All practice questions

Question 2101Question →

For all real numbers $x \neq 2$ , which of the following expressions is equivalent to $\frac{x^4 - 16}{x^3 - 2x^2 + 4x - 8}$ ?

x + 2

x - 2

x^2 + 4

\frac{x + 2}{x - 2}

\pm (x + 2)

Show answer & explanation

Answer: $x + 2$

Answer

x + 2

Factoring the numerator

x^4 - 16

as a difference of squares yields

(x - 2)(x + 2)(x^2 + 4)

. Factoring the denominator

x^3 - 2x^2 + 4x - 8

by grouping yields

(x - 2)(x^2 + 4)

. Dividing the numerator by the denominator allows the shared factor

(x - 2)(x^2 + 4)

to cancel, leaving

x + 2

.

Step-by-Step Solution

1

Factor the numerator

x^4 - 16

x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4)

Apply the difference of squares factorization identity twice.

2

Factor the denominator

x^3 - 2x^2 + 4x - 8

by grouping

x^3 - 2x^2 + 4x - 8 = x^2(x - 2) + 4(x - 2) = (x - 2)(x^2 + 4)

Group the terms in pairs and factor out the common binomial factor

(x - 2)

.

3

Simplify the rational expression by canceling common factors

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

Cancel the non-zero common factor

(x - 2)(x^2 + 4)

present in both numerator and denominator.

Key Concept

Simplifying rational expressions using difference of squares and factoring by grouping

Question 2102Question →

For all real numbers

x

, the algebraic expression

\frac{x^6 - 27}{x^4 + 3x^2 + 9}

can be simplified to the polynomial form

ax^2 + bx + c

, where

a

,

b

, and

c

are real constants. What is the value of

a + b + c

?

Show answer & explanation

Answer: -2

Answer

The simplified expression is

x^2 - 3

, which corresponds to polynomial coefficients

a = 1

,

b = 0

, and

c = -3

. The sum

a + b + c

equals

-2

.

Factoring the numerator

x^6 - 27

as a difference of cubes

(x^2)^3 - 3^3

produces

(x^2 - 3)(x^4 + 3x^2 + 9)

. Canceling the non-zero factor

(x^4 + 3x^2 + 9)

from the numerator and denominator simplifies the expression to

x^2 - 3

. In standard form

ax^2 + bx + c

,

a = 1

,

b = 0

, and

c = -3

. Adding these coefficients gives

1 + 0 + (-3) = -2

.

Step-by-Step Solution

1

Factor the numerator using the difference of cubes formula

x^6 - 27 = (x^2)^3 - 3^3 = (x^2 - 3)(x^4 + 3x^2 + 9)

The expression

x^6 - 27

matches the pattern

u^3 - v^3

with

u = x^2

and

v = 3

.

2

Simplify the rational expression by canceling the common quadratic-biquadratic factor

\frac{(x^2 - 3)(x^4 + 3x^2 + 9)}{x^4 + 3x^2 + 9} = x^2 - 3

Since

x^4 + 3x^2 + 9 > 0

for all real numbers

x

, the denominator is never zero, allowing direct cancellation of the common factor.

3

Match coefficients with

ax^2 + bx + c

and calculate

a + b + c

a = 1, b = 0, c = -3 \implies a + b + c = 1 + 0 + (-3) = -2

Comparing

x^2 - 3 = 1x^2 + 0x - 3

to

ax^2 + bx + c

determines the values of constants

a

,

b

, and

c

.

Key Concept

Difference of Cubes Factoring Identity

Estimated Time:1m 30s

Question 2103Question →

A liquid chemical solution is composed of Chemical A, Chemical B, and Water in the volume ratio $2 : 3 : 5$ , respectively. If $40\text{ liters}$ of Chemical A and $20\text{ liters}$ of Water are added to the mixture, the ratio of Chemical A to Water becomes $1 : 2$ . What was the total volume, in liters, of the original solution?

120

200

300

420

600

Show answer & explanation

Answer: $600$

Answer

600\text{ liters}

Let the original volumes of Chemical A, Chemical B, and Water be

2x

,

3x

, and

5x

liters, respectively. The total volume of the original solution is

2x + 3x + 5x = 10x

liters. After adding

40\text{ liters}

of Chemical A and

20\text{ liters}

of Water, the ratio of Chemical A to Water is given as

1 : 2

. Setting up the proportion

\frac{2x + 40}{5x + 20} = \frac{1}{2}

and cross-multiplying yields

2(2x + 40) = 5x + 20

, which simplifies to

4x + 80 = 5x + 20

. Solving for

x

gives

x = 60

. Substituting

x = 60

into the total volume formula gives

10(60) = 600\text{ liters}

. Thus,

600

is the correct answer.

Step-by-Step Solution

1

Represent the initial volumes of each component in terms of a common multiplier

x

.

Chemical A

= 2x

, Chemical B

= 3x

, Water

= 5x

. Total volume

= 2x + 3x + 5x = 10x

.

Ratios

2 : 3 : 5

mean the actual quantities are proportional to these ratio parts.

2

Set up an equation based on the new volumes of Chemical A and Water.

\frac{2x + 40}{5x + 20} = \frac{1}{2}

Adding

40\text{ L}

of Chemical A and

20\text{ L}

of Water changes their ratio to

1 : 2

.

3

Cross-multiply and solve for

x

.

2(2x + 40) = 1(5x + 20) \implies 4x + 80 = 5x + 20 \implies x = 60.

Solving the linear equation yields the common ratio unit value.

4

Calculate the total original volume using

x = 60

.

\text{Total Volume} = 10x = 10(60) = 600\text{ liters}.

The total solution consists of

10

parts in total.

Key Concept

Three-part ratio algebra and proportion adjustments

Estimated Time:1m 30s

Question 2104Question →

An event production company initially allocates its total equipment budget among Stage Lights, Ambient LED Strips, and Spotlight Towers in the ratio $4 : 5 : 3$ , respectively. Due to venue price adjustments, the budget allocated to Stage Lights is increased by $25\%$ , while the budget allocated to Spotlight Towers is reduced by $20\%$ . If the total overall budget remains unchanged, what is the new ratio of the budget allocated to Stage Lights to Ambient LED Strips to Spotlight Towers?

25 : 23 : 12

25 : 25 : 12

25 : 17 : 12

35 : 22 : 3

5 : 4 : 3

Show answer & explanation

Answer: $25 : 23 : 12$

Answer

The new ratio of the budget allocated to Stage Lights to Ambient LED Strips to Spotlight Towers is

25 : 23 : 12

.

By setting the initial allocations to

4x

,

5x

, and

3x

, the total budget is

12x

. The new allocation for Stage Lights becomes

4x \times 1.25 = 5x

, and for Spotlight Towers it becomes

3x \times 0.80 = 2.4x

. To keep the total budget at

12x

, the Ambient LED Strips allocation must be

12x - 5x - 2.4x = 4.6x

. The ratio

5 : 4.6 : 2.4

scales up to

50 : 46 : 24

, which simplifies to

25 : 23 : 12

.

Step-by-Step Solution

1

Represent the initial component allocations and the total budget in terms of a common variable.

Let the initial allocations be Stage Lights

= 4x

, Ambient LED Strips

= 5x

, and Spotlight Towers

= 3x

. Total budget

= 4x + 5x + 3x = 12x

.

Establishing algebraic quantities allows precise tracking of individual changes relative to the constant total.

2

Calculate the updated allocations for Stage Lights and Spotlight Towers after their respective percentage changes.

New Stage Lights

= 4x \times (1 + 0.25) = 5x

.
New Spotlight Towers

= 3x \times (1 - 0.20) = 2.4x

.

Applying the specified percentage shifts directly to their corresponding original component values.

3

Determine the new allocation for Ambient LED Strips using the fixed total budget condition.

New Ambient LED Strips

= 12x - (5x + 2.4x) = 12x - 7.4x = 4.6x

.

Because the total overall budget remains unchanged at

12x

, the sum of all three new allocations must equal

12x

.

4

Write the new ratio and simplify to integer terms.

Ratio

= 5x : 4.6x : 2.4x = 5 : 4.6 : 2.4 = 50 : 46 : 24 = 25 : 23 : 12

.

Multiplying by

10

clears decimals, and dividing by

2

reduces the ratio to its simplest integer form.

Key Concept

Multi-part ratio adjustment with percentage changes under a constant total constraint

Estimated Time:1m 30s

Question 2105Question →

For all real numbers

x

and

y

such that

x \neq -y

and

x \neq -2y

, which of the following expressions is equivalent to

\frac{x^3 + 2x^2y - xy^2 - 2y^3}{x^2 + 3xy + 2y^2}?

x - y

x + y

x^2 - y^2

x^2 + y^2

x - 2y

Show answer & explanation

Answer: $x - y$

Answer

x - y

Factoring the numerator by grouping gives

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

. Factoring the denominator yields

(x + y)(x + 2y)

. Dividing the numerator by the denominator cancels the common factors

(x + y)

and

(x + 2y)

, leaving

x - y

.

Step-by-Step Solution

1

Factor the numerator by grouping terms

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

Grouping pairs of terms allows factoring out common binomial factors.

2

Factor the quadratic denominator

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

Finding two terms whose sum is

3y

and product is

2y^2

factors the quadratic in

x

.

3

Simplify the rational expression by canceling non-zero common factors

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

Since

x \neq -y

and

x \neq -2y

, the factors

(x + y)

and

(x + 2y)

are non-zero and can be canceled.

Key Concept

Polynomial factoring by grouping, difference of squares, quadratic trinomial factoring, and simplifying rational algebraic expressions.

Estimated Time:1m 30s

Question 2106Question →

The length of a rectangular plot of land is $3$ meters less than twice its width. If the area of the plot is $90$ square meters, what is the perimeter of the plot, in meters?

Show answer & explanation

Answer: 39

Answer

The perimeter of the plot of land is 39 meters.

Setting the length to

2w - 3

gives an area equation of

w(2w - 3) = 90

, which expands and rearranges to

2w^2 - 3w - 90 = 0

. Factoring this quadratic equation yields

(2w - 15)(w + 6) = 0

. Since width must be positive,

w = 7.5

meters, which means the length is

12

meters. The perimeter is

2(12 + 7.5) = 39

meters.

Step-by-Step Solution

1

Express the length in terms of width and set up the area equation.

Let

w

be the width of the rectangle. Length

l = 2w - 3

. Area equation:

w(2w - 3) = 90

.

The area of a rectangle is equal to length multiplied by width.

2

Rearrange into standard quadratic form

ax^2 + bx + c = 0

.

2w^2 - 3w - 90 = 0

Distributing

w

and subtracting

90

from both sides puts the equation in standard quadratic form.

3

Factor the quadratic equation.

(2w - 15)(w + 6) = 0

Finding two numbers with a product of

2 \times (-90) = -180

and a sum of

-3

gives

-15

and

12

.

4

Determine the valid physical dimensions.

w = 7.5

meters and

l = 12

meters.

The root

w = -6

is discarded because physical length cannot be negative. Thus

w = \frac{15}{2} = 7.5

meters.

5

Calculate the perimeter.

Perimeter

= 2(l + w) = 2(12 + 7.5) = 39

meters.

The perimeter of a rectangle is given by

2 \times (\text{length} + \text{width})

.

Key Concept

Solving quadratic word problems via factoring

Estimated Time:1m 30s

Question 2107Question →

A coffee roaster blends Arabica, Robusta, and Liberica beans in the ratio $4 : 3 : 2$ by weight to prepare a master batch. To adjust the flavor profile, the roaster adds $15\text{ kg}$ of Robusta beans and $15\text{ kg}$ of Liberica beans to the batch without changing the amount of Arabica beans. In the modified batch, Arabica beans constitute exactly $\frac{1}{3}$ of the total weight. Which of the following statements must be true? Select all such statements.

Select all that apply

The initial weight of Arabica beans in the master batch was

40\text{ kg}

.

In the modified batch, the ratio of Robusta beans to Liberica beans is

9 : 7

.

In the initial master batch, Robusta beans constituted

\frac{1}{3}

of the total weight.

The total weight of the batch increased by

25\%

after the additions.

In the modified batch, Liberica beans account for

35\%

of the total weight.

Show answer & explanation

Answer: The initial weight of Arabica beans in the master batch was $40\text{ kg}$ .; In the modified batch, the ratio of Robusta beans to Liberica beans is $9 : 7$ .; In the initial master batch, Robusta beans constituted $\frac{1}{3}$ of the total weight.

Answer

The correct statements are those asserting that the initial weight of Arabica beans was 40 kg, that the modified ratio of Robusta to Liberica beans is 9 : 7, and that Robusta beans constituted 1/3 of the total weight in the initial batch.

Solving the proportion equation reveals that the scaling factor for the initial ratio is 10, giving an initial batch weight of 90 kg with 40 kg Arabica, 30 kg Robusta, and 20 kg Liberica. Consequently, Arabica originally weighed 40 kg, Robusta originally made up 30/90 = 1/3 of the batch, and the new Robusta-to-Liberica ratio is 45 : 35 = 9 : 7. All three of these statements are mathematically true.

Step-by-Step Solution

1

Represent initial component weights and total weight using a multiplier variable.

For ratio

4 : 3 : 2

, Arabica =

4x

, Robusta =

3x

, Liberica =

2x

, and initial total weight

W = 9x

.

Ratios define proportional components in terms of a common scalar variable.

2

Set up the equation for the modified batch based on the new total and Arabica ratio.

Added weight =

15 + 15 = 30\text{ kg}

. New total weight =

9x + 30

. Arabica weight remains

4x

. Since Arabica is

\frac{1}{3}

of the new total:

\frac{4x}{9x + 30} = \frac{1}{3} \implies 12x = 9x + 30 \implies 3x = 30 \implies x = 10

.

Equating the unchanged part to the given fraction of the new total allows solving for the scale factor.

3

Calculate all initial and modified quantities.

Initial total

W = 90\text{ kg}

. Initial Arabica =

40\text{ kg}

, Robusta =

30\text{ kg}

, Liberica =

20\text{ kg}

. Modified total =

120\text{ kg}

. Modified Robusta =

45\text{ kg}

, Modified Liberica =

35\text{ kg}

.

Determining exact values enables verification of each statement.

4

Evaluate each given statement.

Initial Arabica =

40\text{ kg}

(True). Modified Robusta : Liberica =

45 : 35 = 9 : 7

(True). Initial Robusta fraction =

\frac{30}{90} = \frac{1}{3}

(True). Percent weight increase =

\frac{30}{90} = 33.33\%

(False). Modified Liberica percentage =

\frac{35}{120} = 29.17\%

(False).

Direct comparison of calculated quantities with statement claims identifies all correct choices.

Key Concept

Ratio scale factors and part-to-whole proportions under quantity modifications

Estimated Time:2m 0s

Question 2108Question →

For all real numbers $x$ such that $x \neq -2$ , $x \neq 2$ , and $x \neq 4$ , which of the following expressions are equivalent to $\frac{x^4 - 16x^2}{x^3 - 4x^2 - 4x + 16}$ ? Select all such expressions.

Select all that apply

\frac{x^3 + 4x^2}{x^2 - 4}

\frac{x^2(x + 4)}{(x - 2)(x + 2)}

\frac{x^2(x - 4)}{x^2 - 4}

\frac{x^2 + 4x}{x - 2}

\frac{x^4 - 4x^2}{x^3 - 4}

Show answer & explanation

Answer: $\frac{x^3 + 4x^2}{x^2 - 4}$ ; $\frac{x^2(x + 4)}{(x - 2)(x + 2)}$

Answer

The equivalent expressions are

\frac{x^3 + 4x^2}{x^2 - 4}

and

\frac{x^2(x + 4)}{(x - 2)(x + 2)}

.

The given expression factors as

\frac{x^2(x - 4)(x + 4)}{(x^2 - 4)(x - 4)}

. Canceling

(x - 4)

leaves

\frac{x^2(x + 4)}{x^2 - 4}

. Expanding the numerator gives

\frac{x^3 + 4x^2}{x^2 - 4}

, and factoring the denominator further gives

\frac{x^2(x + 4)}{(x - 2)(x + 2)}

. Both represent valid equivalent forms of the expression.

Step-by-Step Solution

1

Factor the numerator of the given algebraic expression.

x^4 - 16x^2 = x^2(x^2 - 16) = x^2(x - 4)(x + 4)

Factor out the greatest common factor

x^2

, then apply the difference of squares identity

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

.

2

Factor the denominator by grouping terms.

x^3 - 4x^2 - 4x + 16 = x^2(x - 4) - 4(x - 4) = (x^2 - 4)(x - 4) = (x - 2)(x + 2)(x - 4)

Group the first two terms and last two terms, factor out common binomials, and expand the remaining difference of squares.

3

Simplify the full rational expression by canceling common factors.

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

Since

x \neq 4

, cancel the common factor

(x - 4)

from both numerator and denominator.

4

Compare the simplified form to the given choices to identify all equivalent expressions.

Both

\frac{x^3 + 4x^2}{x^2 - 4}

and

\frac{x^2(x + 4)}{(x - 2)(x + 2)}

match the simplified algebraic forms.

Expanding the numerator or factoring the denominator yields these two equivalent representations.

Key Concept

Simplifying rational algebraic expressions by polynomial factoring and term grouping.

Question 2109Question →

If $r_1$ and $r_2$ are the two distinct real solutions to the quadratic equation $x^2 - 4x - 21 = 0$ , where $r_1 > r_2$ , what is the value of $r_1 + 2r_2$ ?

1

13

-11

-5

17

Show answer & explanation

Answer: $1$

Answer

The correct value of

r_1 + 2r_2

is

1

.

Factoring the quadratic polynomial

x^2 - 4x - 21

gives

(x - 7)(x + 3) = 0

, which yields solutions

x = 7

and

x = -3

. Applying the constraint

r_1 > r_2

identifies

r_1 = 7

and

r_2 = -3

. Substituting these values into

r_1 + 2r_2

yields

7 + 2(-3) = 7 - 6 = 1

.

Step-by-Step Solution

1

Factor the quadratic equation

(x - 7)(x + 3) = 0

Find two numbers that multiply to

-21

and add up to

-4

, which are

-7

and

3

.

2

Solve for the roots of the equation

x = 7

or

x = -3

Set each linear factor equal to zero:

x - 7 = 0 \implies x = 7

and

x + 3 = 0 \implies x = -3

.

3

Assign values to

r_1

and

r_2

based on the condition

r_1 > r_2

r_1 = 7

and

r_2 = -3

Since

7 > -3

,

r_1

must be

7

and

r_2

must be

-3

.

4

Evaluate the expression

r_1 + 2r_2

7 + 2(-3) = 7 - 6 = 1

Substitute

r_1 = 7

and

r_2 = -3

into the given expression.

Key Concept

Solving quadratic equations by factoring and evaluating expressions involving roots.

Estimated Time:1m 30s

Question 2110Question →

Pumps $A$ and $B$ work independently at their respective constant rates to empty a water storage reservoir. Working together, Pump $A$ and Pump $B$ can empty the reservoir in $12$ hours. If Pump $A$ 's rate of pumping is $1.5$ times the rate of Pump $B$ , how many hours would it take Pump $B$ working alone at its constant rate to empty the entire reservoir?

Show answer & explanation

Answer: 30

Answer

It would take Pump B working alone 30 hours to empty the entire reservoir.

Since Pump A's rate is

1.5

times Pump B's rate, their combined rate is

1.5 r_B + r_B = 2.5 r_B

reservoirs per hour. Working together for

12

hours empties

12 \times 2.5 r_B = 30 r_B

reservoirs. Setting

30 r_B = 1

reservoir yields

r_B = \frac{1}{30}

reservoir per hour. Therefore, Pump B working alone requires

30

hours to empty the reservoir.

Step-by-Step Solution

1

Express Pump A's rate in terms of Pump B's rate.

Let

r_B

be the rate of Pump B in reservoirs per hour. Then the rate of Pump A is

r_A = 1.5 r_B

.

Expressing both rates in terms of a single variable simplifies setting up the combined rate equation.

2

Calculate the combined pumping rate.

Combined rate =

r_A + r_B = 1.5 r_B + r_B = 2.5 r_B

.

When two machines work simultaneously, their individual rates add together.

3

Set up the work equation using the given combined time of 12 hours.

\text{Total Work} = \text{Combined Rate} \times 12 = 2.5 r_B \times 12 = 30 r_B = 1 \text{ reservoir}

.

The total work performed equals the combined rate multiplied by the total time taken.

4

Solve for the time required for Pump B to complete the job alone.

\text{Time}_B = \frac{1}{r_B} = 30 \text{ hours}

.

The time needed for Pump B alone to empty

1

reservoir at rate

r_B

is the reciprocal of

r_B

.

Key Concept

Combined Work Rates and Rate Ratios

Alternative Method

Choose a convenient hypothetical capacity for the reservoir. Let Pump B's rate be

2

units per hour. Then Pump A's rate is

1.5 \times 2 = 3

units per hour. Working together, they empty

2 + 3 = 5

units per hour. In

12

hours, the reservoir holds

5 \times 12 = 60

units. Pump B alone at

2

units per hour takes

60 / 2 = 30

hours.

Estimated Time:1m 30s

Question 2111Question →

For all real numbers

x

and

y

such that

x \neq y

, which of the following expressions is equivalent to

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

?

2xy - 2y^2

2xy

-2y^2

2x^2 - 2xy

0

Show answer & explanation

Answer: $2xy - 2y^2$

Answer

The equivalent expression is

2xy - 2y^2

.

Factoring the numerator by grouping gives

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

. Dividing by

(x - y)

yields

x^2 - y^2

. Expanding

(x - y)^2

gives

x^2 - 2xy + y^2

. Subtracting this expansion from

x^2 - y^2

results in

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

.

Step-by-Step Solution

1

Factor the numerator of the rational fraction by grouping terms

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

Grouping terms allows factoring out common binomial components.

2

Simplify the fraction by dividing by

(x - y)

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

Since

x \neq y

,

x - y \neq 0

, so common factors can be canceled.

3

Expand the subtracted term

(x - y)^2

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

Applying the standard perfect square trinomial expansion formula.

4

Subtract the expanded expression from the simplified fraction result

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

Distribute the negative sign across all terms inside the parentheses and combine like terms.

Key Concept

Simplifying algebraic expressions using factoring by grouping, difference of squares, and expanding binomial squares.

Question 2112Question →

A delivery truck completes a trip consisting of two consecutive legs. The ratio of the distance of the first leg to the distance of the second leg is $3 : 1$ . The ratio of the truck's constant speed on the first leg to its constant speed on the second leg is $3 : 2$ . If the average speed of the truck for the entire trip is $40\text{ miles per hour}$ , what was the constant speed of the truck on the second leg, in miles per hour?

16\text{ mph}

25\text{ mph}

30\text{ mph}

32\text{ mph}

45\text{ mph}

Show answer & explanation

Answer: $30\text{ mph}$

Answer

30\text{ mph}

The total distance traveled is

3d + d = 4d

. The time required for the first leg is

\frac{3d}{3v} = \frac{d}{v}

, and the time for the second leg is

\frac{d}{2v}

. Combining these yields a total time of

\frac{3d}{2v}

. Setting total distance over total time equal to the given average speed gives

\frac{4d}{\frac{3d}{2v}} = \frac{8v}{3} = 40\text{ mph}

, which yields

v = 15

. Consequently, the constant speed on the second leg is

2v = 30\text{ mph}

.

Step-by-Step Solution

1

Represent the distances and speeds of both legs using ratio multipliers

Distance leg 1 =

3d

, distance leg 2 =

d

, total distance =

4d

. Speed leg 1 =

3v

, speed leg 2 =

2v

.

Ratios

3:1

for distance and

3:2

for speed allow expressed quantities in terms of single parameters

d

and

v

.

2

Calculate the travel time for each leg using time = distance / speed

Time for leg 1:

t_1 = \frac{3d}{3v} = \frac{d}{v}

. Time for leg 2:

t_2 = \frac{d}{2v}

.

Average speed depends on total distance divided by total time.

3

Compute total time for the entire trip

T_{\text{total}} = t_1 + t_2 = \frac{d}{v} + \frac{d}{2v} = \frac{3d}{2v}

.

Finding a common denominator of

2v

allows combining the two time fractions.

4

Set up the average speed equation and solve for multiplier v

\text{Average Speed} = \frac{4d}{\frac{3d}{2v}} = \frac{8v}{3} = 40 \implies v = 15.

Dividing total distance by total time gives the average speed expression.

5

Calculate the speed of the second leg

Speed of leg 2 =

2v = 2(15) = 30\text{ mph}

.

The question specifically asks for the constant speed on the second leg.

Key Concept

Average rate over multi-leg trips is total distance divided by total time, not the simple arithmetic mean of speeds.

Estimated Time:1m 30s

Question 2113Question →

A civil engineer creates a scale drawing of a rectangular park using a scale where $2.5\text{ inches}$ on the drawing represents an actual distance of $12\text{ feet}$ . On this initial blueprint, the park has a length of $15\text{ inches}$ and a width of $10\text{ inches}$ . If the engineer creates a second blueprint of the same park using a scale where $3\text{ inches}$ represents an actual distance of $16\text{ feet}$ , what is the perimeter, in inches, of the park on the second blueprint?

22.5\text{ inches}

37.5\text{ inches}

40.5\text{ inches}

45\text{ inches}

60\text{ inches}

Show answer & explanation

Answer: $45\text{ inches}$

Answer

45 inches

The correct answer is

45\text{ inches}

. The perimeter on the first blueprint is

2 \times (15 + 10) = 50\text{ inches}

. Since

2.5\text{ inches}

represents

12\text{ feet}

, each inch on the first drawing represents

\frac{12}{2.5} = 4.8\text{ feet}

. Thus, the actual perimeter of the park is

50 \times 4.8 = 240\text{ feet}

. On the second blueprint,

16\text{ feet}

is represented by

3\text{ inches}

, meaning

1\text{ foot}

is represented by

\frac{3}{16}\text{ inches}

. Therefore, the perimeter on the second drawing is

240 \times \frac{3}{16} = 45\text{ inches}

.

Step-by-Step Solution

1

Calculate the perimeter of the park on the first blueprint.

Perimeter on first blueprint =

2 \times (15 + 10) = 50\text{ inches}

.

The perimeter of a rectangle is calculated as

2 \times (\text{length} + \text{width})

.

2

Determine the actual perimeter of the park in feet using the first scale (

2.5\text{ inches} = 12\text{ feet}

).

Actual perimeter =

50\text{ inches} \times \left(\frac{12\text{ feet}}{2.5\text{ inches}}\right) = 240\text{ feet}

.

Multiplying by the ratio of actual feet per blueprint inch gives the true physical perimeter.

3

Convert the actual perimeter to inches on the second blueprint using the second scale (

3\text{ inches} = 16\text{ feet}

).

Perimeter on second blueprint =

240\text{ feet} \times \left(\frac{3\text{ inches}}{16\text{ feet}}\right) = 45\text{ inches}

.

Multiplying the actual distance in feet by the second blueprint scale ratio yields the required blueprint perimeter.

Key Concept

Scale ratios and linear proportional conversions

Estimated Time:1m 30s

Question 2114Question →

If $m$ and $n$ are the two distinct real solutions to the quadratic equation $2x^2 - 8x + c = 0$ , where $c$ is a constant, and $m^2 + n^2 = 12$ , what is the value of $c$ ?

-4

4

8

2

26

Show answer & explanation

Answer: 4

Answer

The value of

c

is

4

.

By Vieta's formulas for the quadratic equation

2x^2 - 8x + c = 0

, the sum of the roots is

m + n = 4

and the product of the roots is

mn = c/2

. Squaring the sum gives

(m + n)^2 = m^2 + 2mn + n^2 = 16

. Substituting

m^2 + n^2 = 12

gives

12 + 2(c/2) = 16

, which simplifies to

12 + c = 16

, so

c = 4

.

Step-by-Step Solution

1

Apply Vieta's formulas to determine the sum and product of the roots in terms of the given quadratic coefficients.

For

2x^2 - 8x + c = 0

, the sum of roots is

m + n = -\frac{-8}{2} = 4

, and the product of roots is

m \cdot n = \frac{c}{2}

.

For a quadratic equation

ax^2 + bx + c = 0

, the sum of roots is

-b/a

and the product of roots is

c/a

.

2

Relate

m^2 + n^2

to the sum and product of roots using the identity

(m + n)^2 = m^2 + n^2 + 2mn

.

m^2 + n^2 = (m + n)^2 - 2mn

Rewriting the sum of squares in terms of

(m+n)

and

mn

allows substitution of the known values.

3

Substitute

m + n = 4

,

mn = \frac{c}{2}

, and

m^2 + n^2 = 12

into the equation and solve for

c

.

12 = (4)^2 - 2\left(\frac{c}{2}\right) \implies 12 = 16 - c \implies c = 4

.

Direct algebraic simplification yields the constant

c

.

Key Concept

Relationship between roots and coefficients of a quadratic equation (Vieta's Formulas) and algebraic identities.

Question 2115Question →

For all real numbers

x

such that

x \neq -3

,

x \neq 0

, and

x \neq 3

, which of the following expressions is equivalent to

\frac{x^4 - 81}{x^3 + 3x^2} \div \frac{x^2 + 9}{x^2}

?

x - 3

x + 3

x^2 - 3

\frac{x - 3}{x}

x - 9

Show answer & explanation

Answer: $x - 3$

Answer

x - 3

Factoring the numerator

x^4 - 81

gives

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

and factoring the denominator

x^3 + 3x^2

gives

x^2(x + 3)

. Multiplying by the reciprocal

\frac{x^2}{x^2 + 9}

allows the factors

(x + 3)

,

(x^2 + 9)

, and

x^2

to cancel completely, leaving

x - 3

.

Step-by-Step Solution

1

Rewrite division of rational expressions as multiplication by the reciprocal

\frac{x^4 - 81}{x^3 + 3x^2} \cdot \frac{x^2}{x^2 + 9}

Dividing by a fraction is equivalent to multiplying by its reciprocal.

2

Factor the numerator

x^4 - 81

using the difference of squares formula twice

x^4 - 81 = (x^2 - 9)(x^2 + 9) = (x - 3)(x + 3)(x^2 + 9)

Recognize

x^4 - 81

as

(x^2)^2 - 9^2

and then factor

x^2 - 9

further into

(x - 3)(x + 3)

.

3

Factor out the greatest common factor from the denominator

x^3 + 3x^2

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

Extract

x^2

from both terms of the binomial.

4

Substitute the factored expressions into the product and cancel common factors

\frac{(x - 3)(x + 3)(x^2 + 9)}{x^2(x + 3)} \cdot \frac{x^2}{x^2 + 9} = x - 3

The common terms

(x + 3)

,

(x^2 + 9)

, and

x^2

cancel out completely from the numerator and denominator for all

x \neq -3, 0, 3

.

Key Concept

Factoring polynomial expressions using difference of squares and common terms to simplify rational expressions.

Estimated Time:1m 30s

Question 2116Question →

For what positive value of the constant $k$ does the quadratic equation $x^2 - 2kx + 3k + 10 = 0$ have exactly one real solution?

Show answer & explanation

Answer: 5

Answer

The positive value of

k

is 5.

A quadratic equation has exactly one real solution when its discriminant

b^2 - 4ac

equals

0

. For

x^2 - 2kx + (3k + 10) = 0

, substituting

a = 1

,

b = -2k

, and

c = 3k + 10

gives

(-2k)^2 - 4(1)(3k + 10) = 0

, which simplifies to

4k^2 - 12k - 40 = 0

. Dividing the entire equation by

4

gives

k^2 - 3k - 10 = 0

. Factoring this quadratic yields

(k - 5)(k + 2) = 0

, giving solutions

k = 5

and

k = -2

. Since the question requires the positive value of

k

, the correct answer is

5

.

Step-by-Step Solution

1

Identify the condition for a quadratic equation to have exactly one real solution.

The discriminant must equal zero (

b^2 - 4ac = 0

).

A quadratic equation

ax^2 + bx + c = 0

has a single repeated real root if and only if its discriminant is zero.

2

Set up the discriminant equation using the coefficients from

x^2 - 2kx + (3k + 10) = 0

.

Discriminant

= (-2k)^2 - 4(1)(3k + 10) = 4k^2 - 12k - 40 = 0

.

Here

a = 1

,

b = -2k

, and

c = 3k + 10

.

3

Divide by 4 and solve the quadratic in terms of

k

by factoring.

k^2 - 3k - 10 = 0 \Rightarrow (k - 5)(k + 2) = 0

, so

k = 5

or

k = -2

.

Factoring the simplified quadratic yields the two potential values of

k

.

4

Choose the value of

k

that satisfies the constraint in the stem.

k = 5

.

The question asks specifically for the positive value of

k

, so

k = -2

is rejected.

Key Concept

Quadratic Discriminant and Factoring

Estimated Time:1m 30s

Question 2117Question →

An industrial bakery uses two commercial mixers, Mixer X and Mixer Y, to prepare large batches of dough. Working continuously at its own constant rate, Mixer X can complete 1 full batch of dough in 4 hours. Mixer Y, working continuously at its own constant rate, can complete 1 full batch of dough in 6 hours.

If Mixer X works alone for 1 hour and is then joined by Mixer Y to complete the rest of the batch together, which of the following statements must be true? Select all such statements.

Select all that apply

The remaining

\frac{3}{4}

of the batch takes 1 hour and 48 minutes for both mixers working together to complete.

Mixer X completes

70\%

of the total batch of dough.

The ratio of the total amount of dough prepared by Mixer X to that prepared by Mixer Y is

7 : 3

.

Both mixers working together complete the remaining

\frac{3}{4}

of the batch in 3 hours and 45 minutes.

Mixer Y completes

40\%

of the total batch of dough.

Show answer & explanation

Answer: The remaining $\frac{3}{4}$ of the batch takes 1 hour and 48 minutes for both mixers working together to complete.; Mixer X completes $70\%$ of the total batch of dough.; The ratio of the total amount of dough prepared by Mixer X to that prepared by Mixer Y is $7 : 3$ .

Answer

The correct statements are those indicating that the remaining work takes 1 hour and 48 minutes, Mixer X completes 70% of the batch, and the ratio of work done by Mixer X to Mixer Y is 7 : 3.

The rate of Mixer X is 1/4 batch per hour and Mixer Y is 1/6 batch per hour. In the first hour, Mixer X completes 1/4 of the batch, leaving 3/4 of the batch. Working together, their combined rate is 1/4 + 1/6 = 5/12 batch per hour. The time needed for the remaining 3/4 batch is (3/4) / (5/12) = 1.8 hours, which equals 1 hour and 48 minutes. Mixer X works for a total of 2.8 hours, completing 2.8 / 4 = 0.7 (or 70%) of the batch. Mixer Y works for 1.8 hours, completing 1.8 / 6 = 0.3 (or 30%) of the batch. The ratio of dough prepared by Mixer X to Mixer Y is 0.7 : 0.3 = 7 : 3. Therefore, the statements asserting 1 hour and 48 minutes remaining time, 70% completed by Mixer X, and a 7 : 3 work ratio are all correct.

Step-by-Step Solution

1

Determine individual work rates and initial work done

Mixer X rate =

\frac{1}{4}

batch/hr, Mixer Y rate =

\frac{1}{6}

batch/hr. In 1 hour, Mixer X completes

\frac{1}{4}

of the batch, leaving

1 - \frac{1}{4} = \frac{3}{4}

of the batch remaining.

Establishing the individual rates allows calculation of remaining work after Mixer X operates alone.

2

Calculate combined rate and time to finish remaining work

Combined rate =

\frac{1}{4} + \frac{1}{6} = \frac{3+2}{12} = \frac{5}{12}

batch/hr. Time required =

\frac{3/4}{5/12} = \frac{3}{4} \times \frac{12}{5} = \frac{9}{5} = 1.8

hours = 1 hour 48 minutes.

When both mixers operate together, their rates add.

3

Calculate total work contributions of each mixer

Mixer X total time =

1 + 1.8 = 2.8

hours

\rightarrow

Work by X =

2.8 \times \frac{1}{4} = 0.7

(

70\%

). Mixer Y total time =

1.8

hours

\rightarrow

Work by Y =

1.8 \times \frac{1}{6} = 0.3

(

30\%

). Work ratio X : Y =

0.7 : 0.3 = 7 : 3

.

Multiplying each mixer's active operating time by its rate determines its share of the total batch.

Key Concept

Combined Work Rates and Ratio of Work Completed

Question 2118Question →

For all real numbers $x$ such that $x \neq -2$ , which of the following expressions are equivalent to $\frac{x^6 - 64}{x^3 + 8}$ ? Select all such expressions.

Select all that apply

x^3 - 8

(x - 2)(x^2 + 2x + 4)

(x - 2)^3

x^3 + 8

x^2 - 8

Show answer & explanation

Answer: $x^3 - 8$ ; $(x - 2)(x^2 + 2x + 4)$

Answer

The expressions equivalent to the given rational expression are

x^3 - 8

and

(x - 2)(x^2 + 2x + 4)

.

Factoring the numerator

x^6 - 64

as a difference of squares yields

(x^3 - 8)(x^3 + 8)

. Dividing by the denominator

(x^3 + 8)

leaves

x^3 - 8

. Further factoring

x^3 - 8

as a difference of cubes gives

(x - 2)(x^2 + 2x + 4)

. Both

x^3 - 8

and

(x - 2)(x^2 + 2x + 4)

are valid equivalent expressions.

Step-by-Step Solution

1

Factor the numerator using the difference of squares identity

x^6 - 64 = (x^3)^2 - 8^2 = (x^3 - 8)(x^3 + 8)

The expression

x^6 - 64

is a difference of two squares.

2

Simplify the fraction by canceling the common non-zero term

(x^3 + 8)

\frac{(x^3 - 8)(x^3 + 8)}{x^3 + 8} = x^3 - 8

Since

x \neq -2

,

x^3 + 8 \neq 0

, so

(x^3 + 8)

can be canceled from numerator and denominator.

3

Factor

x^3 - 8

using the difference of cubes identity

x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)

Applying

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

with

a = x

and

b = 2

.

Key Concept

Simplifying rational expressions by factoring polynomial numerators and denominators using algebraic identities such as difference of squares and difference of cubes.

Question 2119Question →

A laboratory technician prepares a chemical mixture containing water, ethanol, and acid in the volume ratio $5 : 3 : 2$ , respectively. After $12\text{ liters}$ of pure ethanol are added to the mixture, the ratio of the volume of water to the volume of ethanol becomes $1 : 1$ . What was the total volume, in liters, of the chemical mixture before the extra ethanol was added?

60\text{ liters}

48\text{ liters}

72\text{ liters}

18\text{ liters}

30\text{ liters}

Show answer & explanation

Answer: $60\text{ liters}$

Answer

60\text{ liters}

The initial ratio of water, ethanol, and acid is

5 : 3 : 2

, which means the original amounts can be expressed as

5x

,

3x

, and

2x

liters, respectively, giving a total initial volume of

10x

liters. Adding

12\text{ liters}

of ethanol increases the ethanol amount to

3x + 12

, while water remains

5x

. Since the new ratio of water to ethanol is

1 : 1

, we set

5x = 3x + 12

, which solves to

2x = 12

and

x = 6

. The original total volume is therefore

10(6) = 60\text{ liters}

.

Step-by-Step Solution

1

Define initial volumes using a common ratio multiplier

x

Water volume

= 5x

, Ethanol volume

= 3x

, Acid volume

= 2x

, and Total initial volume

= 5x + 3x + 2x = 10x

.

Ratios represent relative parts of a whole, so multiplying each term by

x

gives the actual quantities.

2

Express the new volumes after adding

12\text{ liters}

of pure ethanol

New Ethanol volume

= 3x + 12

, while Water volume remains

5x

.

Ethanol is increased by

12\text{ liters}

, whereas the amount of water remains unchanged.

3

Set up an equation using the new water-to-ethanol ratio of

1 : 1

5x = 3x + 12 \implies 2x = 12 \implies x = 6

.

A ratio of

1 : 1

means the volumes of water and ethanol are equal.

4

Calculate the original total volume of the mixture

Total initial volume

= 10x = 10(6) = 60\text{ liters}

.

Substituting

x = 6

back into the original total volume expression

10x

yields the required initial volume.

Key Concept

Multi-component ratio modification and proportional equivalence

Estimated Time:1m 30s

Question 2120Question →

If $x$ is a real number that satisfies the equation $x^4 - 13x^2 + 36 = 0$ , which of the following values could be the value of $x$ ? Select all such values.

Select all that apply

-3

-2

3

4

9

Show answer & explanation

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

Answer

The correct values are

-3

,

-2

, and

3

.

Factoring the equation yields

(x^2 - 4)(x^2 - 9) = (x - 2)(x + 2)(x - 3)(x + 3) = 0

, giving the four distinct real solutions

x = -3, -2, 2, 3

. The options corresponding to

-3

,

-2

, and

3

represent valid values of

x

.

Step-by-Step Solution

1

Substitute a variable

u = x^2

to express the equation in quadratic form.

u^2 - 13u + 36 = 0

This reduces the fourth-degree polynomial into a standard quadratic equation.

2

Factor the quadratic equation to solve for

u

.

(u - 4)(u - 9) = 0 \implies u = 4 \text{ or } u = 9

Factoring determines the values of

x^2

.

3

Substitute back

x^2 = u

and solve for

x

by taking both positive and negative square roots.

x^2 = 4 \implies x = \pm 2

, and

x^2 = 9 \implies x = \pm 3

Each positive value of

u

yields two real solutions for

x

.

Key Concept

Solving quadratic-form equations by factoring and taking positive and negative square roots.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution