Elementary Algebra Practice Questions — ACT — Page 11

If the expression

\frac{(x^3 y^2)^3 \cdot (x^{-2} y^{-1})^2}{x^k y^3}

is equivalent to $x^2 y$ for all non-zero real numbers $x$ and $y$ , what is the value of the exponent $k$ ?

Show answer & explanation

Answer: 3

Answer

The value of the exponent

k

is 3.

Applying exponent rules simplifies the numerator of the expression to

x^5 y^4

. Dividing this by the denominator

x^k y^3

yields

x^{5-k} y

. Equating the exponent of

x

to 2 in the target expression

x^2 y

gives

5 - k = 2

, which solves to

k = 3

.

Step-by-Step Solution

1

Apply the power of a power rule

(a^m)^n = a^{m \cdot n}

to simplify each factor in the numerator.

(x^3 y^2)^3 = x^9 y^6

and

(x^{-2} y^{-1})^2 = x^{-4} y^{-2}

To raise a product to a power, raise each factor to that power by multiplying the exponents.

2

Multiply the two simplified factors in the numerator together by adding the exponents of like bases.

x^9 y^6 \cdot x^{-4} y^{-2} = x^5 y^4

When multiplying exponential expressions with the same base, add their exponents:

a^m \cdot a^n = a^{m+n}

.

3

Divide the numerator by the denominator by subtracting the exponents of like bases.

\frac{x^5 y^4}{x^k y^3} = x^{5-k} y

When dividing exponential expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator:

\frac{a^m}{a^n} = a^{m-n}

.

4

Set the exponent of

x

in the simplified expression equal to the exponent of

x

in the target expression

x^2 y

, and solve for

k

.

5 - k = 2 \implies k = 3

For the expressions to be equivalent for all non-zero real numbers, the corresponding exponents of like bases must be equal.

Key Concept

Properties of exponents (power of a power, product of powers, and quotient of powers rules)

Estimated Time:1m 30s

Question 202Question →

A manufacturer calculates the net monthly operating profit $P$ (in thousands of dollars) using the expression $\frac{8 - 3x}{2}$ , where $x$ is the number of maintenance hours scheduled. The auxiliary support costs are modeled by the expression $\frac{2x - 1}{3}$ thousand dollars. For the plant to be considered efficient, the net operating profit minus the auxiliary support costs must be at least $3$ thousand dollars. Which of the following inequalities represents the range of maintenance hours, $x$ , that satisfy this efficiency requirement?

x \leq \frac{4}{13}

x \geq \frac{8}{13}

x \leq \frac{8}{13}

x \leq \frac{4}{5}

x \geq \frac{12}{5}

Show answer & explanation

Answer: $x \leq \frac{8}{13}$

Answer

The range of maintenance hours that satisfy the efficiency requirement is

x \leq \frac{8}{13}

.

First, translate the word problem into a mathematical inequality. The net monthly operating profit minus the auxiliary support costs must be at least 3, which translates to:

\frac{8 - 3x}{2} - \frac{2x - 1}{3} \geq 3

To solve this inequality, find the least common denominator (LCD) of 2 and 3, which is 6. Multiply every term in the inequality by 6:

3(8 - 3x) - 2(2x - 1) \geq 18

Distribute the constants on the left side, paying careful attention to distribute the negative sign:

24 - 9x - 4x + 2 \geq 18

Combine like terms:

26 - 13x \geq 18

Subtract 26 from both sides:

-13x \geq -8

Divide both sides by -13. Because we are dividing by a negative number, the direction of the inequality sign must be reversed:

x \leq \frac{8}{13}

Thus, the option stating

x \leq \frac{8}{13}

is correct.

Step-by-Step Solution

1

Translate the word problem into a mathematical inequality.

\frac{8 - 3x}{2} - \frac{2x - 1}{3} \geq 3

The net operating profit minus the auxiliary support costs must be at least 3, which translates to a subtraction operation set greater than or equal to 3.

2

Multiply all terms in the inequality by the least common denominator (LCD) of 2 and 3, which is 6.

3(8 - 3x) - 2(2x - 1) \geq 18

Multiplying by the LCD eliminates the denominators and simplifies the equation for algebraic manipulation.

3

Distribute the constants on the left side of the inequality.

24 - 9x - 4x + 2 \geq 18

Removing the parentheses is required to combine like terms. Be careful to distribute the negative sign:

-2 \times -1 = +2

.

4

Combine like terms on the left side of the inequality.

26 - 13x \geq 18

Simplifying the left-hand side reduces the inequality to a standard two-step linear inequality.

5

Subtract 26 from both sides of the inequality to isolate the variable term.

-13x \geq -8

This isolates the variable term

-13x

on the left side.

6

Divide both sides by -13 and reverse the direction of the inequality sign.

x \leq \frac{8}{13}

Dividing both sides of an inequality by a negative number requires reversing the inequality sign to maintain equivalence.

Key Concept

Solving multi-step linear inequalities, including fraction clearance and reversing the inequality sign when dividing by a negative number.

Question 203Question →

A landscaping company purchases bags of grass seed for $30$ each and bags of fertilizer for $18$ each. For a large project, the company purchased $2$ fewer than $1.5$ times as many bags of fertilizer as bags of grass seed. If the total cost of the grass seed and fertilizer was $420$ , how many bags of fertilizer did the company purchase?

Show answer & explanation

Answer: 10

Answer

The company purchased 10 bags of fertilizer.

The correct answer is 10. By writing the number of fertilizer bags as

f = 1.5g - 2

and substituting this expression into the total cost equation

30g + 18f = 420

, we get

30g + 18(1.5g - 2) = 420

. Simplifying the expression leads to

57g - 36 = 420

, which yields

g = 8

. Substituting

g = 8

back into the relationship for

f

gives

f = 1.5(8) - 2 = 10

.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

g

represent the number of grass seed bags and

f

represent the number of fertilizer bags.

Setting up variables is the first step in translating the word problem into solvable algebraic equations.

2

Translate the relationship between the quantities of bags into an equation.

f = 1.5g - 2

The problem states that the number of fertilizer bags purchased is 2 fewer than 1.5 times the number of grass seed bags purchased.

3

Write the total cost equation using the prices and variables.

30g + 18f = 420

The total cost of

420

is the sum of the cost of the grass seed (

30

per bag) and the fertilizer (

18

per bag).

4

Substitute the equation from Step 2 into the cost equation from Step 3.

30g + 18(1.5g - 2) = 420

Substituting allows us to solve a single linear equation with one variable.

5

Distribute the 18 through the parentheses and combine like terms.

57g - 36 = 420

Distributing gives

18 \times 1.5g = 27g

and

18 \times -2 = -36

. Combining

30g

and

27g

yields

57g

.

6

Isolate the variable term by adding 36 to both sides of the equation.

57g = 456

To solve for

g

, we must isolate the term containing the variable.

7

Divide both sides by 57 to find the value of

g

.

g = 8

This division yields the number of grass seed bags purchased.

8

Substitute the value of

g

back into the relation for

f

to find the final answer.

f = 1.5(8) - 2 = 10

The question asks for the number of fertilizer bags (

f

), not grass seed bags (

g

).

Key Concept

Solving a linear equation by substitution and distributing across linear terms.

Estimated Time:1m 30s

Question 204Question →

A boutique chocolatier packages two types of gift boxes: Standard and Deluxe. Each Standard box contains $4$ dark chocolates, and each Deluxe box contains $8$ dark chocolates. The chocolatier plans to prepare a batch of boxes such that the number of Standard boxes is exactly twice the number of Deluxe boxes. If the chocolatier has a total of $160$ dark chocolates available and uses all of them for this batch, what is the total number of boxes (both Standard and Deluxe combined) they can make?

Show answer & explanation

Answer: 30

Answer

The total number of boxes they can make is 30.

By defining the number of Deluxe boxes as

x

, the number of Standard boxes is

2x

. The equation for the total number of dark chocolates is

4(2x) + 8x = 160

. Solving this gives

16x = 160

, which simplifies to

x = 10

. Thus, there are

10

Deluxe boxes and

20

Standard boxes, for a combined total of

30

boxes.

Step-by-Step Solution

1

Define variables for the number of boxes of each type.

Let

x

be the number of Deluxe boxes and

2x

be the number of Standard boxes.

We are given that the number of Standard boxes is exactly twice the number of Deluxe boxes.

2

Set up an equation representing the total number of dark chocolates used.

4(2x) + 8x = 160

Each Standard box requires

4

dark chocolates, each Deluxe box requires

8

, and the total used is

160

.

3

Solve the equation for

x

.

16x = 160 \Rightarrow x = 10

Combining like terms gives

16x = 160

, and dividing both sides by

16

yields

x = 10

.

4

Calculate the total number of boxes.

x + 2x = 30

The total number of boxes is the sum of Deluxe boxes (

10

) and Standard boxes (

20

).

Key Concept

Translating verbal descriptions of relationships and totals into linear equations to solve algebraic word problems.

Estimated Time:1m 30s

Question 205Question →

An online streaming service offers a basic monthly plan that costs $12$ dollars plus $1.50$ dollars for each premium movie rented. A premium monthly plan costs $20$ dollars and includes $3$ free premium movie rentals, with each additional rental costing $1.00$ dollar. If a subscriber rents $x$ premium movies in a month, where $x > 3$ , which of the following expressions represents the savings, in dollars, of the premium plan compared to the basic plan?

0.50x - 8

5 - 0.50x

0.50x - 5

2.50x - 11

0.50x - 11

Show answer & explanation

Answer: $0.50x - 5$

Answer

0.50x - 5

The cost of the basic plan for

x

rentals is

12 + 1.50x

dollars. Under the premium plan, the first

3

rentals are free, so the subscriber only pays for

x - 3

rentals. The cost of the premium plan is

20 + 1.00(x - 3)

dollars, which simplifies to

17 + x

dollars. The savings of the premium plan compared to the basic plan is the difference between the basic cost and the premium cost:

(12 + 1.50x) - (17 + x) = 0.50x - 5

dollars.

Step-by-Step Solution

1

Write the expression for the cost of the basic monthly plan.

12 + 1.50x

The basic plan has a flat fee of

12

dollars and a rate of

1.50

dollars per movie for all

x

movies rented.

2

Write the expression for the cost of the premium monthly plan.

20 + 1.00(x - 3)

, which simplifies to

17 + x

.

The premium plan has a flat fee of

20

dollars. Since

3

rentals are free, the subscriber only pays

1.00

dollar each for the remaining

x - 3

rentals.

3

Subtract the premium plan cost from the basic plan cost to find the savings.

(12 + 1.50x) - (17 + x) = 0.50x - 5

Savings represents the difference in cost: Basic Cost minus Premium Cost.

Key Concept

Translating verbal descriptions of multi-step costs into linear algebraic expressions and simplifying them.

Estimated Time:1m 30s

Question 206Question →

If $x$ and $y$ are positive real numbers, what is the simplified form of the expression below?

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

2x^{7/5}y

\frac{2y}{x^{1/6}}

2x^{11/6}y

2x^{7/6}y

8x^{11/6}y

Show answer & explanation

Answer: $2x^{11/6}y$

Answer

The simplified form of the expression is

2x^{11/6}y

.

The correct answer is

2x^{11/6}y

. First, simplify the constant coefficients to obtain

\frac{4}{2} = 2

. In the numerator, combine the bases by adding exponents:

x^{1/2} \cdot x^{1/3} = x^{1/2 + 1/3} = x^{5/6}

, and

y^3 \cdot y^{-1} = y^{3 - 1} = y^2

. Next, divide by the terms in the denominator using the quotient rule:

\frac{x^{5/6}}{x^{-1}} = x^{5/6 - (-1)} = x^{11/6}

, and

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

. This yields the fully simplified expression

2x^{11/6}y

.

Step-by-Step Solution

1

Simplify the coefficients of the fraction.

\frac{4}{2} = 2

Dividing the numerical constants in the numerator and denominator simplifies the constant multiplier of the expression.

2

Combine the

x

terms in the numerator using the product rule.

x^{1/2} \cdot x^{1/3} = x^{1/2 + 1/3} = x^{5/6}

When multiplying terms with the same base, add their exponents:

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

.

3

Combine the

y

terms in the numerator using the product rule.

y^3 \cdot y^{-1} = y^{3 + (-1)} = y^2

When multiplying terms with the same base, add their exponents:

3 - 1 = 2

.

4

Simplify the

x

terms in the fraction using the quotient rule.

\frac{x^{5/6}}{x^{-1}} = x^{5/6 - (-1)} = x^{5/6 + 1} = x^{11/6}

When dividing terms with the same base, subtract the denominator's exponent from the numerator's exponent.

5

Simplify the

y

terms in the fraction using the quotient rule.

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

When dividing terms with the same base, subtract the denominator's exponent from the numerator's exponent.

6

Combine the simplified components.

2x^{11/6}y

Multiply the simplified coefficient,

x

term, and

y

term together to get the final simplified expression.

Key Concept

Properties of Exponents in Algebraic Expressions

Estimated Time:1m 30s

Question 207Question →

Two cyclists start at opposite ends of a $90$ -mile trail at the same time and ride toward each other. One cyclist rides at a constant speed that is $3$ miles per hour faster than the other cyclist. If the two cyclists meet after exactly $3$ hours, what is the constant speed, in miles per hour, of the faster cyclist?

Show answer & explanation

Answer: 16.5

Answer

The speed of the faster cyclist is

16.5

miles per hour.

The correct answer of

16.5

is found by setting the speed of the slower cyclist to

s

and the faster cyclist to

s + 3

. Since both cyclists ride toward each other for

3

hours, their combined distance is

3s + 3(s + 3) = 90

. Solving for

s

yields

6s + 9 = 90

, which simplifies to

6s = 81

, or

s = 13.5

. Adding

3

to

13.5

gives the speed of the faster cyclist, which is

16.5

miles per hour.

Step-by-Step Solution

1

Define variables for the speeds of both cyclists in terms of a single variable.

Let

s

be the speed of the slower cyclist. The speed of the faster cyclist is

s + 3

.

Using a single variable simplifies the setup of a solvable linear equation.

2

Write a linear equation using the relationship

\text{Distance} = \text{Speed} \times \text{Time}

.

The equation is

3s + 3(s + 3) = 90

.

The sum of the distances traveled by both cyclists when they meet must equal the total length of the trail, which is

90

miles.

3

Solve the equation for

s

.

6s + 9 = 90 \implies 6s = 81 \implies s = 13.5

.

This yields the speed of the slower cyclist.

4

Calculate the speed of the faster cyclist.

13.5 + 3 = 16.5

.

The question specifically asks for the speed of the faster cyclist, which is represented by

s + 3

.

Key Concept

Translating distance-rate-time relationships from word problems into solvable linear equations.

Alternative Method

An alternative approach is to use the concept of relative speed. Since the two cyclists are moving directly toward each other, their relative speed of approach is the sum of their individual speeds. They cover a total of

90

miles in

3

hours, which means their combined speed is

\frac{90}{3} = 30

miles per hour. If the speed of the faster cyclist is

f

and the slower is

s

, then

f + s = 30

and

f - s = 3

. Adding these two equations gives

2f = 33

, which yields

f = 16.5

miles per hour.

Estimated Time:1m 15s

Question 208Question →

For a certain real number $x$ , the equation $0.4(3x - 5) - 0.15(2x + 8) = 1.3$ is true. What is the value of $4x - 3$ ?

Show answer & explanation

Answer: 17

Answer

The value of the expression is 17.

Distributing the decimals results in

1.2x - 2 - 0.3x - 1.2 = 1.3

. Combining like terms yields

0.9x - 3.2 = 1.3

. Adding

3.2

to both sides results in

0.9x = 4.5

, which simplifies to

x = 5

after dividing by

0.9

. Substituting

x = 5

into the expression

4x - 3

gives

4(5) - 3 = 17

.

Step-by-Step Solution

1

Distribute the decimal factors through the parentheses on the left side of the equation.

1.2x - 2 - 0.3x - 1.2 = 1.3

To eliminate the parentheses and set up terms for simplification.

2

Combine the variable terms and constant terms on the left side of the equation.

0.9x - 3.2 = 1.3

To group like terms and simplify the equation.

3

Isolate the variable term by adding

3.2

to both sides of the equation.

0.9x = 4.5

To gather all constant terms on the right side of the equation.

4

Divide both sides of the equation by

0.9

to solve for

x

.

x = 5

To find the numerical value of the variable.

5

Substitute the value of

x

into the requested expression

4x - 3

.

4(5) - 3 = 17

To evaluate the specific expression requested by the question.

Key Concept

Solving multi-step linear equations with decimals and evaluating algebraic expressions

Question 209Question →

If the expression $\frac{(a^{-3} b^2)^{-2}}{a^4 (b^{-1})^3}$ is simplified to the form $a^x b^y$ , where $a$ and $b$ are positive real numbers, what is the value of the product $xy$ ?

-14

-2

2

10

18

Show answer & explanation

Answer: $-2$

Answer

-2

The correct answer is

-2

. Simplifying the expression step-by-step: first, apply the power of a product and power of a power rules to the numerator to get

a^6 b^{-4}

. Next, simplify the denominator to get

a^4 b^{-3}

. Then, apply the quotient rule to simplify the fraction to

a^{6-4} b^{-4-(-3)} = a^2 b^{-1}

. This gives

x=2

and

y=-1

. The product

xy

is

2 \cdot (-1) = -2

.

Step-by-Step Solution

1

Simplify the numerator using the power of a power rule

(x^p)^q = x^{pq}

and the power of a product rule

(xy)^p = x^p y^p

.

(a^{-3} b^2)^{-2} = a^{(-3)(-2)} b^{(2)(-2)} = a^6 b^{-4}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

2

Simplify the denominator using the power of a power rule.

a^4 (b^{-1})^3 = a^4 b^{(-1)(3)} = a^4 b^{-3}

Multiply the exponents of

b

to simplify the term

(b^{-1})^3

.

3

Divide the simplified numerator by the simplified denominator using the quotient rule

\frac{x^p}{x^q} = x^{p-q}

.

\frac{a^6 b^{-4}}{a^4 b^{-3}} = a^{6-4} b^{-4 - (-3)} = a^2 b^{-1}

Subtract the exponent of the denominator from the exponent of the numerator for each base.

4

Identify the values of

x

and

y

from the simplified form

a^x b^y

and calculate the product

xy

.

x = 2

,

y = -1

, so

xy = 2 \cdot (-1) = -2

Match the simplified exponents with the variables

x

and

y

, then multiply them to find the final value.

Key Concept

Properties of Exponents in Algebraic Expressions

Alternative Method

Alternatively, you can rewrite the negative exponents as positive exponents by moving them across the fraction bar first. The numerator

(a^{-3} b^2)^{-2}

becomes

\frac{1}{(a^{-3} b^2)^2} = \frac{1}{a^{-6} b^4} = \frac{a^6}{b^4}

. The denominator

a^4 (b^{-1})^3

becomes

\frac{a^4}{b^3}

. Dividing these gives

\frac{a^6}{b^4} \cdot \frac{b^3}{a^4} = \frac{a^2}{b^1} = a^2 b^{-1}

. This yields

x = 2

and

y = -1

, leading to the product

xy = -2

.

Estimated Time:1m 30s

Question 210Question →

A local courier service offers two delivery options inside the city limits: Standard and Express. A Standard delivery costs a flat fee of $15.00 plus$ 0.75 per mile. An Express delivery costs a flat fee of $25.00 plus$ 0.50 per mile. If the cost of a Standard delivery is exactly $5.00 less than the cost of an Express delivery, what is the total cost of the Express delivery?

$30.00

$35.00

$45.00

$55.00

$60.00

Show answer & explanation

Answer: $35.00

Answer

The total cost of the Express delivery is $35.00.

The correct answer is

35.00. First, represent the costs as algebraic functions of the distance

m

:

15 + 0.75m

for Standard and

25 + 0.50m

for Express. Set up the equation reflecting that the Standard cost is

5.00 less than the Express cost:

15 + 0.75m = (25 + 0.50m) - 5

. Simplifying this equation gives

15 + 0.75m = 20 + 0.50m

. Subtracting

0.50m

and

15

from both sides results in

0.25m = 5

, which yields

m = 20

. Finally, plug

m = 20

into the Express cost function:

25 + 0.50(20) = 35.00

dollars.

Step-by-Step Solution

1

Define the variables and write the cost functions for both options.

Let

m

represent the number of miles. The Standard delivery cost is

S = 15 + 0.75m

, and the Express delivery cost is

E = 25 + 0.50m

.

Translating the verbal descriptions into algebraic expressions establishes the basis for solving the problem.

2

Set up the equation using the given relationship that Standard is 5.00 dollars less than Express.

15 + 0.75m = (25 + 0.50m) - 5

The phrase 'Standard cost is 5.00 dollars less than Express cost' translates to

S = E - 5

.

3

Solve the equation for

m

.

15 + 0.75m = 20 + 0.50m \implies 0.25m = 5 \implies m = 20

miles.

Solving for

m

determines the delivery distance that satisfies the cost relationship.

4

Calculate the total cost of the Express delivery by substituting the value of

m

into the Express cost function.

E = 25 + 0.50(20) = 25 + 10 = 35

dollars.

The question specifically asks for the total cost of the Express delivery, not the distance or the Standard delivery cost.

Key Concept

Translating verbal descriptions of costs and differences into algebraic equations and solving them.

Alternative Method

You can test the choices. For example, if the Express cost is

35.00, then the distance

m

is found by solving

25 + 0.50m = 35 \implies 0.50m = 10 \implies m = 20

. At

20

miles, the Standard cost is

15 + 0.75(20) = 30

. Since

30

is indeed exactly

5

less than

35$, this choice is correct.

Estimated Time:1m 30s

Question 211Question →

If $x$ and $y$ are non-zero real numbers such that the expression $\frac{(x^2 y^a)^3}{x^5 y^{-2}}$ is equivalent to $x^b y^{14}$ for some integers $a$ and $b$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a + b

is

5

.

Simplifying the expression

\frac{(x^2 y^a)^3}{x^5 y^{-2}}

using the exponent rules yields

x^{6-5} y^{3a-(-2)} = x^1 y^{3a+2}

. Equating this to

x^b y^{14}

shows that

b = 1

and

3a + 2 = 14

. Solving for

a

gives

a = 4

. Thus, the sum

a + b

is

4 + 1 = 5

.

Step-by-Step Solution

1

Simplify the numerator of the given expression.

x^6 y^{3a}

Applying the power of a product rule

(uv)^n = u^n v^n

and the power of a power rule

(u^m)^n = u^{mn}

to

(x^2 y^a)^3

results in

(x^2)^3 (y^a)^3 = x^6 y^{3a}

.

2

Simplify the quotient by subtracting exponents with the same base.

x^1 y^{3a+2}

Using the quotient rule

\frac{u^m}{u^n} = u^{m-n}

, the base

x

term becomes

x^{6-5} = x^1

, and the base

y

term becomes

y^{3a - (-2)} = y^{3a+2}

.

3

Equate the exponents of like bases to find the values of

a

and

b

.

b = 1

and

a = 4

Comparing

x^1 y^{3a+2}

to

x^b y^{14}

gives

b = 1

and

3a + 2 = 14

. Solving

3a + 2 = 14

yields

3a = 12

, which simplifies to

a = 4

.

4

Sum the values of

a

and

b

.

5

Adding

a

and

b

yields

4 + 1 = 5

.

Key Concept

Applying properties of exponents, including the power of a product, power of a power, and quotient rules, to simplify algebraic expressions.

Estimated Time:1m 30s

Question 212Question →

A botanist is monitoring the heights of two bamboo plants. Plant A is $20$ inches tall and grows at a constant rate of $1.5$ inches per day. Plant B is $12$ inches tall and grows at a constant rate of $2.5$ inches per day. After how many days will Plant B be exactly $6$ inches taller than Plant A?

Show answer & explanation

Answer: 14

Answer

The correct answer is 14 days.

The correct answer is

14

days because when we translate the relationship, we get the equation

12 + 2.5d = 20 + 1.5d + 6

. Solving this equation yields

d = 14

.

Step-by-Step Solution

1

Define the variable

d

as the number of days and write the expressions for the heights of both plants.

Plant A's height is

20 + 1.5d

inches, and Plant B's height is

12 + 2.5d

inches.

To represent the growth of each plant algebraically over time.

2

Set up an equation representing that Plant B's height is

6

inches more than Plant A's height.

12 + 2.5d = (20 + 1.5d) + 6

To translate the verbal relationship into a mathematical equation.

3

Simplify the equation and solve for

d

.

d = 14

To find the number of days that satisfies the given condition.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 15s

Question 213Question →

A fitness tracker records a user's daily steps. On Monday, the user walked a certain number of steps. On Tuesday, they walked $1.5$ times the number of steps they walked on Monday. On Wednesday, they walked $2,500$ fewer steps than they did on Tuesday. If the user walked a total of $21,500$ steps over these three days, how many steps did they walk on Monday?

4,750

6,000

6,500

8,000

9,000

Show answer & explanation

Answer: 6,000

Answer

6,000

The correct answer is

6,000

. By letting

x

represent the steps on Monday, we express Tuesday's steps as

1.5x

and Wednesday's steps as

1.5x - 2,500

. Summing these expressions gives the equation

x + 1.5x + 1.5x - 2,500 = 21,500

. Combining like terms yields

4x - 2,500 = 21,500

. Adding

2,500

to both sides gives

4x = 24,000

, and dividing by

4

results in

x = 6,000

.

Step-by-Step Solution

1

Define the variable and write expressions for each day's steps.

Let

x

be the number of steps walked on Monday. Tuesday's steps are

1.5x

, and Wednesday's steps are

1.5x - 2,500

.

This translates the word problem statements into algebraic terms using a single variable.

2

Set up the linear equation representing the total steps.

x + 1.5x + (1.5x - 2,500) = 21,500

The sum of the steps over the three days must equal the given total of

21,500

steps.

3

Combine like terms and solve for

x

.

4x - 2,500 = 21,500 \Rightarrow 4x = 24,000 \Rightarrow x = 6,000

Simplifying the equation isolates the variable to find Monday's step count.

Key Concept

Formulating and solving a single-variable linear equation from a word problem context.

Estimated Time:1m 15s

Question 214Question →

A delivery drone starts a flight with a battery charge of $95\%$ . The battery charge decreases at a constant rate of $1.5\%$ per minute of flight time. During the flight, the drone lands on a charging station for $15$ minutes, during which its battery charge increases at a constant rate of $2.4\%$ per minute. After this charging period, the drone resumes its flight. If the drone ends its flight with a battery charge of $65\%$ , and its total flight time (excluding the time spent charging) was $t$ minutes, what is the value of $t$ ?

Show answer & explanation

Answer: 44

Answer

44

The correct answer is found by setting up a linear equation that models the drone's battery charge changes. The drone starts with

95\%

battery, loses

1.5\%

per minute for

t

minutes, and gains

36\%

from charging (

15 \times 2.4\%

). Setting this equal to the final charge of

65\%

gives the equation

95 - 1.5t + 36 = 65

. Simplifying gives

131 - 1.5t = 65

, which leads to

-1.5t = -66

and

t = 44

.

Step-by-Step Solution

1

Calculate the total percentage of battery charge gained while charging.

36\%

The drone charges at a rate of

2.4\%

per minute for

15

minutes, so the total gain is

15 \times 2.4\% = 36\%

.

2

Set up a linear equation for the final battery charge.

95 - 1.5t + 36 = 65

The final battery charge (

65\%

) is the initial charge (

95\%

) minus the battery consumed during flight (

1.5\%

per minute for

t

minutes) plus the charge gained (

36\%

).

3

Combine constant terms on the left side of the equation.

131 - 1.5t = 65

Adding

95

and

36

simplifies the expression on the left side.

4

Isolate the variable term by subtracting

131

from both sides.

-1.5t = -66

Subtracting

131

from both sides leaves only the variable term on the left.

5

Divide both sides by

-1.5

to solve for

t

.

t = 44

Dividing

-66

by

-1.5

isolates

t

to find the total flight time.

Key Concept

Solving linear equations in a real-world context by setting up an algebraic equation.

Estimated Time:1m 30s

Question 215Question →

A company's net profit is calculated by subtracting its monthly operating costs from its monthly revenue. In a certain month, the company's revenue is $260,000$ dollars minus twice its monthly operating costs. If the company wants its monthly net profit to be at least $110,000$ dollars, which of the following inequalities represents all possible values of its monthly operating costs, $C$ , in dollars?

C \ge 50,000

C \le 50,000

C \le 75,000

C \ge 75,000

C \le 150,000

Show answer & explanation

Answer: $C \le 50,000$

Answer

The inequality

C \le 50,000

The correct inequality is obtained by setting up the net profit as the revenue (

260,000 - 2C

) minus the operating costs (

C

), which simplifies to

260,000 - 3C

. Setting this profit to be at least

110,000

gives the inequality

260,000 - 3C \ge 110,000

. Subtracting

260,000

from both sides results in

-3C \ge -150,000

. Finally, dividing both sides by

-3

and reversing the inequality sign because of the division by a negative number yields

C \le 50,000

.

Step-by-Step Solution

1

Define the expression for net profit by subtracting the operating costs,

C

, from the revenue, which is

260,000 - 2C

.

Net profit =

(260,000 - 2C) - C = 260,000 - 3C

Net profit is defined as revenue minus operating costs.

2

Set up the inequality where the net profit is at least

110,000

.

260,000 - 3C \ge 110,000

The phrase 'at least' translates to the 'greater than or equal to' inequality sign (

\ge

).

3

Subtract

260,000

from both sides of the inequality.

-3C \ge -150,000

Isolate the term containing the variable

C

.

4

Divide both sides by

-3

and reverse the inequality sign.

C \le 50,000

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 216Question →

For all real values of $x$ and $y$ where the expression is defined, simplify the algebraic fraction:

\frac{(x + y)^3 x^{-2} y}{(x+y) (x y^{-1})^{-2}}

Which expression is equivalent to this fraction?

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

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

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

(x+y)^2 y^2

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

Show answer & explanation

Answer: $\frac{(x+y)^2}{y}$

Answer

The expression

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

The correct expression is derived by first simplifying the denominator

(xy^{-1})^{-2}

to

x^{-2}y^2

. We then group and simplify like terms:

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

,

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

, and

\frac{y}{y^2} = \frac{1}{y}

. Multiplying these yields the simplified expression.

Step-by-Step Solution

1

Simplify the term

(x y^{-1})^{-2}

in the denominator

x^{-2} y^2

Apply the power of a product rule:

(ab)^n = a^n b^n

, which gives

x^{-2} (y^{-1})^{-2}

. Then, use the power of a power rule:

(y^{-1})^{-2} = y^{(-1) \cdot (-2)} = y^2

.

2

Rewrite the original fraction with the simplified denominator

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

Substitute the simplified term back into the expression to align similar bases.

3

Simplify the fraction by dividing terms with like bases using the quotient rule

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

Divide each component:

\frac{(x+y)^3}{x+y} = (x+y)^{3-1} = (x+y)^2

,

\frac{x^{-2}}{x^{-2}} = x^{-2 - (-2)} = x^0 = 1

, and

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

. Multiplying these results gives

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

.

Key Concept

Properties of exponents in algebraic expressions, including power of a product, power of a power, and quotient rules.

Question 217Question →

An online bookstore offers two subscription options for downloading e-books. Store A charges a monthly membership fee of $6.00$ plus $1.25$ per e-book download. Store B has no membership fee and charges $2.75$ per e-book download. A customer downloads the same number of e-books from each store in a single month and spends a total of $46.00$ across both stores. How many e-books did the customer download from each store?

4

7

10

11.5

13

Show answer & explanation

Answer: 10

Answer

10 e-books from each store

The correct answer represents the number of downloads from each store that satisfies the total cost equation. Let

x

represent the number of books downloaded from each store. The cost at Store A is

6.00 + 1.25x

and the cost at Store B is

2.75x

. Since the total spent is

46.00

, we write the equation

(6.00 + 1.25x) + 2.75x = 46.00

. Combining like terms gives

6.00 + 4.00x = 46.00

. Subtracting

6.00

from both sides gives

4.00x = 40.00

, and dividing by

4.00

yields

x = 10

.

Step-by-Step Solution

1

Define the variable and write the expressions for the cost of each store.

Let

x

be the number of e-books downloaded from each store. Store A cost is

6.00 + 1.25x

, and Store B cost is

2.75x

.

Establishing algebraic expressions represents the verbal statements mathematically.

2

Set up the total cost equation by summing the costs of both stores and setting it equal to the total spent.

(6.00 + 1.25x) + 2.75x = 46.00

The customer spends a combined total of

46.00

across both bookstores.

3

Combine like terms and solve for the variable

x

.

6.00 + 4.00x = 46.00 \implies 4.00x = 40.00 \implies x = 10

Isolating the variable determines the number of e-books downloaded from each store.

Key Concept

Solving linear equations in one variable derived from real-world scenarios.

Estimated Time:1m 30s

Question 218Question →

A technician uses a linear model to estimate the time, $t$ hours, required to complete a project. The estimate satisfies the equation:

\frac{2}{3}(t - 12) + \frac{5}{6}(t + 6) = 18

What is the value of $t$ ?

Show answer & explanation

Answer: 14

Answer

The value of

t

is

14

.

To solve the linear equation, we first multiply all terms by the least common multiple of the denominators, which is

6

. This simplifies the equation to

4(t - 12) + 5(t + 6) = 108

. Distributing the terms gives

4t - 48 + 5t + 30 = 108

. Combining like terms yields

9t - 18 = 108

. Adding

18

to both sides gives

9t = 126

. Finally, dividing by

9

gives

t = 14

.

Step-by-Step Solution

1

Multiply both sides of the equation by

6

to eliminate the denominators.

4(t - 12) + 5(t + 6) = 108

Multiplying by the least common multiple of the denominators clears the fractions, making the linear equation easier to solve.

2

Apply the distributive property to expand the terms.

4t - 48 + 5t + 30 = 108

Distributing the constants allows us to group like terms.

3

Combine the variable terms and the constant terms on the left side.

9t - 18 = 108

Simplifying the expression is a necessary step before isolating the variable.

4

Add

18

to both sides of the equation.

9t = 126

Adding the constant to both sides isolates the variable term on the left side.

5

Divide both sides of the equation by

9

to solve for

t

.

t = 14

Dividing by the coefficient of the variable yields the final solution.

Key Concept

Solving linear equations with fractional coefficients by clearing denominators and isolating the variable.

Question 219Question →

A community theater group sells tickets for a play. Student tickets cost $12$ each, and adult tickets cost $18$ each. For a Friday night performance, the theater sold a total of $150$ tickets. The revenue from the adult tickets sold was $360$ more than the revenue from the student tickets sold. What is the absolute difference between the number of student tickets and the number of adult tickets sold for this performance?

78

72

54

30

6

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. By setting the number of student tickets to

s

, the number of adult tickets is

150 - s

. Setting up the revenue relationship gives

18(150 - s) = 12s + 360

. Solving this equation gives

s = 78

student tickets and

150 - 78 = 72

adult tickets. The absolute difference between the two values is

78 - 72 = 6

.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

s

be the number of student tickets sold. Since the total number of tickets sold was

150

, the number of adult tickets sold is

150 - s

.

Establishing single-variable representations allows us to model the entire situation using one equation.

2

Write an equation representing the revenue relationship.

The revenue from student tickets is

12s

and the revenue from adult tickets is

18(150 - s)

. The equation is:

18(150 - s) = 12s + 360

.

The problem states that the adult ticket revenue was

360

more than the student ticket revenue.

3

Solve the linear equation for

s

.

2700 - 18s = 12s + 360 \implies 2340 = 30s \implies s = 78

.

Solving the equation gives the exact number of student tickets sold.

4

Calculate the number of adult tickets sold.

Adult tickets sold =

150 - 78 = 72

.

Subtracting the number of student tickets from the total ticket count yields the number of adult tickets.

5

Find the absolute difference between the two quantities.

|78 - 72| = 6

.

The question asks for the absolute difference between the number of student tickets and the number of adult tickets sold.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 220Question →

If $w$ and $z$ are positive real numbers and $k$ is a constant such that the expression $\frac{(w^{3/2} z^{-1})^4}{(w z^k)^2}$ is equivalent to $w^4 z^6$ , what is the value of $k$ ?

Show answer & explanation

Answer: -5

Answer

The value of

k

is

-5

.

Applying the exponent rules, the expression simplifies to

w^4 z^{-4-2k}

. Equating the exponent of

z

to the exponent in the target expression

w^4 z^6

gives

-4-2k = 6

, which solves to

k = -5

.

Step-by-Step Solution

1

Apply the power of a product rule to the expression in the numerator

(w^{3/2} z^{-1})^4 = w^6 z^{-4}

When raising a product to a power, multiply the exponent of each factor by the outer exponent:

(x^a y^b)^c = x^{ac} y^{bc}

.

2

Apply the power of a product rule to the expression in the denominator

(w z^k)^2 = w^2 z^{2k}

Multiply the exponent of each factor in the denominator by

2

.

3

Divide the numerator by the denominator using the quotient rule for exponents

\frac{w^6 z^{-4}}{w^2 z^{2k}} = w^4 z^{-4-2k}

When dividing terms with the same base, subtract the exponent in the denominator from the exponent in the numerator:

\frac{x^a}{x^b} = x^{a-b}

.

4

Equate the exponent of

z

in the simplified expression to the exponent of

z

in the target expression

-4 - 2k = 6

Since the simplified expression is equivalent to

w^4 z^6

, the exponents of the corresponding variable bases must be equal.

5

Solve the linear equation for

k

k = -5

Add

4

to both sides of the equation to get

-2k = 10

, then divide both sides by

-2

.

Key Concept

Properties of Exponents in Algebraic Expressions

Elementary Algebra

Answer

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