All practice questions

Question 2041Question →

For all real numbers $x$ and $y$ such that $x \neq y$ and $x \neq -y$ , which of the following is equivalent to the algebraic expression $\frac{x^3(x + 2y) - y^3(y + 2x)}{x^2 - y^2}$ ?

x^2 + y^2

(x + y)^2

(x - y)^2

x^2 + 2xy - y^2

x + y

Show answer & explanation

Answer: $(x + y)^2$

Answer

The simplified expression is

(x + y)^2

.

Expanding and grouping the terms in the numerator gives

(x^4 - y^4) + 2xy(x^2 - y^2) = (x^2 - y^2)(x^2 + 2xy + y^2) = (x^2 - y^2)(x + y)^2

. Canceling the common factor

(x^2 - y^2)

from both the numerator and the denominator leaves

(x + y)^2

.

Step-by-Step Solution

1

Expand the numerator terms

x^3(x + 2y) - y^3(y + 2x) = x^4 + 2x^3y - y^4 - 2xy^3

Apply the distributive property to remove parentheses in the numerator.

2

Group the terms in the numerator to factor

(x^4 - y^4) + (2x^3y - 2xy^3) = (x^2 - y^2)(x^2 + y^2) + 2xy(x^2 - y^2)

Use difference of squares on

x^4 - y^4

and factor out the greatest common factor

2xy

from the remaining terms.

3

Factor out the common term

(x^2 - y^2)

from the numerator

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

Recognize that

x^2 + 2xy + y^2

is the perfect square binomial

(x + y)^2

.

4

Simplify the rational expression by canceling common factors

\frac{(x^2 - y^2)(x + y)^2}{x^2 - y^2} = (x + y)^2

Divide numerator and denominator by

(x^2 - y^2)

, which is non-zero since

x \neq \pm y

.

Key Concept

Factoring high-degree algebraic expressions by grouping terms, recognizing difference of squares, and applying perfect square binomial identities.

Estimated Time:2m 0s

Question 2042Question →

If $x$ and $y$ are real numbers such that $x < 0 < y$ and $x^2 > y^2$ , which of the following statements must be true? Select all such statements.

Select all that apply

x^3 < y^3

\sqrt{x^2} + x = 0

x + y < 0

\sqrt{x^2 + y^2} = |x| + y

(-x)^2 = -x^2

Show answer & explanation

Answer: $x^3 < y^3$ ; $\sqrt{x^2} + x = 0$ ; $x + y < 0$

Answer

The correct statements are

x^3 < y^3

,

\sqrt{x^2} + x = 0

, and

x + y < 0

.

The statement

x^3 < y^3

is correct because cubing a negative number keeps it negative while cubing a positive number keeps it positive. The statement

\sqrt{x^2} + x = 0

is correct because

\sqrt{x^2} = |x| = -x

for negative numbers. The statement

x + y < 0

is correct because

x^2 > y^2

implies

|x| > y

, meaning the negative component

x

has a larger absolute magnitude than the positive component

y

.

Step-by-Step Solution

1

Analyze the signs of odd powers for

x < 0 < y

x^3 < 0

and

y^3 > 0

, which guarantees

x^3 < y^3

.

Odd powers preserve the original sign of the base.

2

Apply the definition of principal square roots to negative values

\sqrt{x^2} = |x| = -x

, so

\sqrt{x^2} + x = -x + x = 0

.

The square root symbol

\sqrt{}

denotes the principal (non-negative) root.

3

Compare absolute values using

x^2 > y^2

|x| > y \implies -x > y \implies x + y < 0

.

Since

x < 0

, its magnitude

|x|

is

-x

, which dominates the positive value

y

.

4

Evaluate the false options against exponent and radical rules

\sqrt{x^2 + y^2} \neq |x| + y

due to non-distributivity of roots, and

(-x)^2 = x^2 \neq -x^2

.

Radicals do not distribute over sums, and even powers eliminate negative signs.

Key Concept

Properties of real exponents, radical expressions, and absolute values for negative bases

Estimated Time:1m 30s

Question 2043Question →

A laboratory tests two solar panels, Panel A and Panel B, under identical conditions. The probability that Panel A operates at peak efficiency on any given day is $0.75$ , and the probability that Panel B operates at peak efficiency on any given day is $0.60$ . The daily efficiency outcomes of the two panels are independent events. Which of the following statements must be true? Select all such statements.

Select all that apply

The probability that both panels operate at peak efficiency on a given day is

0.45

.

The probability that neither panel operates at peak efficiency on a given day is

0.35

.

The probability that at least one panel operates at peak efficiency on a given day is

0.90

.

The conditional probability that Panel A operates at peak efficiency, given that Panel B operates at peak efficiency, is

0.45

.

The probability that Panel A operates at peak efficiency and Panel B does not operate at peak efficiency on a given day is

0.30

.

Show answer & explanation

Answer: The probability that both panels operate at peak efficiency on a given day is $0.45$ .; The probability that at least one panel operates at peak efficiency on a given day is $0.90$ .; The probability that Panel A operates at peak efficiency and Panel B does not operate at peak efficiency on a given day is $0.30$ .

Answer

The correct statements are: the probability that both panels operate at peak efficiency is

0.45

; the probability that at least one panel operates at peak efficiency is

0.90

; and the probability that Panel A operates at peak efficiency while Panel B does not is

0.30

.

The statements confirming that both panels operate at peak efficiency (

0.45

), that at least one operates at peak efficiency (

0.90

), and that Panel A operates while Panel B does not (

0.30

) are mathematically sound applications of independent event rules.

Step-by-Step Solution

1

Identify given probabilities and independence condition

P(A) = 0.75

,

P(B) = 0.60

, and events

A

and

B

are independent.

Establishes the given parameter values.

2

Calculate joint probability of both events occurring

P(A \cap B) = P(A) \times P(B) = 0.75 \times 0.60 = 0.45

.

For independent events, joint probability equals the product of individual probabilities.

3

Determine complement probabilities and probability of neither event occurring

P(A^c) = 1 - 0.75 = 0.25

,

P(B^c) = 1 - 0.60 = 0.40

, so

P(A^c \cap B^c) = 0.25 \times 0.40 = 0.10

.

Complements of independent events are also independent.

4

Calculate the union probability (at least one panel at peak efficiency)

P(A \cup B) = 1 - P(A^c \cap B^c) = 1 - 0.10 = 0.90

.

The event 'at least one' is the logical complement of 'neither'.

5

Evaluate conditional probability

P(A \mid B)

and difference probability

P(A \cap B^c)

P(A \mid B) = P(A) = 0.75

and

P(A \cap B^c) = 0.75 \times 0.40 = 0.30

.

Independence implies

P(A \mid B) = P(A)

and

P(A \cap B^c) = P(A) P(B^c)

.

Key Concept

Probability rules for independent events, complement rule, and conditional probability definition

Estimated Time:1m 30s

Question 2044Question →

A municipal authority allocates water from a central reservoir to three sectors: Agriculture, Industry, and Residential. Initially, the ratio of the volume of water allocated to Agriculture to that of Industry is $5 : 3$ , and the ratio of the volume allocated to Industry to that of Residential is $4 : 5$ . During a drought, the total supply is reallocated such that Agriculture's allocation is decreased by $20\%$ , Industry's allocation is decreased by $10\%$ , and Residential's allocation is increased by $12\%$ . After these adjustments, what is the ratio of Agriculture's new allocation to Residential's new allocation?

20 : 21

5 : 7

4 : 3

21 : 20

15 : 14

Show answer & explanation

Answer: $20 : 21$

Answer

The ratio of Agriculture's new allocation to Residential's new allocation is

20 : 21

.

To find the new ratio, first unify the given ratios Agriculture : Industry (

5 : 3

) and Industry : Residential (

4 : 5

) by finding a common multiplier for Industry (12). This yields an overall initial ratio of

20 : 12 : 15

. Applying a

20\%

decrease to Agriculture gives

20 \times 0.80 = 16

, and applying a

12\%

increase to Residential gives

15 \times 1.12 = 16.8

. The ratio of the new allocations is

16 : 16.8

, which simplifies to

160 : 168 = 20 : 21

.

Step-by-Step Solution

1

Express the three initial allocations in a unified three-part ratio.

Agriculture : Industry =

5 : 3 = 20 : 12

, Industry : Residential =

4 : 5 = 12 : 15

. Therefore, Agriculture : Industry : Residential =

20 : 12 : 15

.

Industry is the common element linking both ratios, so its ratio component must be equalized (LCM of 3 and 4 is 12).

2

Assign algebraic representations to the initial allocations based on the unified ratio.

Let Agriculture's initial allocation be

20x

, Industry's initial allocation be

12x

, and Residential's initial allocation be

15x

.

This allows for exact percentage calculations on consistent base values.

3

Calculate the updated allocations after applying the specified percentage adjustments.

Agriculture's new allocation =

20x \times (1 - 0.20) = 16x

. Residential's new allocation =

15x \times (1 + 0.12) = 16.8x

.

A

20\%

decrease reduces a quantity to

80\%

of its original value, and a

12\%

increase expands it to

112\%

of its original value.

4

Compute and simplify the ratio of Agriculture's new allocation to Residential's new allocation.

\frac{\text{Agriculture}_{\text{new}}}{\text{Residential}_{\text{new}}} = \frac{16x}{16.8x} = \frac{160}{168} = \frac{20}{21}.

Multiplying both terms by 10 eliminates decimals, and dividing both by their greatest common divisor (8) yields the simplified integer ratio

20 : 21

.

Key Concept

Combining relative ratios through a common quantity and calculating proportional adjustments.

Alternative Method

Instead of setting a variable

x

, assume a concrete initial volume for Industry equal to

12

units. Consequently, Agriculture is

20

units and Residential is

15

units. Agriculture's new volume is

20 - 4 = 16

units, and Residential's new volume is

15 + 1.8 = 16.8

units. The ratio

\frac{16}{16.8} = \frac{20}{21}

is obtained immediately.

Estimated Time:2m 0s

Question 2045Question →

If $x$ is the solution to the linear equation $\frac{4x - 3}{3} - \frac{x + 2}{4} = \frac{5x}{6}$ , which of the following statements about $x$ must be true? Select all such statements.

Select all that apply

x

is a multiple of 3

2x - 5 > 6

x^2 - 5x = 6

x

is a prime number

x < 5

Show answer & explanation

Answer: $x$ is a multiple of 3; $2x - 5 > 6$ ; $x^2 - 5x = 6$

Answer

The statements establishing that

x

is a multiple of 3, that

2x - 5 > 6

, and that

x^2 - 5x = 6

are all true.

Solving the linear equation yields

x = 6

. Substituting

x = 6

into each statement shows that

x

is a multiple of 3 (

6 = 3 \times 2

),

2x - 5 > 6

evaluates to

7 > 6

, and

x^2 - 5x = 6

evaluates to

36 - 30 = 6

. All three statements are correct.

Step-by-Step Solution

1

Clear denominators by multiplying both sides by the least common multiple.

12 \cdot \left(\frac{4x - 3}{3} - \frac{x + 2}{4}\right) = 12 \cdot \left(\frac{5x}{6}\right) \implies 4(4x - 3) - 3(x + 2) = 2(5x)

The least common multiple of 3, 4, and 6 is 12.

2

Distribute factors and combine like terms.

16x - 12 - 3x - 6 = 10x \implies 13x - 18 = 10x

Distributing

-3

into

(x + 2)

yields

-3x - 6

.

3

Isolate

x

on one side of the equation.

13x - 10x = 18 \implies 3x = 18 \implies x = 6

Subtract

10x

and add 18 to isolate the variable term.

4

Evaluate each given statement using

x = 6

.

Statement 1:

6

is a multiple of 3 (True). Statement 2:

2(6) - 5 = 7 > 6

(True). Statement 3:

6^2 - 5(6) = 6

(True). Statement 4: 6 is prime (False). Statement 5:

6 < 5

(False).

Test each option against the calculated solution

x = 6

.

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

Question 2046Question →

If $k$ and $m$ are integers such that $(-k)^5 m < 0$ and $k - m$ is an odd integer, which of the following expressions must be a positive even integer?

k^2 m^2

(-k)^2 m

(k - m)^2

-k m

k^3 + m^3

Show answer & explanation

Answer: $k^2 m^2$

Answer

The expression

k^2 m^2

must be a positive even integer.

Simplifying

(-k)^5 m < 0

gives

-k^5 m < 0 \implies k^5 m > 0

. This confirms

k \neq 0

and

m \neq 0

, so both

k^2

and

m^2

are positive integers, making

k^2 m^2 > 0

. Additionally,

k - m

being odd requires one variable to be even and the other odd. Squaring an even integer yields an even integer, and multiplying by any integer keeps it even. Hence, the expression

k^2 m^2

is guaranteed to be a positive even integer.

Step-by-Step Solution

1

Analyze the sign condition

(-k)^5 m < 0

.

Since

(-k)^5 = -k^5

, the inequality becomes

-k^5 m < 0

, which means

k^5 m > 0

. This implies that neither

k

nor

m

is zero, and both

k

and

m

have the same sign (either both positive or both negative).

Raising a negative quantity to an odd power retains the negative sign.

2

Analyze the parity condition

k - m

is odd.

The difference between two integers is odd if and only if one integer is even and the other is odd.

Even minus odd (or odd minus even) produces an odd result.

3

Evaluate the sign and parity of

k^2 m^2

.

Since

k \neq 0

and

m \neq 0

,

k^2 > 0

and

m^2 > 0

, so

k^2 m^2 > 0

(positive). Since one of

k

or

m

is even, its square is also even, so the product

k^2 m^2

must be even. Thus,

k^2 m^2

is guaranteed to be a positive even integer.

The product of non-zero squares is positive, and any integer multiple of an even number is even.

Key Concept

Parity rules under subtraction/multiplication and sign rules under odd powers.

Question 2047Question →

If $x = 4$ , what is the value of $\sqrt{9^x + 9^x + 9^x + 9^x}$ ?

81

162

324

648

1,296

Show answer & explanation

Answer: 162

Answer

162

Combining the four identical terms under the square root gives

\sqrt{4 \cdot 9^4}

. Splitting the root using product rules yields

\sqrt{4} \cdot \sqrt{9^4} = 2 \cdot 9^2 = 2 \cdot 81 = 162

.

Step-by-Step Solution

1

Substitute

x = 4

into the given radical expression.

The expression becomes

\sqrt{9^4 + 9^4 + 9^4 + 9^4}

.

Direct substitution of the given variable value.

2

Combine the four identical terms under the radical.

9^4 + 9^4 + 9^4 + 9^4 = 4 \cdot 9^4

.

Repeated addition of

4

identical terms is equivalent to multiplication by

4

.

3

Apply the product rule for square roots,

\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}

.

\sqrt{4 \cdot 9^4} = \sqrt{4} \cdot \sqrt{9^4} = 2 \cdot 9^2.

Both

4

and

9^4

are perfect squares.

4

Evaluate the numerical value.

2 \cdot 81 = 162.

Simplifying the arithmetic expression.

Key Concept

Combining like terms under radical sign and applying product rules of square roots

Estimated Time:1m 15s

Question 2048Question →

A sequence of 50 numerical measurements $x_1, x_2, \dots, x_{50}$ is collected. Each measurement $x_i$ is rounded to the nearest tenth to produce a rounded value $r_i$ . The sum of the 50 rounded values, $\sum_{i=1}^{50} r_i$ , is equal to $250.0$ . If $S$ represents the true sum of the unrounded measurements $\sum_{i=1}^{50} x_i$ , what is the maximum possible percent error of the rounded sum relative to the true sum $S$ , rounded to the nearest hundredth of a percent?

0.99\%

1.00\%

1.01\%

2.00\%

2.04\%

Show answer & explanation

Answer: $1.01\%$

Answer

The maximum possible percent error of the rounded sum relative to the true sum is

1.01\%

.

When rounding numbers to the nearest tenth, the maximum error for each number is

0.05

. For 50 numbers, the maximum possible error in the sum is

50 \times 0.05 = 2.5

. The true sum

S

therefore lies in the range

[247.5, 252.5]

. To maximize the percent error relative to

S

, defined as

\frac{|250.0 - S|}{S} \times 100\%

, we use the maximum numerator

2.5

and the smallest possible denominator

S = 247.5

. This yields

\frac{2.5}{247.5} \times 100\% \approx 1.01\%

.

Step-by-Step Solution

1

Determine the maximum rounding error for a single term

Maximum error per measurement is

|x_i - r_i| \le 0.05

When rounding to the nearest tenth, any value within

0.05

of the rounded value rounds to that tenth.

2

Calculate the maximum cumulative error for the sequence sum

Maximum total error

= 50 \times 0.05 = 2.5

The maximum difference between the true sum

S

and the rounded sum

250.0

occurs when all individual rounding errors accumulate in the same direction.

3

Find the range of possible true sum values

S

247.5 \le S \le 252.5

Subtracting and adding the maximum error of

2.5

from the rounded sum

250.0

establishes the bounds for

S

.

4

Set up and maximize the percent error expression

Max percent error occurs at minimum

S = 247.5

, giving

\frac{2.5}{247.5} \times 100\% \approx 1.0101\%

Percent error relative to

S

is given by

\frac{|250.0 - S|}{S} \times 100\%

. To maximize this ratio, we divide the maximum numerator

2.5

by the smallest positive denominator

S = 247.5

.

Key Concept

Error propagation in sequence sums and optimizing percent error base values

Question 2049Question →

Events $A$ and $B$ are mutually exclusive, with $P(A) = 0.25$ and $P(B) = 0.40$ . Event $C$ is independent of both event $A$ and event $B$ , with $P(C) = 0.50$ . What is the probability that event $C$ occurs and at least one of events $A$ or $B$ occurs?

0.050

0.125

0.325

0.650

0.825

Show answer & explanation

Answer: $0.325$

Answer

The probability that event

C

occurs and at least one of events

A

or

B

occurs is

0.325

.

The correct answer is

0.325

. First, since

A

and

B

are mutually exclusive events, the probability of at least one of them occurring is

P(A \text{ or } B) = P(A) + P(B) = 0.25 + 0.40 = 0.65

. Second, because event

C

is independent of both events, the probability that

C

occurs AND at least one of

A

or

B

occurs is given by the multiplication rule for independent events:

P(C) \times P(A \text{ or } B) = 0.50 \times 0.65 = 0.325

.

Step-by-Step Solution

1

Calculate the probability of the union of mutually exclusive events

A

and

B

.

P(A \text{ or } B) = P(A) + P(B) = 0.25 + 0.40 = 0.65

Since

A

and

B

are mutually exclusive,

P(A \cap B) = 0

, so their combined probability is simply the sum of their individual probabilities.

2

Calculate the joint probability of event

C

and event

(A \text{ or } B)

.

P(C \text{ and } (A \text{ or } B)) = P(C) \times P(A \text{ or } B) = 0.50 \times 0.65 = 0.325

Event

C

is independent of both

A

and

B

, which implies

C

is independent of

(A \text{ or } B)

. Therefore, the joint probability is found by multiplying their individual probabilities.

Key Concept

Probability rules for mutually exclusive events (addition rule) and independent events (multiplication rule).

Question 2050Question →

Let $a$ , $b$ , and $c$ be non-zero integers such that $\frac{a}{b} < 0$ , $a^3 b c > 0$ , and $a + b$ is an odd integer. Which of the following statements must be true? Select all such statements.

Select all that apply

c < 0

a b

is an even integer

a c > 0

a^2 + c

is an even integer

b + c < 0

Show answer & explanation

Answer: $c < 0$ ; $a b$ is an even integer

Answer

The statements '

c < 0

' and '

a b

is an even integer' must be true.

The statement '

c < 0

' must be true because

\frac{a}{b} < 0

forces

a b < 0

, and since

a^3 b c = a^2 (a b) c

with

a^2 > 0

,

c

must be negative to yield a positive product. The statement '

a b

is an even integer' must be true because an odd sum

a + b

requires one variable to be even and the other to be odd, making their product even.

Step-by-Step Solution

1

Determine the sign relationship between

a

and

b

.

a

and

b

have opposite signs, so

a b < 0

.

The quotient

\frac{a}{b} < 0

implies the numerator and denominator have different signs.

2

Determine the sign of

c

using

a^3 b c > 0

.

c < 0

.

Rewrite

a^3 b c

as

a^2 \cdot (a b) \cdot c

. Since

a \neq 0

,

a^2 > 0

. Since

a b < 0

, the product

a^2 (a b) < 0

. For the entire product

a^2 (a b) c

to be positive,

c

must be negative.

3

Analyze the parity of

a

and

b

from

a + b

being odd.

One of

a

or

b

is even and the other is odd, so

a b

must be even.

An odd sum of two integers requires one even and one odd addend. The product of an even integer and any integer is always even.

4

Test the remaining options for counterexamples.

The statements '

a c > 0

', '

a^2 + c

is an even integer', and '

b + c < 0

' can be false under valid assignments.

For example, if

a = 2

,

b = -1

, and

c = -3

, then

\frac{a}{b} = -2 < 0

,

a^3 b c = 8(-1)(-3) = 24 > 0

, and

a + b = 1

(odd). Here,

a c = -6 < 0

,

a^2 + c = 4 - 3 = 1

(odd), and

b + c = -4 < 0

, but setting

b = 5, a = -2, c = -1

gives

b + c = 4 > 0

.

Key Concept

Deducing sign and parity properties of integers

Estimated Time:1m 30s

Question 2051Question →

On the real number line, the distance between a real number $k$ and $-3$ is strictly less than $7$ , and the distance between $k$ and $5$ is at least $4$ . Which of the following inequalities represents the complete set of all possible values of $k$ ?

-10 < k \le 1

-10 \le k \le 1

-10 < k < 1

-4 < k \le 1

1 \le k < 4

Show answer & explanation

Answer: $-10 < k \le 1$

Answer

-10 < k \le 1

The correct inequality

-10 < k \le 1

properly combines the strict bound from the distance to

-3

(which gives

-10 < k < 4

) with the non-strict bound from the distance to

5

(which gives

k \le 1

or

k \ge 9

). Taking their intersection gives

-10 < k \le 1

.

Step-by-Step Solution

1

Express the first condition using absolute value notation and solve for

k

.

|k - (-3)| < 7 \implies |k + 3| < 7 \implies -7 < k + 3 < 7 \implies -10 < k < 4

.

Distance on a number line between

x

and

y

is given by

|x - y|

.

2

Express the second condition using absolute value notation and solve for

k

.

|k - 5| \ge 4 \implies k - 5 \le -4

or

k - 5 \ge 4 \implies k \le 1

or

k \ge 9

.

The phrase 'at least 4' means greater than or equal to 4.

3

Find the overlap (intersection) of the two solution sets.

(-10 < k < 4) \cap (k \le 1 \text{ or } k \ge 9) = -10 < k \le 1

.

Since

k

must be less than

4

, the region

k \ge 9

contains no valid solutions, leaving only

-10 < k \le 1

.

Key Concept

Absolute value as distance on the number line and solving compound absolute value inequalities.

Estimated Time:1m 30s

Question 2052Question →

A biomanufacturing facility operates two harvesting lines, Line A and Line B, to collect a refined protein compound suspended in a liquid growth medium into a single storage tank.

- Line A processes liquid at a constant rate of $120\text{ liters per hour}$ , and its output contains protein and medium in a volume ratio of $1 : 4$ .
- Line B processes liquid at a constant rate of $180\text{ liters per hour}$ , and its output contains protein and medium in a volume ratio of $1 : 9$ .

Both lines run simultaneously for exactly $5\text{ hours}$ into the empty storage tank. Which of the following statements about the resulting liquid mixture in the storage tank must be true? Indicate all such statements.

Select all that apply

The total volume of protein collected in the storage tank is

210\text{ liters}

.

Protein accounts for exactly

14\%

of the total liquid volume in the storage tank.

The ratio of total protein to total medium in the storage tank is

7 : 43

.

The total volume of protein collected in the storage tank is

250\text{ liters}

.

Protein accounts for exactly

15\%

of the total liquid volume in the storage tank.

Show answer & explanation

Answer: The total volume of protein collected in the storage tank is $210\text{ liters}$ .; Protein accounts for exactly $14\%$ of the total liquid volume in the storage tank.; The ratio of total protein to total medium in the storage tank is $7 : 43$ .

Answer

The correct statements are those asserting that the total volume of protein collected is 210 liters, that protein accounts for exactly 14% of the total liquid volume, and that the ratio of total protein to total medium is 7 to 43.

The total protein collected is

210\text{ liters}

(

120\text{ L}

from Line A and

90\text{ L}

from Line B). Dividing this by the overall volume of

1500\text{ liters}

gives

14\%

. Subtracting protein from total volume gives

1290\text{ liters}

of medium, yielding a protein-to-medium ratio of

210 : 1290 = 7 : 43

. Therefore, the three statements asserting

210\text{ liters}

of protein, a

14\%

concentration, and a

7:43

ratio are all correct.

Step-by-Step Solution

1

Calculate the total liquid volume produced by each line in 5 hours.

Line A volume =

120\text{ L/hr} \times 5\text{ hr} = 600\text{ liters}

. Line B volume =

180\text{ L/hr} \times 5\text{ hr} = 900\text{ liters}

. Total combined volume =

600 + 900 = 1500\text{ liters}

.

Total volume per line is the product of its constant flow rate and duration.

2

Convert the part-to-part ratios to part-to-whole fractions to find the protein volume from each line.

Line A ratio

1:4

means protein is

\frac{1}{1+4} = \frac{1}{5}

of the volume. Protein A =

\frac{1}{5} \times 600 = 120\text{ L}

. Line B ratio

1:9

means protein is

\frac{1}{1+9} = \frac{1}{10}

of the volume. Protein B =

\frac{1}{10} \times 900 = 90\text{ L}

. Total protein =

120 + 90 = 210\text{ liters}

.

Ratios of

a:b

correspond to a component fraction of

\frac{a}{a+b}

of the total mixture.

3

Determine the percentage concentration of protein in the final mixture.

\text{Percentage} = \frac{210\text{ L}}{1500\text{ L}} \times 100\% = 14\%

.

The overall concentration is total protein volume divided by overall mixture volume.

4

Determine the simplified ratio of total protein to total medium.

Total medium volume =

1500 - 210 = 1290\text{ L}

. Ratio of protein to medium =

210 : 1290 = 7 : 43

.

Dividing both parts of

210 : 1290

by their greatest common divisor (

30

) yields

7 : 43

.

Key Concept

Combining rates and converting part-to-part ratios to part-to-whole fractions

Question 2053Question →

Let $m$ and $n$ be integers such that $m < 0$ , $n > 0$ , $(-1)^m = -1$ , and $m + n$ is an even integer. Which of the following expressions must be a positive even integer?

m \cdot n

(-m)^n

n - m

(m + n)^2

m^2 - n

Show answer & explanation

Answer: $n - m$

Answer

The expression

n - m

must be a positive even integer.

The expression

n - m

subtracts a negative odd integer from a positive odd integer, which equals adding two positive odd integers. The sum of two positive odd integers is always a positive even integer.

Step-by-Step Solution

1

Determine the parity and sign of

m

.

Since

m < 0

and

(-1)^m = -1

,

m

must be a negative odd integer.

An odd exponent on

-1

yields

-1

.

2

Determine the parity and sign of

n

.

Since

n > 0

and

m + n

is even,

n

must be a positive odd integer.

The sum of two integers is even if and only if both integers have the same parity. Since

m

is odd,

n

must also be odd.

3

Evaluate the sign and parity of

n - m

.

n - m = n + (-m)

. Since

n \ge 1

and

-m \ge 1

,

n - m \ge 2

(strictly positive). Also, odd minus odd is always even.

Combining the sign rules (

n > 0

and

-m > 0

) with the even-odd subtraction rule confirms

n - m

is always a positive even integer.

Key Concept

Even-Odd Properties and Sign Rules

Question 2054Question →

The continuous operating times of a model of industrial drone batteries are normally distributed with a mean of $220$ minutes and a standard deviation of $15$ minutes. A battery is designated as "high-efficiency" if its operating time places it in the top $16\%$ of all tested batteries. Based on the $68\text{--}95\text{--}99.7$ empirical rule for normal distributions, what is the minimum operating time, in minutes, required for a battery to be designated as high-efficiency?

205

220

235

236

250

Show answer & explanation

Answer: $235$

Answer

235

minutes

According to the empirical rule for normal distributions,

68\%

of all observations fall within

1

standard deviation of the mean (

220 \pm 15

, or between

205

and

235

). Because normal distributions are symmetric, the remaining

32\%

of observations are split evenly between the upper and lower tails (

16\%

in each tail). The upper tail containing the top

16\%

of battery operating times starts at

1

standard deviation above the mean, which is

220 + 15 = 235

minutes.

Step-by-Step Solution

1

Identify the given parameters of the normal distribution.

Mean

\mu = 220

minutes and standard deviation

\sigma = 15

minutes.

The problem specifies a normal distribution defined by these two parameters.

2

Apply the

68\text{--}95\text{--}99.7

empirical rule to determine the percentile threshold.

Approximately

68\%

of the distribution falls within

[\mu - \sigma, \mu + \sigma]

. The unshaded area (

100\% - 68\% = 32\%

) is split symmetrically, with

16\%

below

\mu - \sigma

and

16\%

above

\mu + \sigma

.

By symmetry of the normal curve, the top

16\%

corresponds precisely to values at or above

1

standard deviation above the mean (

z = +1

).

3

Calculate the raw score corresponding to

z = +1

.

Operating time

= \mu + 1\sigma = 220 + 1(15) = 235

minutes.

Adding one standard deviation to the mean yields the minimum score required to be in the upper tail containing

16\%

of the population.

Key Concept

Empirical Rule (68-95-99.7 Rule) and Symmetry of Normal Distributions

Question 2055Question →

Two software security tools, Tool X and Tool Y, operate independently to scan code repositories for vulnerabilities. The probability that Tool X detects a specific type of security flaw is $0.80$ , and the probability that Tool Y detects the same flaw is $0.75$ . What is the probability that exactly one of the two tools detects the flaw?

Show answer & explanation

Answer: 0.35

Answer

The probability that exactly one of the two tools detects the flaw is

0.35

.

The scenario requires finding the probability that exactly one tool detects the flaw. For independent events

X

and

Y

, 'exactly one' consists of two mutually exclusive events: (1) Tool X succeeds while Tool Y fails, which has probability

0.80 \times (1 - 0.75) = 0.80 \times 0.25 = 0.20

, and (2) Tool Y succeeds while Tool X fails, which has probability

0.75 \times (1 - 0.80) = 0.75 \times 0.20 = 0.15

. Summing these mutually exclusive probabilities gives

0.20 + 0.15 = 0.35

. Alternatively, one can subtract the probability of both tools succeeding (

0.80 \times 0.75 = 0.60

) from the probability of at least one tool succeeding (

0.80 + 0.75 - 0.60 = 0.95

), yielding

0.95 - 0.60 = 0.35

.

Step-by-Step Solution

1

Determine the complement probabilities for each independent tool failing to detect the flaw.

P(\text{Not X}) = 1 - 0.80 = 0.20

and

P(\text{Not Y}) = 1 - 0.75 = 0.25

.

The probability of an event not occurring is equal to 1 minus the probability that it occurs.

2

Calculate the joint probability of Tool X detecting the flaw and Tool Y failing to detect it.

P(\text{X and Not Y}) = 0.80 \times 0.25 = 0.20

.

Because the tools operate independently, the joint probability is the product of their individual probabilities.

3

Calculate the joint probability of Tool Y detecting the flaw and Tool X failing to detect it.

P(\text{Y and Not X}) = 0.75 \times 0.20 = 0.15

.

Tool independence allows multiplying the individual probabilities of detection and non-detection.

4

Sum the probabilities of the two mutually exclusive scenarios representing 'exactly one tool detects the flaw'.

P(\text{Exactly One}) = 0.20 + 0.15 = 0.35

.

The events 'X only' and 'Y only' cannot happen simultaneously, so their probabilities add directly.

Key Concept

Independence and Mutual Exclusivity Rules in Compound Probability

Question 2056Question →

Three water pumps, $P$ , $Q$ , and $R$ , operate at constant individual rates. The ratio of the rate of pump $P$ to the rate of pump $Q$ is $2 : 3$ . When all three pumps operate simultaneously, their combined rate is $3$ times the rate of pump $P$ alone. If pump $R$ working alone can drain a full reservoir in $24$ hours, how many hours would pump $Q$ working alone take to drain the same full reservoir?

8

hours

12

hours

16

hours

32

hours

56

hours

Show answer & explanation

Answer: $8$ hours

Answer

8 hours

The correct answer is 8 hours. By setting the rate of pump Q as

\frac{3}{2}

times the rate of pump P, the combined rate of all three pumps is

r_P + \frac{3}{2} r_P + r_R = \frac{5}{2} r_P + r_R

. Setting this equal to

3 r_P

shows that pump R's rate is

\frac{1}{2} r_P

. Since pump R takes 24 hours (

r_R = \frac{1}{24}

), pump P's rate is

\frac{1}{12}

(taking 12 hours), and pump Q's rate is

\frac{3}{2} \times \frac{1}{12} = \frac{1}{8}

(taking 8 hours).

Step-by-Step Solution

1

Express the rates of pumps P and Q in terms of a common variable.

Let

r_P

,

r_Q

, and

r_R

be the rates of pumps

P

,

Q

, and

R

in reservoirs per hour. Given

r_P : r_Q = 2 : 3

, we have

r_Q = \frac{3}{2} r_P

.

Relating pump rates using the given ratio simplifies the system of equations to one variable.

2

Set up the combined rate equation and solve for

r_R

in terms of

r_P

.

r_P + r_Q + r_R = 3 r_P \implies r_P + \frac{3}{2} r_P + r_R = 3 r_P \implies \frac{5}{2} r_P + r_R = 3 r_P \implies r_R = \frac{1}{2} r_P

.

The total rate is the sum of individual rates, allowing us to express pump R's rate in terms of pump P's rate.

3

Calculate

r_P

and

r_Q

using the given rate for pump R.

Since pump

R

takes

24

hours alone,

r_R = \frac{1}{24}

. Therefore,

\frac{1}{2} r_P = \frac{1}{24} \implies r_P = \frac{1}{12}

. Then

r_Q = \frac{3}{2} \times \frac{1}{12} = \frac{1}{8}

.

Knowing pump R's explicit numerical rate allows finding the numerical rates for pumps P and Q.

4

Determine the time required for pump Q alone to drain the reservoir.

\text{Time for } Q = \frac{1}{r_Q} = \frac{1}{1/8} = 8 \text{ hours}.

The time required to complete one full job is the reciprocal of the rate.

Key Concept

Combined Work Rates and Ratio Relationships

Estimated Time:2m 0s

Question 2057Question →

A venue offers two types of event packages: Standard and Deluxe. The total cost of 3 Standard packages and 2 Deluxe packages is $410. The total cost of 2 Standard packages and 5 Deluxe packages is$ 640. What is the cost, in dollars, of 1 Deluxe package?

Show answer & explanation

Answer: 100

Answer

100

Setting up the linear system

3S + 2D = 410

and

2S + 5D = 640

allows us to multiply the equations by 2 and 3 respectively, obtaining

6S + 4D = 820

and

6S + 15D = 1920

. Subtracting the two equations eliminates

S

and gives

11D = 1100

, leading to

D = 100

.

Step-by-Step Solution

1

Define variables and establish the system of linear equations.

Let

S

represent the cost of a Standard package and

D

represent the cost of a Deluxe package.
Equation 1:

3S + 2D = 410

Equation 2:

2S + 5D = 640

Translating the scenario into mathematical equations forms a 2x2 system of linear equations.

2

Use elimination to eliminate variable

S

.

Multiply Equation 1 by 2:

6S + 4D = 820

Multiply Equation 2 by 3:

6S + 15D = 1920

Creating matching coefficients for

S

allows elimination by subtraction.

3

Subtract the transformed equations and solve for

D

.

(6S + 15D) - (6S + 4D) = 1920 - 820 \implies 11D = 1100 \implies D = 100

Subtracting cancels out

S

, leaving a single linear equation in terms of

D

.

Key Concept

Solving 2x2 Systems of Linear Equations via Elimination

Alternative Method

Use the substitution method: Solve for

S

in terms of

D

from the first equation (

S = \frac{410 - 2D}{3}

) and substitute this into the second equation (

2\left(\frac{410 - 2D}{3}\right) + 5D = 640

). Multiplying both sides by 3 yields

820 - 4D + 15D = 1920

, which simplifies to

11D = 1100

, so

D = 100

.

Estimated Time:1m 30s

Question 2058Question →

A data processing center uses two server clusters, Cluster $X$ and Cluster $Y$ , operating at constant individual processing rates. The ratio of the rate of Cluster $X$ to the rate of Cluster $Y$ is $3 : 5$ . Cluster $X$ alone can process a standard dataset of size $D$ gigabytes in 20 hours.

If Cluster $X$ and Cluster $Y$ work together for 4 hours at their initial rates, and then Cluster $X$ 's rate is increased by $33\frac{1}{3}\%$ while Cluster $Y$ 's rate is increased by $20\%$ , how many additional hours will it take for the two clusters working together at their new rates to complete the remaining portion of dataset $D$ ?

Show answer & explanation

Answer: 2.8

Answer

It will take 2.8 additional hours for the two clusters working together at their new rates to complete the remaining portion of dataset

D

.

Representing Cluster

X

's rate as

3k

and Cluster

Y

's rate as

5k

establishes the dataset size

D = 20 \times 3k = 60k

. During the first 4 hours, both clusters process

4 \times (3k + 5k) = 32k

GB, leaving

28k

GB remaining. After rate increases, Cluster

X

's rate becomes

4k

and Cluster

Y

's rate becomes

6k

, resulting in a new combined rate of

10k

. Dividing the remaining

28k

GB by

10k

GB/hr yields

2.8

hours.

Step-by-Step Solution

1

Define variables for the initial rates and dataset size based on the given ratio.

Let the processing rate of Cluster

X

be

r_X = 3k

GB/hr and Cluster

Y

be

r_Y = 5k

GB/hr for some constant

k > 0

. Since Cluster

X

alone completes dataset

D

in 20 hours,

D = 3k \times 20 = 60k

GB.

Relating the ratio of individual rates to total work defines all quantities in terms of a single parameter

k

.

2

Calculate the amount of work finished during the initial joint operation.

The combined initial rate is

r_X + r_Y = 3k + 5k = 8k

GB/hr. Working together for 4 hours completes

8k \times 4 = 32k

GB.

Working simultaneously means their processing rates add together.

3

Find the remaining work and the updated processing rates after adjustments.

Remaining dataset volume =

60k - 32k = 28k

GB. Cluster

X

's new rate =

3k \times \left(1 + \frac{1}{3}\right) = 4k

GB/hr. Cluster

Y

's new rate =

5k \times 1.20 = 6k

GB/hr. New combined rate =

4k + 6k = 10k

GB/hr.

Modifying the individual rates changes the combined throughput for the remaining task.

4

Compute the additional time required to process the remaining dataset.

Additional time =

\frac{28k \text{ GB}}{10k \text{ GB/hr}} = 2.8

hours.

Dividing the remaining work volume by the new combined rate gives the exact required time.

Key Concept

Combined work rates, ratio proportionality, and percentage rate adjustments.

Question 2059Question →

If $a$ and $b$ are real numbers such that $|a - 3| = 5$ and $|2b + 1| = 9$ , what is the minimum possible value of $|a - b|$ ?

1

2

3

4

6

Show answer & explanation

Answer: 3

Answer

The minimum possible value of

|a - b|

is

3

.

Solving

|a - 3| = 5

gives two possible values for

a

:

a = 8

and

a = -2

. Solving

|2b + 1| = 9

gives two possible values for

b

:

b = 4

and

b = -5

. Evaluating the distance

|a - b|

for all four pairs

(a,b)

gives

|8 - 4| = 4

,

|8 - (-5)| = 13

,

|-2 - 4| = 6

, and

|-2 - (-5)| = 3

. The minimum possible value is

3

.

Step-by-Step Solution

1

Solve the absolute value equation

|a - 3| = 5

for all possible values of

a

.

a - 3 = 5 \implies a = 8

or

a - 3 = -5 \implies a = -2

. Thus,

a \in \{-2, 8\}

.

An absolute value equation

|x| = k

splits into two linear equations:

x = k

and

x = -k

.

2

Solve the absolute value equation

|2b + 1| = 9

for all possible values of

b

.

2b + 1 = 9 \implies 2b = 8 \implies b = 4

or

2b + 1 = -9 \implies 2b = -10 \implies b = -5

. Thus,

b \in \{-5, 4\}

.

An absolute value equation

|2b + 1| = 9

has two cases:

2b + 1 = 9

and

2b + 1 = -9

.

3

Calculate

|a - b|

for all four possible pairs of

(a, b)

.

For

(8, 4): |8 - 4| = 4

.
For

(8, -5): |8 - (-5)| = 13

.
For

(-2, 4): |-2 - 4| = 6

.
For

(-2, -5): |-2 - (-5)| = 3

.

To find the minimum possible value of

|a - b|

, every valid combination of

a

and

b

must be tested.

4

Identify the minimum value among the calculated absolute differences.

The minimum calculated value is

3

.

Comparing

4, 13, 6,

and

3

yields

3

as the smallest value.

Key Concept

Solving absolute value equations and finding distances between points on the real number line

Estimated Time:1m 30s

Question 2060Question →

If $x$ and $y$ are real numbers such that $x < -1$ and $0 < y < 1$ , which of the following statements must be true? Select all such statements.

Select all that apply

x^2 > y^2

\sqrt{x^2 y^4} = -x y^2

x^3 > y^3

\sqrt{x^2} + y = x + y

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

Show answer & explanation

Answer: $x^2 > y^2$ ; $\sqrt{x^2 y^4} = -x y^2$

Answer

The correct statements are the inequality asserting that the square of the first variable is greater than the square of the second, and the identity simplifying the square root of the product of the powers to negative the product of the first variable and the square of the second.

The statement comparing squared values is correct because any real number less than -1 has an absolute value greater than 1, so its square is strictly greater than 1, while any positive number less than 1 has a square strictly less than 1. The radical identity statement is correct because taking the square root of

x^2

yields

|x|

, which evaluates to

-x

when

x

is negative.

Step-by-Step Solution

1

Analyze the given bounds for both variables.

For the first variable,

x < -1

, which implies

|x| > 1

,

x

is negative,

x^2 > 1

, and

x^3 < -1

. For the second variable,

0 < y < 1

, which implies

y > 0

,

y^2 < 1

, and

y^3 > 0

.

Establishing explicit bounds on magnitudes and signs is necessary to evaluate powers and absolute value roots.

2

Evaluate the inequality comparing the squared terms.

Since

x^2 > 1

and

y^2 < 1

, it follows directly that

x^2 > y^2

.

Transitive comparison across the threshold value of 1 proves the inequality holds.

3

Simplify the radical expression

\sqrt{x^2 y^4}

.

\sqrt{x^2 y^4} = \sqrt{x^2} \cdot \sqrt{y^4} = |x| \cdot y^2

. Since

x < 0

,

|x| = -x

, so the expression simplifies to

-x y^2

.

The principal square root of

x^2

must equal the absolute value

|x|

, which requires a sign flip when

x

is negative.

4

Verify remaining candidate expressions for potential fallacies.

Odd powers preserve negative signs so

x^3 < y^3

;

\sqrt{x^2}

equals

-x \neq x

; and expanding

(x+y)^2

produces a nonzero cross-term

2xy

.

Eliminating false options confirms that only two statements are universally true.

Key Concept

Principal square roots and even/odd power behaviors under negative variable bounds

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

Alternative Method

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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

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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

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

Answer