All practice questions

Question 301Question →

Let $x$ and $y$ be real numbers such that $|x^2 - 4| = 2y$ and $|y - 3| \le 1$ . Which of the following could be the value of $x$ ? Select all such values.

Select all that apply

-3

0

2

3

4

Show answer & explanation

Answer: $-3$ ; $0$ ; $3$

Answer

The values that

x

could take are

-3

,

0

, and

3

.

The inequality

|y - 3| \le 1

restricts

y

to the closed interval

[2, 4]

. Consequently,

2y

lies in

[4, 8]

, which means

4 \le |x^2 - 4| \le 8

. Breaking this compound inequality into cases yields

x = 0

(from

x^2 = 0

) or

x \in [-\sqrt{12}, -\sqrt{8}] \cup [\sqrt{8}, \sqrt{12}]

. Since

\sqrt{8} \approx 2.83

and

\sqrt{12} \approx 3.46

, the integer values

-3

and

3

fall in these intervals, alongside

0

. Thus,

-3

,

0

, and

3

are all valid choices.

Step-by-Step Solution

1

Determine the acceptable range for

y

from the given absolute value inequality.

Solving

|y - 3| \le 1

gives

-1 \le y - 3 \le 1

, which simplifies to

2 \le y \le 4

.

The inequality

|y - c| \le r

represents all values of

y

within distance

r

from

c

.

2

Relate the range of

y

to the absolute value expression

|x^2 - 4|

.

Since

2 \le y \le 4

, multiplying by

2

yields

4 \le 2y \le 8

. Therefore,

4 \le |x^2 - 4| \le 8

.

Substitute

2y = |x^2 - 4|

into the inequality derived for

y

.

3

Analyze the two cases for the absolute value equation

|x^2 - 4|

.

Case 1:

4 \le x^2 - 4 \le 8 \implies 8 \le x^2 \le 12

, which gives

x \in [-\sqrt{12}, -\sqrt{8}] \cup [\sqrt{8}, \sqrt{12}]

. Case 2:

4 \le -(x^2 - 4) \le 8 \implies 4 \le 4 - x^2 \le 8 \implies -4 \le -x^2 \le 4 \implies -4 \le x^2 \le 0

. Since

x^2 \ge 0

for all real numbers,

x^2 = 0 \implies x = 0

.

Absolute value

|A|

splits into

A

when

A \ge 0

and

-A

when

A < 0

.

4

Evaluate the test values against the valid domains for

x

.

For

x = -3

,

x^2 = 9

, which satisfies

8 \le 9 \le 12

. For

x = 0

,

x^2 = 0

, which satisfies

x^2 = 0

. For

x = 3

,

x^2 = 9

, which satisfies

8 \le 9 \le 12

. The values

x = 2

and

x = 4

give

x^2 = 4

and

x^2 = 16

, neither of which fall into the valid intervals.

Checking test options confirms which values fall within the solution intervals.

Key Concept

Absolute Value Equations and System Constraints on Real Numbers

Question 302Question →

Let $x$ and $y$ be positive integers such that $\gcd(x, y) = 14$ and $\text{lcm}(x, y) = 420$ . Which of the following values could be the sum $x + y$ ? Select all such values.

Select all that apply

154

182

210

238

435

Show answer & explanation

Answer: 154; 182; 238

Answer

The possible values for the sum

x + y

are 154, 182, and 238.

By writing

x = 14a

and

y = 14b

with

\gcd(a, b) = 1

, the relation

\text{lcm}(x, y) = 14ab = 420

requires

ab = 30

. The positive coprime factor pairs of 30 are

(1, 30)

,

(2, 15)

,

(3, 10)

, and

(5, 6)

. Multiplying these pairs by 14 gives the possible sums 434, 238, 182, and 154. Therefore, the options equal to 154, 182, and 238 are all correct.

Step-by-Step Solution

1

Express

x

and

y

in terms of their greatest common divisor.

Let

x = 14a

and

y = 14b

, where

a

and

b

are positive integers such that

\gcd(a, b) = 1

.

Factoring out the greatest common divisor leaves coprime quotient factors

a

and

b

.

2

Relate the least common multiple to

a

and

b

.

\text{lcm}(x, y) = 14ab = 420 \implies ab = \frac{420}{14} = 30$.

The least common multiple of two numbers sharing a GCD of

g

is given by

g \cdot a \cdot b

.

3

Find all coprime pairs

(a, b)

with

a \le b

whose product is 30.

The prime factorization of 30 is

2 \times 3 \times 5

. The valid coprime pairs

(a, b)

are

(1, 30)

,

(2, 15)

,

(3, 10)

, and

(5, 6)

.

Since

\gcd(a, b) = 1

, all factors of 30 split into pairs of coprime integers.

4

Calculate the corresponding values of

x

,

y

, and their sum

x + y

for each pair.

Pair (1, 30):

x = 14, y = 420 \implies x + y = 434

.
Pair (2, 15):

x = 28, y = 210 \implies x + y = 238

.
Pair (3, 10):

x = 42, y = 140 \implies x + y = 182

.
Pair (5, 6):

x = 70, y = 84 \implies x + y = 154

.

Multiplying each pair by the GCD of 14 yields the original integers

x

and

y

.

Key Concept

Relationship between GCD, LCM, and prime factorization of quotient factors

Question 303Question →

If $x$ satisfies the equation $\frac{x + 5}{3} - 2 = \frac{x - 1}{4}$ , what is the value of $x$ ?

-21

-7

1

7

17

Show answer & explanation

Answer: 1

Answer

1

Multiplying the equation

\frac{x + 5}{3} - 2 = \frac{x - 1}{4}

by the common denominator

12

eliminates fractions to produce

4(x + 5) - 24 = 3(x - 1)

. Expanding both sides yields

4x + 20 - 24 = 3x - 3

, which simplifies to

4x - 4 = 3x - 3

. Subtracting

3x

from both sides and adding

4

to both sides gives

x = 1

.

Step-by-Step Solution

1

Multiply every term on both sides of the equation by the least common denominator of 3 and 4, which is 12.

12 \cdot \left(\frac{x + 5}{3}\right) - 12 \cdot 2 = 12 \cdot \left(\frac{x - 1}{4}\right)

, which simplifies to

4(x + 5) - 24 = 3(x - 1)

.

Clearing fractional denominators simplifies the equation into integer-coefficient linear form.

2

Expand both sides by distributing the numeric multipliers.

4x + 20 - 24 = 3x - 3

, which combines like terms to

4x - 4 = 3x - 3

.

Distributing coefficients removes grouping symbols so variable and constant terms can be combined.

3

Isolate the variable term

x

on one side.

Subtract

3x

from both sides to get

x - 4 = -3

, then add

4

to both sides to get

x = 1

.

Standard algebraic reduction requires grouping all terms containing the unknown variable on one side and numerical constants on the other.

Key Concept

Solving single-variable linear equations containing fractional terms by clearing denominators

Estimated Time:1m 0s

Question 304Question →

A logistics company received a shipment of identical freight containers. On Monday, the crew unloaded $\frac{2}{7}$ of the total shipment. On Tuesday, they unloaded $\frac{3}{5}$ of the remaining containers. On Wednesday, they unloaded $\frac{1}{2}$ of the containers that remained after Tuesday's work. If 30 containers remained unloaded at the end of Wednesday, what was the total number of containers in the original shipment?

80

105

140

210

525

Show answer & explanation

Answer: 210

Answer

The total number of containers in the original shipment was 210.

The correct answer is 210. Working forward, after Monday

\frac{5}{7}

of the initial shipment

N

remains. On Tuesday,

\frac{2}{5}

of that remainder stays unloaded, which equals

\frac{2}{5} \times \frac{5}{7}N = \frac{2}{7}N

. On Wednesday,

\frac{1}{2}

of that remainder stays unloaded, yielding

\frac{1}{2} \times \frac{2}{7}N = \frac{1}{7}N

. Setting

\frac{1}{7}N = 30

gives

N = 210

.

Step-by-Step Solution

1

Determine the fraction of containers remaining after Monday.

Since

\frac{2}{7}

of the total shipment

N

was unloaded on Monday, the fraction remaining is

1 - \frac{2}{7} = \frac{5}{7}

of

N

.

Subtracting the fraction unloaded on Monday from 1 gives the remaining fraction.

2

Calculate the fraction of containers remaining after Tuesday.

On Tuesday,

\frac{3}{5}

of the remaining

\frac{5}{7}N

was unloaded, which is

\frac{3}{5} \times \frac{5}{7}N = \frac{3}{7}N

. The remaining fraction after Tuesday is

\frac{5}{7}N - \frac{3}{7}N = \frac{2}{7}N

.

Alternatively, if

\frac{3}{5}

of the remainder was unloaded, then

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

of the remainder was left:

\frac{2}{5} \times \frac{5}{7}N = \frac{2}{7}N

.

3

Calculate the fraction of containers remaining after Wednesday.

On Wednesday,

\frac{1}{2}

of the remaining

\frac{2}{7}N

was unloaded, leaving

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

of that remainder. Thus, the final fraction remaining is

\frac{1}{2} \times \frac{2}{7}N = \frac{1}{7}N

.

Multiplying the remaining fraction after Tuesday by the fraction left unhandled on Wednesday yields the overall fraction of the original shipment remaining.

4

Solve for the total initial number of containers

N

.

Set

\frac{1}{7}N = 30

, which gives

N = 30 \times 7 = 210

.

Equating the calculated final remaining fraction to the given numerical count allows solving for the total original quantity.

Key Concept

Sequential Fraction of Remaining Quantities

Question 305Question →

If $k$ and $m$ are integers such that $k < 0 < m$ , $(-1)^k = -1$ , and $(-1)^m = 1$ , which of the following expressions MUST be a negative odd integer?

k - m

k + m

k m

m - k

k^2 - m

Show answer & explanation

Answer: $k - m$

Answer

The expression

k - m

MUST be a negative odd integer.

Given

k < 0 < m

,

k

is negative and

m

is positive. The relation

(-1)^k = -1

shows

k

is odd, while

(-1)^m = 1

shows

m

is even. Subtracting a positive even integer

m

from a negative odd integer

k

yields

k - m

, which must be less than 0 (negative) and odd (odd minus even).

Step-by-Step Solution

1

Determine the parity and sign of

k

k

is a negative odd integer.

k < 0

specifies that

k

is negative, and

(-1)^k = -1

implies that the exponent

k

must be odd.

2

Determine the parity and sign of

m

m

is a positive even integer.

m > 0

specifies that

m

is positive, and

(-1)^m = 1

implies that the exponent

m

must be even.

3

Evaluate the sign and parity of

k - m

k - m

is strictly negative and odd.

Since

k < 0

and

m > 0

,

k - m = k + (-m) < 0

. By parity rules,

\text{odd} - \text{even} = \text{odd}

.

Key Concept

Even-Odd Properties and Sign Rules

Question 306Question →

A chemist mixes $x$ ounces of a 40% acid solution with $y$ ounces of a 70% acid solution to produce a 20-ounce mixture that is 52% acid. What is the value of $x - y$ ?

-4

4

8

12

20

Show answer & explanation

Answer: 4

Answer

4

Setting up the linear system

x + y = 20

and

0.40x + 0.70y = 10.4

leads to

x = 12

and

y = 8

. Subtracting

y

from

x

yields

12 - 8 = 4

.

Step-by-Step Solution

1

Set up the system of linear equations based on total solution volume and pure acid content.

Total volume equation:

x + y = 20

. Pure acid equation:

0.40x + 0.70y = 0.52(20) = 10.4

.

The sum of the component volumes equals the total volume, and the sum of the pure acid contents equals the total acid content.

2

Multiply the acid equation by 10 to eliminate decimals.

4x + 7y = 104

.

Clearing decimals simplifies the subsequent elimination calculation.

3

Solve for

y

using the elimination method.

Multiply

x + y = 20

by 4 to get

4x + 4y = 80

. Subtract this from

4x + 7y = 104

:

(4x + 7y) - (4x + 4y) = 104 - 80 \implies 3y = 24 \implies y = 8

.

Eliminating the variable

x

isolates

y

.

4

Solve for

x

and compute

x - y

.

x = 20 - 8 = 12

. Therefore,

x - y = 12 - 8 = 4

.

Substitute

y = 8

back into the first equation and calculate the requested expression.

Key Concept

Systems of Linear Equations in Mixture Problems

Question 307Question →

On the real number line, the distance between two real numbers $x$ and $y$ is $d$ . If the midpoint of $x$ and $y$ is $7$ and $|x - 3| = 2|y - 3|$ , what is the maximum possible value of $d$ ?

12

16

24

28

32

Show answer & explanation

Answer: 24

Answer

The maximum possible value of

d

is 24.

The midpoint condition dictates that

x + y = 14

, so

y = 14 - x

and the distance between them is

d = |x - y| = 2|x - 7|

. Substituting

y = 14 - x

into

|x - 3| = 2|y - 3|

gives

|x - 3| = 2|11 - x|

. Solving the positive case

x - 3 = 2(x - 11)

gives

x = 19

and

y = -5

, resulting in distance

d = |19 - (-5)| = 24

. Solving the negative case

x - 3 = -2(x - 11)

gives

x = 25/3

and

y = 17/3

, resulting in distance

d = 8/3

. Thus, 24 is the maximum possible value of

d

.

Step-by-Step Solution

1

Express

y

in terms of

x

using the midpoint formula

Since the midpoint of

x

and

y

is

7

,

\frac{x + y}{2} = 7

, which gives

y = 14 - x

. The distance

d = |x - y| = |x - (14 - x)| = |2x - 14| = 2|x - 7|

.

Relating

y

to

x

reduces the problem to a single variable.

2

Substitute

y = 14 - x

into the absolute value equation

|x - 3| = 2|(14 - x) - 3| \implies |x - 3| = 2|11 - x| = 2|x - 11|

.

Setting up the single-variable absolute value equation allows finding all possible values for

x

.

3

Solve the absolute value equation for all possible cases

Case 1:

x - 3 = 2(x - 11) \implies x - 3 = 2x - 22 \implies x = 19

. Then

y = 14 - 19 = -5

.
Case 2:

x - 3 = -2(x - 11) \implies x - 3 = -2x + 22 \implies 3x = 25 \implies x = \frac{25}{3}

. Then

y = 14 - \frac{25}{3} = \frac{17}{3}

.

Absolute value equations

|A| = B

split into

A = B

and

A = -B

.

4

Calculate the distance

d

for each case and select the maximum

For Case 1 (

x = 19, y = -5

):

d = |19 - (-5)| = 24

.
For Case 2 (

x = 25/3, y = 17/3

):

d = |25/3 - 17/3| = 8/3

.
The maximum possible value of

d

is

24

.

Comparing the distance values determined in each case identifies the maximum distance.

Key Concept

Absolute Value as Distance and Multi-Case Equations on the Real Number Line

Estimated Time:2m 30s

Question 308Question →

For how many integer values of $n$ in the interval $-15 \le n \le 15$ is the value of the expression $(-1)^{n^2 + n} - (-1)^{3n}$ equal to $2$ ?

Show answer & explanation

Answer: 16

Answer

The correct answer is 16.

Because

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

is the product of two consecutive integers, it is guaranteed to be even for every integer

n

. Consequently,

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

. Substituting this into the given equation yields

1 - (-1)^{3n} = 2

, which reduces to

(-1)^{3n} = -1

. A power of

-1

equals

-1

if and only if the exponent is odd, so

3n

must be odd, which requires

n

itself to be odd. In the interval

[-15, 15]

, there are 16 odd integers: 8 negative odd integers and 8 positive odd integers.

Step-by-Step Solution

1

Determine the parity of

n^2 + n

The expression

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

represents the product of two consecutive integers. Because one of any two consecutive integers is even, their product is always even. Therefore,

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

for all integers

n

.

Simplifying the exponent with a known parity rule reduces the expression to a constant.

2

Isolate

(-1)^{3n}

in the equation

Substituting

1

into the original equation gives

1 - (-1)^{3n} = 2

, which simplifies to

(-1)^{3n} = -1

.

Isolating the exponential term reveals the sign condition required for the equality to hold.

3

Find the parity condition for

n

For

(-1)^{3n}

to equal

-1

, the exponent

3n

must be an odd integer. Since

3

is odd, the product

3n

is odd if and only if

n

is odd.

Applying the product parity rule (

\text{odd} \times \text{odd} = \text{odd}

) relates the condition on

3n

back to

n

.

4

Count the odd integers in the interval

[-15, 15]

The odd integers in the interval are

-15, -13, -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, 13, 15

. There are 16 such integers.

Counting all qualifying values within the specified range yields the final numeric answer.

Key Concept

Parity rules for consecutive integers and exponents of negative numbers

Question 309Question →

For all real numbers $x$ such that $x \neq -3$ , which of the following expressions is equivalent to $\frac{2x^2 - 18}{4x + 12}$ ?

\frac{x - 3}{2}

\frac{x + 3}{2}

\frac{(x - 3)^2}{2(x + 3)}

\frac{2x - 3}{4}

\frac{x - 9}{2}

Show answer & explanation

Answer: $\frac{x - 3}{2}$

Answer

The expression

\frac{x - 3}{2}

is equivalent to the given rational expression.

Factoring the numerator gives

2(x^2 - 9) = 2(x - 3)(x + 3)

and factoring the denominator gives

4(x + 3)

. Canceling the non-zero common terms

2(x + 3)

leaves

\frac{x - 3}{2}

.

Step-by-Step Solution

1

Factor out the greatest common factor from the numerator and denominator.

Numerator:

2x^2 - 18 = 2(x^2 - 9)

; Denominator:

4x + 12 = 4(x + 3)

.

Factoring out common numerical coefficients simplifies the expression and reveals algebraic patterns.

2

Apply the difference of squares formula to factor

x^2 - 9

.

x^2 - 9 = (x - 3)(x + 3)

, so the numerator becomes

2(x - 3)(x + 3)

.

The algebraic identity

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

allows complete factoring of the numerator.

3

Cancel common factors shared by the numerator and denominator.

\frac{2(x - 3)(x + 3)}{4(x + 3)} = \frac{2(x - 3)}{4} = \frac{x - 3}{2}.

Since

x \neq -3

, the factor

(x + 3)

is non-zero and can be safely canceled along with reducing the constant ratio

\frac{2}{4}

to

\frac{1}{2}

.

Key Concept

Simplifying rational algebraic expressions by factoring common numerical factors and applying the difference of squares identity.

Question 310Question →

If $x$ satisfies the linear equation $\frac{x + 2}{2} + \frac{x - 1}{3} = 4$ , what is the value of $3x - 2$ ?

10

4

-2

26.5

7.6

Show answer & explanation

Answer: $10$

Answer

The value of

3x - 2

is

10

.

To solve

\frac{x + 2}{2} + \frac{x - 1}{3} = 4

, multiply the entire equation by the common denominator 6 to clear fractions, yielding

3(x + 2) + 2(x - 1) = 24

. Distributing gives

3x + 6 + 2x - 2 = 24

, which simplifies to

5x + 4 = 24 \implies 5x = 20 \implies x = 4

. Substituting

x = 4

into

3x - 2

gives

3(4) - 2 = 10

.

Step-by-Step Solution

1

Find a common denominator to clear the fractions from the equation.

The least common multiple of 2 and 3 is 6. Multiplying both sides of the equation by 6 gives

6 \cdot \left(\frac{x + 2}{2}\right) + 6 \cdot \left(\frac{x - 1}{3}\right) = 6 \cdot 4

, which simplifies to

3(x + 2) + 2(x - 1) = 24

.

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

2

Expand terms and solve for

x

.

3x + 6 + 2x - 2 = 24 \implies 5x + 4 = 24 \implies 5x = 20 \implies x = 4

.

Combining like terms isolates the variable

x

.

3

Substitute

x = 4

into the target expression

3x - 2

.

3(4) - 2 = 12 - 2 = 10

.

The question asks for the value of the algebraic expression

3x - 2

, not

x

itself.

Key Concept

Solving linear equations with fractional coefficients by clearing denominators and evaluating target algebraic expressions.

Estimated Time:1m 15s

Question 311Question →

Complete the passage below by filling in the blanks with the words that best preserve the semantic contrast dictated by the structural pivot word.

Fill in the blanks below

While early behavioral economists posited that human financial decision-making was inherently , empirical studies published over the past decade have demonstrated that consumer behavior is surprisingly , often adhering to consistent, albeit unconventional, cognitive heuristics.

Show answer & explanation

Answer

Blank 1 is correctly filled by words denoting randomness or lack of order (such as 'capricious' or 'erratic'), while Blank 2 is filled by words denoting order and pattern adherence (such as 'systematic' or 'predictable').

The introductory pivot 'While' sets up a contrast between historical assumptions and modern empirical findings. The clause describing empirical evidence notes that consumer behavior adheres to 'consistent' heuristics, requiring a word signifying order (such as 'systematic' or 'predictable') for the second blank. To complete the contrast, the first blank must express the opposite quality—unpredictability or randomness—requiring words such as 'capricious' or 'erratic'.

Step-by-Step Solution

1

Identify the primary contrast pivot word in the sentence.

The subordinating conjunction 'While' at the start of the sentence establishes a direct contrast between the belief in the first clause and the empirical findings in the second clause.

Contrast pivots signal that the idea in the first clause must oppose the idea in the second clause.

2

Determine the required meaning of the second blank using descriptive context clues.

The trailing modifier 'often adhering to consistent, albeit unconventional, cognitive heuristics' indicates that consumer behavior actually follows reliable patterns.

The clue 'consistent' directly defines the second blank as expressing order or predictability, fitting words like 'systematic' or 'predictable'.

3

Apply the contrast pivot 'While' to solve for the first blank.

Since modern findings show behavior is predictable ('systematic'), early assumptions must have viewed it as lacking pattern or logic ('capricious' or 'erratic').

The pivot requires an antonymic relationship between the core descriptive terms in both blanks.

Key Concept

Utilizing structural contrast signals such as 'while' to establish opposing semantic relationships between text completion blanks.

Estimated Time:1m 30s

Question 312Question →

At the beginning of Year 1, a commercial logistics company allocated its total storage space between two facilities, Facility X and Facility Y, such that Facility X contained $60\%$ of the total space and Facility Y contained the remaining $40\%$ . During Year 1, the storage space of Facility X was increased by $x\%$ , while the storage space of Facility Y was increased by $25\%$ . During Year 2, the storage space of Facility X was decreased by $20\%$ from its Year 1 level, while the storage space of Facility Y was increased by $x\%$ from its Year 1 level. If the total combined storage space of both facilities at the end of Year 2 was $17.6\%$ greater than the total combined storage space at the beginning of Year 1, what is the value of $x$ ?

15

18.5

19.6

20

25

Show answer & explanation

Answer: 20

Answer

The value of

x

is 20.

Let the initial total space be

A

. Facility X initially has

0.60A

space and Facility Y has

0.40A

space. After Year 1, Facility X space becomes

0.60A(1 + x/100)

and Facility Y space becomes

0.40A(1.25) = 0.50A

. At the end of Year 2, Facility X space decreases by

20\%

to

0.48A(1 + x/100)

, while Facility Y space increases by

x\%

to

0.50A(1 + x/100)

. Factoring out

(1 + x/100)

, the total final combined space is

(0.48A + 0.50A)(1 + x/100) = 0.98A(1 + x/100)

. Setting this equal to

1.176A

(a

17.6\%

increase over

A

) yields

1 + x/100 = 1.176 / 0.98 = 1.20

, which gives

x = 20

.

Step-by-Step Solution

1

Define initial storage spaces for Facility X and Facility Y in terms of total initial area

A

.

Facility X initial space =

0.60A

, Facility Y initial space =

0.40A

.

Establishing explicit algebraic representations based on the given

60\% / 40\%

ratio.

2

Calculate the storage space of each facility after Year 1 changes.

Facility X after Year 1 =

0.60A \left(1 + \frac{x}{100}\right)

; Facility Y after Year 1 =

0.40A (1 + 0.25) = 0.50A

.

Facility X increased by

x\%

and Facility Y increased by

25\%

of its original base.

3

Calculate the storage space of each facility at the end of Year 2.

Facility X final =

0.60A \left(1 + \frac{x}{100}\right) (1 - 0.20) = 0.48A \left(1 + \frac{x}{100}\right)

; Facility Y final =

0.50A \left(1 + \frac{x}{100}\right)

.

Facility X decreased by

20\%

from its Year 1 amount, and Facility Y increased by

x\%

from its Year 1 amount.

4

Sum the final spaces of both facilities and set equal to the total space given (

17.6\%

overall increase over

A

).

Total final space =

0.48A \left(1 + \frac{x}{100}\right) + 0.50A \left(1 + \frac{x}{100}\right) = 0.98A \left(1 + \frac{x}{100}\right) = 1.176A

.

The total space at the end of Year 2 is

100\% + 17.6\% = 117.6\%

of initial total space

A

.

5

Solve the linear equation for

x

.

1 + \frac{x}{100} = \frac{1.176}{0.98} = 1.20 \implies \frac{x}{100} = 0.20 \implies x = 20

.

Dividing both sides by

0.98A

isolates the percentage factor.

Key Concept

Successive Percent Change and Multi-Base Percentage Problems

Estimated Time:2m 30s

Question 313Question →

Read the passage below:

Early proponents of radical linguistic determinism argued that human cognitive architecture is thoroughly (i)________ by native language structures, asserting that concepts lacking overt grammatical encoding remain entirely inaccessible. Subsequent cross-cultural empirical trials, however, demonstrated that speakers of languages lacking dedicated color terms could nevertheless distinguish fine chromatic gradations with precision, effectively (ii)________ the assertion that vocabulary imposes an impassable barrier on perception. Consequently, modern cognitive scientists generally view language not as a restrictive blueprint, but rather as a flexible lens that merely (iii)________ attentional focus rather than strictly constraining cognitive capacity.

Which of the following sets of terms best completes the passage so that cross-sentence contextual coherence is maintained?

(i) circumscribed, (ii) repudiating, (iii) channels

(i) circumscribed, (ii) vindicating, (iii) channels

(i) augmented, (ii) repudiating, (iii) obfuscates

(i) instantiated, (ii) prefiguring, (iii) channels

(i) circumscribed, (ii) repudiating, (iii) annihilates

Show answer & explanation

Answer: (i) circumscribed, (ii) repudiating, (iii) channels

Answer

The passage is correctly completed by the terms '(i) circumscribed, (ii) repudiating, (iii) channels'.

The combination '(i) circumscribed, (ii) repudiating, (iii) channels' establishes full thematic and structural coherence across the passage. Blank (i) requires a term indicating restriction ('circumscribed') to match 'entirely inaccessible'. Blank (ii) requires a word signifying refutation ('repudiating') because the transition word 'however' indicates that the studies disproved the barrier claim. Blank (iii) requires a term denoting gentle direction ('channels') to complete the contrast between a rigid constraint and a 'flexible lens'.

Step-by-Step Solution

1

Analyze Blank (i) using sentence-internal context clues.

The clause states that concepts lacking grammatical encoding remain 'entirely inaccessible', indicating that cognitive architecture is bounded or restricted by language structure. 'Circumscribed' (meaning restricted or confined) fits this requirement.

Establishing the initial premise of radical linguistic determinism requires a word denoting restriction.

2

Analyze Blank (ii) using cross-sentence contrast clues.

The second sentence introduces a pivot with 'however' and shows speakers could distinguish fine colors despite lacking dedicated terms. This empirical finding refutes the claims of determinism, requiring a word like 'repudiating' (disproving/rejecting).

The structural contrast marker 'however' signals that the empirical evidence undermined the deterministic claim.

3

Analyze Blank (iii) by evaluating the modern synthesis contrast.

The third sentence contrasts 'a restrictive blueprint' with 'a flexible lens that merely _____ attentional focus'. The term 'channels' (directs or guides) completes the contrast with 'strictly constraining'.

The passage requires a term that reflects mild guidance rather than absolute restriction.

Key Concept

Cross-Sentence Contextual Coherence & Discourse Pivot Tracking

Question 314Question →

Passage:
In 1794, German physicist Ernst Chladni published a controversial treatise asserting that meteorites—masses of iron and stone that fall to Earth—originated in outer space rather than from terrestrial volcanic eruptions or atmospheric aggregations of dust, as contemporary natural philosophers maintained. Chladni reasoned that the extreme velocity and scorched exterior crusts of recovered iron masses indicated an extra-terrestrial origin outside Earth's atmosphere. Furthermore, he calculated that if these objects were produced by atmospheric processes, they would require chemical components not present in the upper atmosphere. Despite Chladni’s rigorous compilation of historical eyewitness reports and chemical samples, the scientific establishment initially rejected his hypothesis, deeming eyewitness accounts of falling rocks to be unscientific folklore. It was only after a massive meteorite shower at L'Aigle, France, in 1803—thoroughly documented by the French Academy of Sciences under Jean-Baptiste Biot—that Chladni's extraterrestrial theory gained widespread scientific acceptance.

According to the passage, Chladni based his assertion that recovered iron masses originated outside Earth's atmosphere on which of the following explicit details?

The extreme velocity and scorched outer crusts observed on the iron masses

The detailed scientific documentation compiled during the 1803 L'Aigle meteorite shower

The presence of chemical elements in meteorites that perfectly matched upper atmospheric dust

The complete absence of prior eyewitness accounts attributing fallen stones to terrestrial volcanic eruptions

Laboratory experiments simulating the high-pressure aggregation of atmospheric dust particles

Show answer & explanation

Answer: The extreme velocity and scorched outer crusts observed on the iron masses

Answer

Chladni based his assertion on the extreme velocity and scorched outer crusts observed on the iron masses.

The passage explicitly states in the second sentence that Chladni deduced an extraterrestrial origin based on the extreme velocity and scorched exterior crusts of recovered iron masses.

Step-by-Step Solution

1

Identify the key focus of the question stem

The question asks for the specific explicit detail Chladni used to argue that iron masses originated outside Earth's atmosphere.

Explicit detail retrieval requires matching the specific premise attributed to Chladni in the passage.

2

Locate the corresponding sentence in the passage

The second sentence states: 'Chladni reasoned that the extreme velocity and scorched exterior crusts of recovered iron masses indicated an extra-terrestrial origin outside Earth's atmosphere.'

This direct sentence directly provides Chladni's specific evidence.

3

Match the passage detail with the correct paraphrased option

The option citing 'The extreme velocity and scorched outer crusts observed on the iron masses' directly paraphrases the text.

It accurately captures the facts without adding outside assumptions or distorting sentence modifiers.

Key Concept

Explicit Detail Retrieval

Estimated Time:1m 30s

Question 315Question →

Consider the following system of linear equations:

\begin{aligned} 2x + 3y &= 12 \\ 4x + 6y &= 24 \end{aligned}

Which of the following statements about this system must be true? Select all that apply.

Select all that apply

The system has infinitely many solutions.

The graphs of the two equations represent the exact same line in the

xy

-plane.

The ordered pair

(3, 2)

is a solution to the system.

The system is inconsistent and has no solution.

The

y

-intercept of the line is

(6, 0)

.

Show answer & explanation

Answer: The system has infinitely many solutions.; The graphs of the two equations represent the exact same line in the $xy$ -plane.; The ordered pair $(3, 2)$ is a solution to the system.

Answer

The correct statements are that the system has infinitely many solutions, the graphs represent the exact same line, and

(3, 2)

is a solution to the system.

Dividing

4x + 6y = 24

by

2

produces

2x + 3y = 12

, showing that both equations represent the exact same line. Therefore, the system has infinitely many solutions. Substituting

x = 3

and

y = 2

yields

2(3) + 3(2) = 12

, confirming that

(3, 2)

is one of the infinitely many valid solutions.

Step-by-Step Solution

1

Analyze the relationship between the two linear equations

Dividing the second equation

4x + 6y = 24

by

2

gives

2x + 3y = 12

, which is identical to the first equation.

Comparing coefficients and constants determines whether equations in a system are dependent, independent, or inconsistent.

2

Determine the number of solutions and geometric structure

Because the equations are mathematically equivalent, they describe the same line in the coordinate plane and have infinitely many intersection points.

Identical linear equations form a dependent system with infinitely many solutions.

3

Test the ordered pair

(3, 2)

Evaluating

2(3) + 3(2) = 6 + 6 = 12

confirms that

(3, 2)

lies on the line.

Any point satisfying one equation satisfies the entire system of equivalent equations.

4

Determine the axis intercepts

Setting

y=0

gives

2x=12 \Rightarrow x=6

(the

x

-intercept is

(6,0)

); setting

x=0

gives

3y=12 \Rightarrow y=4

(the

y

-intercept is

(0,4)

).

Checking coordinates of axis intersections prevents mislabeling

x

- and

y

-intercepts.

Key Concept

Dependent Systems of Linear Equations

Question 316Question →

A positive integer $n$ has the prime factorization $n = 2^a \cdot 3^b \cdot 5^c$ , where $a$ , $b$ , and $c$ are positive integers. If $n$ is divisible by $36$ and is a divisor of $5{}400$ , which of the following values could be the total number of positive divisors of $n$ ? Select all such values.

Select all that apply

18

20

27

32

40

Show answer & explanation

Answer: 18; 27; 32

Answer

18, 27, and 32 are all possible total numbers of positive divisors for

n

.

Prime factorization gives

36 = 2^2 \cdot 3^2

and

5{}400 = 2^3 \cdot 3^3 \cdot 5^2

. For

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

to be a multiple of 36 and a divisor of 5,400 with positive integer exponents,

a \in \{2,3\}

,

b \in \{2,3\}

, and

c \in \{1,2\}

. The total number of divisors is

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

. The possible values for this product are 18, 24, 27, 32, 36, and 48. Among the choices, 18, 27, and 32 are valid values.

Step-by-Step Solution

1

Find the prime factorizations of the boundary numbers 36 and 5,400.

36 = 2^2 \cdot 3^2

and

5{}400 = 2^3 \cdot 3^3 \cdot 5^2

.

Establishing the prime factor bounds determines the range of possible values for exponents

a

,

b

, and

c

.

2

Determine the constraints on exponents

a

,

b

, and

c

.

Since

36 \mid n

,

a \ge 2

,

b \ge 2

, and

c \ge 1

(given

c

is a positive integer). Since

n \mid 5{}400

,

a \le 3

,

b \le 3

, and

c \le 2

. Thus,

a \in \{2, 3\}

,

b \in \{2, 3\}

, and

c \in \{1, 2\}

.

Divisibility rules require prime factor exponents of a multiple to be greater than or equal to those of the divisor, and exponents of a divisor to be less than or equal to those of the multiple.

3

Calculate all possible total divisor counts using the formula

d(n) = (a+1)(b+1)(c+1)

.

Possible factor values are

(a+1) \in \{3, 4\}

,

(b+1) \in \{3, 4\}

, and

(c+1) \in \{2, 3\}

. Evaluating all combinations yields:

3 \cdot 3 \cdot 2 = 18

,

3 \cdot 3 \cdot 3 = 27

,

3 \cdot 4 \cdot 2 = 24

,

3 \cdot 4 \cdot 3 = 36

,

4 \cdot 4 \cdot 2 = 32

, and

4 \cdot 4 \cdot 3 = 48

.

The total number of positive integer divisors is found by adding 1 to each exponent in the prime factorization and multiplying the results.

4

Compare the calculated divisor counts with the options provided.

The values 18, 27, and 32 appear in the calculated set of possible total divisors.

Direct matching identifies all valid options.

Key Concept

Prime factor exponent bounds and total number of positive divisors formula

Question 317Question →

If $x$ is a real number such that $|x^2 - 9| = 5|x - 3|$ , which of the following could be the value of $x$ ? Select all such values.

Select all that apply

-8

-2

2

3

8

Show answer & explanation

Answer: $-8$ ; $2$ ; $3$

Answer

The values of

x

that satisfy the equation are

-8

,

2

, and

3

.

Factoring the left side of

|x^2 - 9| = 5|x - 3|

yields

|x - 3||x + 3| = 5|x - 3|

. Setting the common factor

|x - 3| = 0

gives

x = 3

. Dividing both sides by the non-zero quantity

|x - 3|

leaves

|x + 3| = 5

, which splits into

x + 3 = 5 \implies x = 2

and

x + 3 = -5 \implies x = -8

. Therefore,

-8

,

2

, and

3

are all valid solutions.

Step-by-Step Solution

1

Apply the product rule for absolute values to factor the left-hand side.

|x^2 - 9| = |(x - 3)(x + 3)| = |x - 3| \cdot |x + 3|

, rewriting the equation as

|x - 3| \cdot |x + 3| = 5|x - 3|

.

The absolute value of a product is equal to the product of the individual absolute values.

2

Evaluate the case where the shared factor is zero:

|x - 3| = 0

.

x - 3 = 0 \implies x = 3

. Both sides equal

0

, making

x = 3

a valid solution.

Dividing by

|x - 3|

without checking if it can be zero would cause the loss of the root

x = 3

.

3

Evaluate the case where

|x - 3| \neq 0

by dividing both sides of the equation by

|x - 3|

.

|x + 3| = 5

.

Since

|x - 3| > 0

, we can safely divide both sides by this non-zero quantity.

4

Solve the remaining absolute value equation

|x + 3| = 5

.

x + 3 = 5 \implies x = 2

, and

x + 3 = -5 \implies x = -8

.

An expression inside an absolute value equal to a positive number

k

can equal either

k

or

-k

.

Key Concept

Factoring absolute value expressions using

|ab| = |a||b|

and systematically considering all cases to avoid dropping zero-roots or negative solutions.

Question 318Question →

For all non-zero real numbers $x$ and $y$ , which of the following expressions is equivalent to $\frac{12x^4y^2 - 8x^2y^3}{4x^2y}$ ?

3x^2y - 2y^2

3x^2y - 8x^2y^3

3x^6y^3 - 2x^4y^4

3x^2y + 2y^2

3x^2 - 2y

Show answer & explanation

Answer: $3x^2y - 2y^2$

Answer

3x^2y - 2y^2

Dividing each term in the numerator by

4x^2y

yields

\frac{12x^4y^2}{4x^2y} - \frac{8x^2y^3}{4x^2y} = 3x^2y - 2y^2

, which matches the correct expression.

Step-by-Step Solution

1

Split the rational algebraic expression into two separate fractions by distributing the denominator.

\frac{12x^4y^2}{4x^2y} - \frac{8x^2y^3}{4x^2y}

When dividing a polynomial by a monomial, each term of the numerator must be divided independently by the denominator.

2

Simplify the first term

\frac{12x^4y^2}{4x^2y}

.

\frac{12}{4} \cdot x^{4-2} \cdot y^{2-1} = 3x^2y

Divide the numerical coefficients and apply the exponent quotient rule

a^m / a^n = a^{m-n}

for each variable.

3

Simplify the second term

\frac{8x^2y^3}{4x^2y}

.

\frac{8}{4} \cdot x^{2-2} \cdot y^{3-1} = 2(1)y^2 = 2y^2

Divide the coefficients and apply the exponent quotient rule, noting that

x^0 = 1

for non-zero

x

.

4

Combine the simplified terms with the original subtraction operator.

3x^2y - 2y^2

Combine terms to form the final simplified expression.

Key Concept

Simplifying algebraic fractions by distributing a monomial denominator across numerator terms and applying exponent rules.

Estimated Time:45s

Question 319Question →

Two positive integers $m$ and $n$ satisfy $\gcd(m, n) = 20$ and $\text{lcm}(m, n) = 4{}200$ . It is given that $m$ is divisible by $7$ and has exactly $24$ positive integer divisors. If $n$ is a multiple of $3$ , what is the value of $n$ ?

60

200

300

420

600

Show answer & explanation

Answer: $60$

Answer

The value of

n

is

60

.

Prime factorization reveals that

\gcd(m,n) = 2^2 \cdot 3^0 \cdot 5^1 \cdot 7^0

and

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

. The condition that

n

is a multiple of

3

fixes the exponent of

3

in

n

to

1

(so

m

has exponent

0

). The condition that

m

is a multiple of

7

fixes the exponent of

7

in

m

to

1

(so

n

has exponent

0

). For

m = 2^a \cdot 3^0 \cdot 5^c \cdot 7^1

, its divisor count equation

2(a+1)(c+1) = 24

simplifies to

(a+1)(c+1) = 12

. Since

a \in \{2, 3\}

and

c \in \{1, 2\}

, the only valid solution is

a=3

and

c=2

. This leaves

n

with exponents

2

for prime

2

,

1

for prime

3

,

1

for prime

5

, and

0

for prime

7

, giving

n = 2^2 \cdot 3^1 \cdot 5^1 \cdot 7^0 = 60

.

Step-by-Step Solution

1

Express the GCD and LCM in prime factorized form.

\gcd(m, n) = 20 = 2^2 \cdot 3^0 \cdot 5^1 \cdot 7^0

and

\text{lcm}(m, n) = 4{}200 = 2^3 \cdot 3^1 \cdot 5^2 \cdot 7^1

.

The prime factor exponents of

m

and

n

must have minimums equal to the GCD exponents and maximums equal to the LCM exponents.

2

Determine the exponents of prime factors

3

and

7

for

m

and

n

.

Since

n

is divisible by

3

, exponent of

3

in

n

is

1

, so exponent of

3

in

m

is

0

. Since

m

is divisible by

7

, exponent of

7

in

m

is

1

, so exponent of

7

in

n

is

0

.

Each prime exponent in

\text{lcm}(m, n)

must belong to at least one of the numbers.

3

Use the divisor count of

m

to find its remaining exponents for primes

2

and

5

.

Let

m = 2^a \cdot 3^0 \cdot 5^c \cdot 7^1

, where

a \in \{2, 3\}

and

c \in \{1, 2\}

. The number of positive divisors is

(a+1)(0+1)(c+1)(1+1) = 2(a+1)(c+1) = 24

, giving

(a+1)(c+1) = 12

. Testing

a=2 \implies c=3

(invalid range). Testing

a=3 \implies c=2

(valid). Thus

m = 2^3 \cdot 3^0 \cdot 5^2 \cdot 7^1 = 1{}400

.

The total number of positive integer divisors of

p_1^{e_1} p_2^{e_2} \cdots

is given by

(e_1+1)(e_2+1)\cdots

.

4

Determine the prime exponents for

n

and calculate its value.

Since

a=3

,

n

gets

2^2

. Since

c=2

,

n

gets

5^1

. Together with

3^1

and

7^0

,

n = 2^2 \cdot 3^1 \cdot 5^1 \cdot 7^0 = 60

.

For each prime

p

, the exponent in

n

must match the bound opposite to

m

to satisfy both

\gcd

and

\text{lcm}

.

Key Concept

Prime exponent analysis of GCD and LCM alongside the divisor count formula

Question 320Question →

A shipping company calculates the total cost $C$ , in dollars, to deliver a package of weight $w$ pounds using the linear relationship $C = kw + b$ , where $k$ and $b$ are positive constants. The delivery cost for a $4$ -pound package is $\$19$ , and the delivery cost for a $9$ -pound package is $\$39$ . If a customer pays a total delivery cost of $\$71$ for a single package, what is the weight of the package, in pounds?

13

15

17

18

21

Show answer & explanation

Answer: $17$

Answer

The weight of the package is

17

pounds.

Using the two points

(4, 19)

and

(9, 39)

, the rate of change

k

is calculated as

\frac{39 - 19}{9 - 4} = \frac{20}{5} = 4

dollars per pound. Substituting

k = 4

into

4(4) + b = 19

yields

b = 3

. Setting the linear equation

4w + 3 = 71

and solving for

w

gives

4w = 68

, so

w = 17

pounds.

Step-by-Step Solution

1

Set up a system of linear equations using the given data points

(4, 19)

and

(9, 39)

.

4k + b = 19

and

9k + b = 39

.

The cost model follows

C = kw + b

for weight

w

.

2

Subtract the first equation from the second equation to solve for

k

.

(9k + b) - (4k + b) = 39 - 19 \implies 5k = 20 \implies k = 4

.

Subtracting eliminates the constant

b

to determine the unit rate per pound.

3

Substitute

k = 4

into

4k + b = 19

to solve for

b

.

4(4) + b = 19 \implies 16 + b = 19 \implies b = 3

.

Finding

b

establishes the full linear equation model:

C = 4w + 3

.

4

Substitute

C = 71

into the linear equation

4w + 3 = 71

and solve for

w

.

4w = 71 - 3 \implies 4w = 68 \implies w = 17

.

Solving for

w

yields the required weight corresponding to a

\$71

delivery cost.

Key Concept

Linear Modeling and Single-Variable Linear Equations

Estimated Time:1m 30s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Answer

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Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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