directly cleared orders,

\frac{3}{8}

are executed immediately, so

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

are placed in the queue. The fraction of total incoming orders queued is

\frac{5}{8} \times \frac{2}{3} = \frac{5}{12}

.

Fraction of a fraction requires multiplying the operational fraction by the current subtotal.

3

Determine the fraction of total incoming orders that remain unprocessed at the end of Stage 3.

Since

\frac{1}{3}

of queued orders are processed,

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

of the queued orders remain unprocessed. The fraction of total orders unprocessed is

\frac{2}{3} \times \frac{5}{12} = \frac{5}{18}

.

Multiply the remaining fraction in Stage 3 by the total queued fraction from Stage 2.

4

Calculate the total number of incoming orders (

N

) and the target flagged orders.

Setting

\frac{5}{18}N = 250

gives

N = 250 \times \frac{18}{5} = 900

total incoming orders. The number of orders flagged for manual review is

\frac{1}{6} \times 900 = 150

.

Equate the given numerical quantity to its fractional equivalent to find the total, then compute the required part.

Key Concept

Multi-step fraction calculations involving fractions of remaining quantities and working backward from a known part to the whole.

Question 1333Question →

A boutique owner increases the original price of a handbag by $25\%$ . One month later, the owner reduces the updated price by $20\%$ . Which of the following statements about the price of the handbag must be true? Select all such statements.

Select all that apply

The final price of the handbag is equal to its original price.

The overall net percent change in the price of the handbag is

0\%

.

The dollar amount of the

25\%

increase is equal to the dollar amount of the

20\%

discount.

The final price of the handbag is

5\%

greater than its original price.

The dollar amount of the

20\%

discount is less than the dollar amount of the

25\%

increase.

Show answer & explanation

Answer: The final price of the handbag is equal to its original price.; The overall net percent change in the price of the handbag is $0\%$ .; The dollar amount of the $25\%$ increase is equal to the dollar amount of the $20\%$ discount.

Answer

The correct statements are: the final price is equal to the original price, the overall net percent change is 0%, and the dollar amount of the 25% increase equals the dollar amount of the 20% discount.

Applying a 25% increase to an initial amount

P

yields

1.25P

, representing a

0.25P

increase. Taking 20% off of

1.25P

calculates to

0.20 \times 1.25P = 0.25P

. Subtracting

0.25P

from

1.25P

returns the price to

1.00P

. Therefore, the final price is equal to the original price, the overall percent change is 0%, and the two dollar changes are equal.

Step-by-Step Solution

1

Define a variable for the original price and calculate the price after the 25% increase.

Let the original price be

P

. The price after a

25\%

increase is

P + 0.25P = 1.25P

. The dollar increase is

0.25P

.

Establishing the intermediate price is necessary to apply the subsequent discount correctly.

2

Calculate the price after the 20% discount applied to the updated price.

The discount amount is

20\%

of

1.25P

, which is

0.20 \times 1.25P = 0.25P

. The final price is

1.25P - 0.25P = 1.00P

.

Percentage discounts must be evaluated using the price immediately preceding the discount as the base value.

3

Evaluate the net change and compare dollar amounts.

Final price = Original price (

P

). Net percent change =

\frac{1.00P - P}{P} \times 100\% = 0\%

. Dollar increase (

0.25P

) = Dollar decrease (

0.25P

).

Comparing final values to initial values confirms which statements are true.

Key Concept

Successive Percentage Changes and Base Value Shifts

Question 1334Question →

On Monday, a community kitchen starts with a bulk container of rice. First, $\frac{1}{3}$ of the initial supply of rice is used for lunch. Next, $\frac{1}{4}$ of the remaining rice in the container is used for dinner. After preparing both meals, exactly $60$ kilograms of rice remain in the container. How many kilograms of rice were in the container initially?

84 kg

90 kg

120 kg

144 kg

180 kg

Show answer & explanation

Answer: 120 kg

Answer

120 kg

The correct option is 120 kg. After using

\frac{1}{3}

of the initial supply for lunch,

\frac{2}{3}

of the supply remains. Using

\frac{1}{4}

of this remaining amount for dinner leaves

\frac{3}{4}

of the

\frac{2}{3}

remaining. Multiplying these fractions gives

\frac{3}{4} \times \frac{2}{3} = \frac{1}{2}

of the original supply. Since

\frac{1}{2}

of the original supply equals

60

kg, the initial total supply was

60 \times 2 = 120

kg.

Step-by-Step Solution

1

Determine the fraction of rice remaining after lunch.

Since

\frac{1}{3}

of the initial supply is used for lunch,

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

of the initial supply remains.

The remaining portion after lunch is the base for the next fractional reduction.

2

Determine the fraction of rice used for dinner and the fraction remaining overall.

Dinner uses

\frac{1}{4}

of the remaining

\frac{2}{3}

, which is

\frac{1}{4} \times \frac{2}{3} = \frac{1}{6}

of the initial total supply. The total fraction used is

\frac{1}{3} + \frac{1}{6} = \frac{1}{2}

. Thus, the fraction remaining after dinner is

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

of the initial total.

Sequential fractional reductions require multiplying the second fraction by the remaining fraction from the first step.

3

Solve for the initial total mass of rice.

Let

R

be the initial mass in kilograms. Then

\frac{1}{2} R = 60 \implies R = 120

kg.

Setting the calculated remaining fraction equal to the given numerical remaining quantity yields the starting total.

Key Concept

Sequential Fraction of Remaining Quantities

Estimated Time:1m 30s

Question 1335Question →

Let $N$ be a positive integer with exactly $12$ positive divisors. If the sum of the distinct prime factors of $N$ is $12$ and $N$ is not divisible by $4$ , what is the least possible value of $N$ ?

Show answer & explanation

Answer: 126

Answer

126

The least possible value of

N

is 126. The only sets of distinct prime factors summing to 12 are

\{5, 7\}

and

\{2, 3, 7\}

. Using

\{2, 3, 7\}

produces smaller candidates. Because

N

is not divisible by 4, the exponent of 2 must be 1. The total divisor condition

(1+1)(b+1)(c+1) = 12

requires

(b+1)(c+1) = 6

, giving exponent pairs

(1, 2)

or

(2, 1)

for bases 3 and 7. Assigning the exponent 2 to 3 and 1 to 7 minimizes

N

, giving

2^1 \cdot 3^2 \cdot 7^1 = 126

.

Step-by-Step Solution

1

Find all sets of distinct prime factors that sum to 12.

The possible sets of distinct prime factors are

\{5, 7\}

and

\{2, 3, 7\}

.

Testing combinations of prime numbers (

2, 3, 5, 7, 11, \dots

):

5 + 7 = 12

and

2 + 3 + 7 = 12

are the only valid sets of distinct primes summing to 12.

2

Analyze the set of prime factors

\{2, 3, 7\}

under the condition that

N

is not divisible by 4.

The prime factorization is

N = 2^a \cdot 3^b \cdot 7^c

, where

a = 1

.

Since 2 is a prime factor of

N

,

a \ge 1

. Because

N

is not divisible by 4 (

2^2

),

a

must be strictly less than 2. Thus,

a = 1

.

3

Determine the exponents

b

and

c

using the total number of positive divisors.

The number of divisors is

(1+1)(b+1)(c+1) = 12

, which simplifies to

(b+1)(c+1) = 6

. The possible integer pairs

(b, c)

for

b, c \ge 1

are

(1, 2)

and

(2, 1)

.

The divisor count formula for

N = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}

is

(e_1+1)(e_2+1)\dots(e_k+1) = 12

.

4

Calculate values of

N

for these exponent pairs.

For

(b, c) = (1, 2)

,

N = 2^1 \cdot 3^1 \cdot 7^2 = 294

. For

(b, c) = (2, 1)

,

N = 2^1 \cdot 3^2 \cdot 7^1 = 126

.

To make

N

as small as possible, assign the larger exponent to the smaller prime base (

3^2 \cdot 7^1 < 3^1 \cdot 7^2

).

5

Compare with candidate values from the alternative set of prime factors

\{5, 7\}

.

For

\{5, 7\}

,

N = 5^3 \cdot 7^2 = 6125

or

5^5 \cdot 7^1 = 21875

, both of which are much larger than 126.

The combination

(a+1)(b+1) = 12

yields exponents of 3 and 2 (or 5 and 1), resulting in much higher prime powers.

Key Concept

Divisor count formula combined with prime factorization constraints

Question 1336Question →

In analyzing the ice-sheet dynamics of Jupiter's moon Europa, astrobiologists originally posited that the outer lithospheric shell remained entirely rigid except when disrupted by high-velocity impacts. However, recent thermal modeling of convective ice plumes suggests that the deep sub-surface ice shell displays a surprisingly plastic response to tidal forces. Far from remaining brittle or undergoing sudden, catastrophic fracturing, the cryogenic ice gradually deforms and flows under low, continuous gravitational stress without losing its solid-phase continuity. Consequently, planetary geologists must revise their rheological models to account for this creeping internal movement, recognizing that the material yields dynamically over vast temporal scales rather than maintaining an unyielding structural monolith.

In the context of the passage, the word "plastic" most nearly means

composed of synthetic polymeric materials

capable of continuous deformation without fracturing

returning immediately to an original shape once stress is removed

prone to sudden structural failure under low thermal thresholds

fluctuating rapidly over brief temporal intervals

Show answer & explanation

Answer: capable of continuous deformation without fracturing

Answer

capable of continuous deformation without fracturing

The passage explicitly defines the ice's behavior as deforming and flowing gradually under sustained stress without breaking or fracturing. In physics and geology, 'plastic' describes a material capable of permanent deformation or flow when subjected to stress, making 'capable of continuous deformation without fracturing' the correct contextual definition.

Step-by-Step Solution

1

Locate the target word in the passage and examine its immediate sentence context.

The text states that 'the deep sub-surface ice shell displays a surprisingly plastic response to tidal forces.'

Contextual clues in the surrounding sentences define how the author uses the target term.

2

Analyze structural elaboration and contrast clues in surrounding sentences.

The passage elaborates: 'Far from remaining brittle or undergoing sudden, catastrophic fracturing, the cryogenic ice gradually deforms and flows under low, continuous gravitational stress without losing its solid-phase continuity.'

This establishes that 'plastic' means ductile or able to flow/deform continuously rather than snapping or breaking.

3

Evaluate option definitions against the contextual meaning.

The meaning 'capable of continuous deformation without fracturing' precisely captures the passage's description of gradual deformation and creeping flow without structural failure.

This avoids the trap of common non-contextual definitions (synthetic polymer) or scientific near-synonym traps (elasticity).

Key Concept

Passage-Based Word in Context Analysis

Question 1337Question →

Although mid-century municipal planners envisioned sprawling low-density suburban developments as a paragon of structural efficiency and social harmony, contemporary urban theorists increasingly regard such zoning designs as remarkably ________, pointing to their exorbitant infrastructure maintenance costs and ecological fragility.

Which of the following words best completes the blank?

untenable

harmonious

efficient

innovative

ubiquitous

Show answer & explanation

Answer: untenable

Answer

The correct completion is 'untenable' because the transition 'Although' establishes a contrast between the past view of suburbs as efficient and harmonious and the modern view highlighting financial and ecological drawbacks.

The introductory transition 'Although' signals a reversal of sentiment between the two clauses. Mid-century planners viewed suburban development positively ('paragon of structural efficiency and social harmony'). In contrast, contemporary theorists view it negatively, as elaborated by 'exorbitant infrastructure maintenance costs and ecological fragility'. The word 'untenable' (meaning unsustainable or indefensible) accurately provides the required negative contrast.

Step-by-Step Solution

1

Identify structural contrast signals and sentence pivot points

The sentence begins with the concessionary pivot 'Although', signaling that the perspective of contemporary urban theorists contrasts with that of mid-century municipal planners.

Contrast words dictate that the blank must take a semantic valence opposite to the initial clause.

2

Analyze the tone and meaning of the initial premise

Mid-century planners saw suburban development as a 'paragon of structural efficiency and social harmony' (highly positive).

This establishes the baseline idea that must be reversed in the main clause.

3

Evaluate the context clues following the blank

The main clause supports the blank by referencing 'exorbitant infrastructure maintenance costs and ecological fragility' (negative qualities).

The missing word must mean unsustainable, flawed, or incapable of being maintained.

4

Select the option that strictly satisfies the contrast requirement

'Untenable' means indefensible or unsustainable, perfectly contrasting with 'paragon of structural efficiency'.

It fulfills both the structural contrast dictated by 'Although' and the contextual elaboration regarding high costs and fragility.

Key Concept

Identifying structural contrast clues to determine word polarity in Text Completion items

Estimated Time:1m 0s

Question 1338Question →

While the bioethicist's latest paper was condemned by critics as a __________ departure from her previous publications, closer analysis reveals that it simply elaborates upon principles she outlined years earlier. Which two of the following options, when inserted into the blank, produce completed sentences that are equivalent in meaning?

Select all that apply

anomalous

divergent

dogmatic

doctrinaire

reactionary

gradual

Show answer & explanation

Answer: anomalous; divergent

Answer

The correct options are 'anomalous' and 'divergent', as both words indicate a deviation or departure from past work, producing sentences with equivalent meaning.

The sentence contrasts critics' perception of the paper with its true nature. Critics see it as a sharp break or departure, while it actually builds on existing work. Both 'anomalous' (deviating from what is standard or expected) and 'divergent' (differing or deviating from a course) fit the sentence context and produce coherent, equivalent sentences.

Step-by-Step Solution

1

Analyze structural pivot signals and sentence polarity.

The initial concession signal 'While' sets up a contrast between what critics claim (that the paper is a sharp break or departure) and what closer analysis shows (that it merely elaborates on existing principles).

Understanding structural signals dictates whether the blank requires a word aligned with continuation or contrast.

2

Determine the required contextual meaning for the blank.

The blank modifies 'departure' and must describe something unexpected, deviating, or atypical relative to her prior trajectory.

The sentence requires words that convey an unexpected shift or deviation from a baseline.

3

Evaluate answer choices for semantic equivalence and contextual fit.

'Anomalous' (deviating from normal/expected) and 'divergent' (differing or deviating from a standard) both fit the blank logically and yield identical sentence meanings.

Sentence Equivalence requires both contextual fit and semantic equivalence between the completed sentences.

Key Concept

Contextual Synonym Pair Selection in Sentence Equivalence

Question 1339Question →

Read the paleoclimatology passage below. Based on the structural transition signals and contextual contrast markers, which word best completes the blank?

Fill in the blanks below

For decades, paleoclimatologists operating under the equilibrium paradigm assumed that mid-Holocene megadroughts were driven exclusively by slow, predictable shifts in Earth's orbital precession. However, high-resolution speleothem records have recently upended this consensus: far from representing a mere anomaly in an otherwise stable regime, localized volcanic aerosol forcing frequently triggered abrupt shifts in atmospheric circulation. Indeed, by demonstrating that episodic volcanic events could override long-term orbital trends, these findings show that brief atmospheric shocks played a decisive, rather than secondary, role in shaping regional hydrology.

Show answer & explanation

Answer

incidental (or accepted contextual equivalents: inconsequential, negligible, peripheral)

The passage relies on two key structural transition signals: the concessive/contrast frame 'far from representing a mere {{blank_1}} anomaly...' and the subsequent continuation signal 'Indeed... played a decisive, rather than secondary, role.' The phrase 'far from a mere...' indicates that the word in the blank must express insignificance, minor importance, or secondary status. Therefore, terms like 'incidental', 'inconsequential', 'negligible', or 'peripheral' correctly complete the semantic and structural logic of the sentence.

Step-by-Step Solution

1

Analyze structural contrast signals in the sentence containing the blank

Identified the pivot signal 'However' following the traditional paradigm, and the immediate structural reversal marker 'far from representing a mere [blank] anomaly...'

The construction 'far from representing a mere X' signals that the target word X must describe a diminished, minor, or non-essential status that the subsequent clause refutes.

2

Evaluate continuation and elaboration markers in the surrounding passage context

Observed the colon elaboration ('triggered abrupt shifts') and the continuation signal 'Indeed... played a decisive, rather than secondary, role'.

The explicit contrast between a 'decisive' role and a 'secondary' role reinforces that the refuted view considered volcanic forcing minor or tangential.

3

Synthesize contextual clues to select the appropriate vocabulary term

Determined that the blank requires a word meaning secondary, minor, or occurring as a chance/unimportant feature, such as 'incidental'.

'Incidental' perfectly satisfies the structural requirement of depicting a minor/tangential anomaly negated by the phrase 'far from'.

Key Concept

Deciphering Meaning via Structural Clues and Contrast/Continuation Signals

Question 1340Question →

A residential solar power system generated $4{}000$ kilowatt-hours ( $\text{kWh}$ ) of electricity during its first year of operation. In the second year, electricity generation decreased by $10\%$ . In the third year, electricity generation increased by $10\%$ relative to the second year's generation. How many kilowatt-hours of electricity did the solar system generate in the third year?

3{}240

3{}600

3{}960

4{}000

4{}400

Show answer & explanation

Answer: $3{}960$

Answer

The solar power system generated

3{}960\text{ kWh}

of electricity in the third year.

In the first year, generation was

4{}000\text{ kWh}

. A

10\%

decrease yields

4{}000 \times (1 - 0.10) = 3{}600\text{ kWh}

for the second year. In the third year, a

10\%

increase over the second year's generation yields

3{}600 \times (1 + 0.10) = 3{}960\text{ kWh}

.

Step-by-Step Solution

1

Calculate the electricity generated in the second year after a

10\%

decrease from the first year.

4{}000 - (0.10 \times 4{}000) = 4{}000 - 400 = 3{}600\text{ kWh}

A

10\%

decrease means retaining

90\%

of the original

4{}000\text{ kWh}

base.

2

Calculate the electricity generated in the third year after a

10\%

increase over the second year's total.

3{}600 + (0.10 \times 3{}600) = 3{}600 + 360 = 3{}960\text{ kWh}

The

10\%

increase applies to the new base value of

3{}600\text{ kWh}

, not the initial

4{}000\text{ kWh}

.

Key Concept

Successive Percent Changes and Base Shifts

Estimated Time:45s

Answer

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