Elementary Algebra Practice Questions — ACT — Page 6

Question 101Question →

A company allocates its monthly advertising budget between online advertisements and print media. Last month, the company spent $300$ more than half of its total budget on online advertisements, and $\frac{1}{6}$ of its total budget on print media. The remaining $700$ of the budget was spent on administrative fees. If $B$ represents the total budget in dollars, what is the value of $\frac{B}{10} + 150$ ?

270

300

450

405

315

Show answer & explanation

Answer: 450

Answer

The correct value is 450.

Evaluating the total budget equation yields

B = 3000

. Substituting this value into the expression

\frac{B}{10} + 150

gives

450

. This is correct because the individual allocations (online ads, print media, and administrative fees) sum to the total budget, and the fraction arithmetic is performed correctly.

Step-by-Step Solution

1

Define the variables and write the expressions for each category of expenses.

Let

B

be the total budget. Online advertisements budget is

\frac{1}{2}B + 300

, print media budget is

\frac{1}{6}B

, and administrative fees are

700

.

Translating verbal descriptions into algebraic expressions is necessary to build the equation.

2

Set up the linear equation representing the sum of all expenses equaling the total budget

B

.

B = \left(\frac{1}{2}B + 300\right) + \frac{1}{6}B + 700

The total budget is the sum of its individual components.

3

Group like terms and solve for the total budget

B

.

Combine constant terms:

300 + 700 = 1000

. Combine fraction terms:

\frac{1}{2}B + \frac{1}{6}B = \frac{3}{6}B + \frac{1}{6}B = \frac{4}{6}B = \frac{2}{3}B

. The equation becomes

B = \frac{2}{3}B + 1000

. Subtracting

\frac{2}{3}B

from both sides gives

\frac{1}{3}B = 1000

, which yields

B = 3000

.

This isolates the variable

B

to find the total budget value.

4

Evaluate the required expression

\frac{B}{10} + 150

using the solved value of

B

.

\frac{3000}{10} + 150 = 300 + 150 = 450

The question asks for the value of this specific expression rather than

B

itself.

Key Concept

Solving linear equations in one variable, including translating word problems with fractional terms and evaluating algebraic expressions.

Alternative Method

Instead of solving the equation algebraically, we could test the options to find the total budget

B

. Since each option represents the value of

V = \frac{B}{10} + 150

, we can express

B

as

B = 10(V - 150)

. For the correct option of

450

, we get

B = 10(450 - 150) = 3000

. Substituting

B = 3000

back into the original word problem description: half the budget plus

300

is

1500 + 300 = 1800

; one-sixth of the budget is

500

; the remaining is

3000 - 1800 - 500 = 700

, which matches the given administrative fees.

Estimated Time:2m 0s

Question 102Question →

An online retailer determines that the cost to ship a package of weight $w$ pounds is given by the linear expression $C(w) = kw + b$ , where $k$ and $b$ are constants. Shipping a $3$ -pound package costs $11.50$ dollars, and shipping an $8$ -pound package costs $24.00$ dollars. If the total shipping cost for two packages is $47.00$ dollars, and one of the packages weighs $5$ pounds, what is the weight, in pounds, of the other package?

Show answer & explanation

Answer: 10.6

Answer

The weight of the other package is 10.6 pounds.

The correct weight of 10.6 pounds is found by setting up a linear cost function

C(w) = 2.5w + 4

using the two data points, calculating the cost of the 5-pound package as

16.50

dollars, subtracting this from the total cost of

47.00

dollars to get

30.50

dollars, and solving

2.5w + 4 = 30.50

for the weight.

Step-by-Step Solution

1

Set up the linear system from the given costs

3k + b = 11.50

and

8k + b = 24.00

To determine the relationship between weight and shipping cost.

2

Solve for the slope

k

k = 2.50

Subtracting the first equation from the second eliminates

b

.

3

Solve for the intercept

b

b = 4.00

Substitute

k = 2.50

back into the first equation.

4

Determine the cost of the 5-pound package

C(5) = 16.50

dollars

Evaluate the linear expression

2.50(5) + 4.00

.

5

Determine the remaining cost for the second package

C(w_2) = 30.50

dollars

Subtract the cost of the first package from the total cost (

47.00 - 16.50

).

6

Solve the linear equation for the second package's weight

w_2 = 10.6

Solve

2.50w_2 + 4.00 = 30.50

for

w_2

.

Key Concept

Solving Linear Equations

Question 103Question →

Jordan starts a walk with $2{}500$ steps already recorded on a fitness tracker. Jordan then walks at a constant rate of $120$ steps per minute. If the fitness tracker shows a total of $7{}900$ steps at the end of the walk, for how many minutes did Jordan walk?

Show answer & explanation

Answer: 45

Answer

Jordan walked for

45

minutes.

The correct answer is

45

. The scenario describes a linear relationship with a constant rate of

120

steps per minute and a starting baseline of

2{}500

steps. Let

m

represent the number of minutes Jordan walked. The equation is

2{}500 + 120m = 7{}900

. Subtracting

2{}500

from both sides gives

120m = 5{}400

, and dividing by

120

yields

m = 45

.

Step-by-Step Solution

1

Set up the algebraic equation based on the word problem details.

2{}500 + 120m = 7{}900

The total steps are the sum of the starting steps (

2{}500

) and the product of the rate (

120

steps/min) and time (

m

minutes).

2

Isolate the variable term by subtracting

2{}500

from both sides of the equation.

120m = 5{}400

Subtracting the initial steps gives the steps accumulated solely during the walk.

3

Solve for

m

by dividing both sides of the equation by

120

.

m = 45

Dividing the total steps walked by the rate per minute yields the duration of the walk in minutes.

Key Concept

Translating verbal descriptions of constant rates and starting values into linear equations

Alternative Method

Solve arithmetically by subtracting the baseline steps from the final count (

7{}900 - 2{}500 = 5{}400

steps) and dividing the remaining steps by the walking rate (

5{}400 \div 120 = 45

minutes).

Estimated Time:45s

Question 104Question →

Match each unsimplified algebraic expression on the left with its equivalent simplified form on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings match

3x(x - 2y) - 2x(x - 3y)

to

x^2

;

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

to

2x^2 - 5xy + 3y^2

;

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

to

-x^2y

; and

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

to

4xy

.

Each unsimplified expression is correctly matched to its simplified equivalent by expanding parenthetical terms (taking care to distribute negative signs) and combining like terms.

Step-by-Step Solution

1

Distribute and combine like terms for the first expression

3x(x - 2y) - 2x(x - 3y)

.

The expression simplifies to

x^2

.

First distribute the coefficients to get

3x^2 - 6xy - 2x^2 + 6xy

. Then combine

3x^2 - 2x^2 = x^2

and

-6xy + 6xy = 0

.

2

Distribute and combine like terms for the second expression

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

.

The expression simplifies to

2x^2 - 5xy + 3y^2

.

Distribute the coefficients to get

2x^2 - 2xy - 3xy + 3y^2

. Note that distributing the negative sign of

-3

to

-y^2

results in

+3y^2

. Then combine

-2xy - 3xy = -5xy

.

3

Distribute and combine like terms for the third expression

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

.

The expression simplifies to

-x^2y

.

Distribute to get

x^3 - 2x^2y - x^3 + x^2y

. Note that distributing

-x

to

-xy

gives

+x^2y

. The

x^3 - x^3

terms cancel, leaving

-2x^2y + x^2y = -x^2y

.

4

Expand and simplify the fourth expression

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

.

The expression simplifies to

4xy

.

Expand both squared binomials:

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

. Distribute the negative sign to get

x^2 + 2xy + y^2 - x^2 + 2xy - y^2

. Combine terms to cancel

x^2

and

y^2

, leaving

4xy

.

Key Concept

Simplifying Expressions and Combining Like Terms

Estimated Time:1m 30s

Question 105Question →

For any positive real number $y$ , the expression $\frac{(y^3)^{1/2}}{y^{2/3}}$ is equivalent to which of the following?

y^{7/6}

y^{1/6}

y^{5/6}

y^{17/6}

y^{13/6}

Show answer & explanation

Answer: $y^{5/6}$

Answer

The simplified expression is

y^{5/6}

Applying the power of a power rule to the numerator gives

(y^3)^{1/2} = y^{3/2}

. Then, applying the quotient rule to divide by

y^{2/3}

requires subtracting the exponents:

\frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}

. This results in the equivalent expression

y^{5/6}

.

Step-by-Step Solution

1

Apply the power of a power rule to the numerator

(y^3)^{1/2}

.

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

When raising a power to a power, multiply the exponents.

2

Apply the quotient of powers rule to divide

y^{3/2}

by

y^{2/3}

.

y^{3/2 - 2/3}

When dividing powers with the same base, subtract the exponent in the denominator from the exponent in the numerator.

3

Subtract the fractions in the exponent by finding a common denominator.

y^{9/6 - 4/6} = y^{5/6}

A common denominator for 2 and 3 is 6. Rewrite the fractions and subtract their numerators.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 106Question →

If $x$ satisfies the equation $\frac{3(x - 2)}{5} - \frac{2x - 1}{3} = \frac{1}{15}$ , what is the value of the expression $2x + 7$ ?

Show answer & explanation

Answer: -21

Answer

The final value of the expression is

-21

.

Solving the equation by first multiplying both sides by the least common denominator of 15 yields the simplified equation

9(x - 2) - 5(2x - 1) = 1

. Distributing and combining terms yields

-x - 13 = 1

, which gives

x = -14

. Substituting this value into the expression

2x + 7

yields

-21

.

Step-by-Step Solution

1

Multiply the entire equation by the least common multiple of the denominators (

15

) to eliminate all fractions.

9(x - 2) - 5(2x - 1) = 1

Multiplying by the least common multiple of 5 and 3 eliminates the fractions and simplifies the equation.

2

Distribute the coefficients (

9

and

-5

) to their respective terms inside the parentheses.

9x - 18 - 10x + 5 = 1

Distributive property allows us to remove parentheses. Note that distributing

-5

to

-1

results in

+5

.

3

Combine the variable terms (

9x

and

-10x

) and the constant terms (

-18

and

5

) on the left side of the equation.

-x - 13 = 1

Combining like terms simplifies the expression to prepare for isolating the variable.

4

Isolate the variable term

-x

by adding

13

to both sides, then solve for

x

by multiplying by

-1

.

x = -14

Adding

13

yields

-x = 14

, and multiplying by

-1

isolates

x

to find its value.

5

Substitute

x = -14

into the given expression

2x + 7

.

2(-14) + 7 = -21

The question asks for the value of the expression

2x + 7

, not just

x

itself.

Key Concept

Solving multi-step linear equations involving fractions, distributive property with negative signs, and evaluating algebraic expressions.

Alternative Method

Instead of multiplying by the least common multiple first, you can separate the fractions:

\frac{3}{5}x - \frac{6}{5} - \frac{2}{3}x + \frac{1}{3} = \frac{1}{15}

. Combining the

x

terms gives

(\frac{9}{15} - \frac{10}{15})x = -\frac{1}{15}x

. Combining the constant terms gives

-\frac{18}{15} + \frac{5}{15} = -\frac{13}{15}

. The equation becomes

-\frac{1}{15}x - \frac{13}{15} = \frac{1}{15}

. Multiplying the entire equation by

15

yields

-x - 13 = 1

, which gives

x = -14

, and substituting into

2x + 7

yields

-21

.

Estimated Time:2m 0s

Question 107Question →

A digital marketing firm runs advertisements on two platforms: SocialMedia and SearchEngine. The cost to run an advertisement on SocialMedia is $15$ dollars per day, and the cost to run an advertisement on SearchEngine is $25$ dollars per day. Last month, the firm ran advertisements on both platforms for a combined total of $60$ days. The total amount spent on SocialMedia advertisements was $350$ dollars more than half the total amount spent on SearchEngine advertisements. For how many days last month did the firm run advertisements on SocialMedia?

Show answer & explanation

Answer: 40

Answer

The firm ran advertisements on SocialMedia for 40 days.

Setting up the system of equations based on the problem description gives

S + E = 60

and

15S = 0.5(25E) + 350

. Substituting the first equation into the second yields

15S = 12.5(60 - S) + 350

. Solving this linear equation results in

S = 40

days.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

S

be the number of days the firm ran advertisements on SocialMedia, and

E

be the number of days they ran advertisements on SearchEngine.

Establishing variables is the first step in translating a word problem into algebraic equations.

2

Express the relationship for the total number of days.

S + E = 60

, which simplifies to

E = 60 - S

.

This allows us to express one variable in terms of the other, making it easier to solve the system by substitution.

3

Translate the cost relationship statement into an algebraic equation.

15S = 12.5E + 350

The cost of running advertisements on SocialMedia is

15S

. The cost on SearchEngine is

25E

. Half of the SearchEngine cost is

12.5E

. Adding

350

to half of the SearchEngine cost gives the SocialMedia cost.

4

Substitute the expression for

E

into the cost equation and solve for

S

.

15S = 12.5(60 - S) + 350 \Rightarrow 15S = 750 - 12.5S + 350 \Rightarrow 27.5S = 1100 \Rightarrow S = 40

.

Solving the resulting linear equation yields the number of days spent on SocialMedia ads.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:2m 0s

Question 108Question →

A rectangular region has a width of $2x - 3$ meters and a length of $3x + 1$ meters. A square piece with a side length of $x - 2$ meters is removed from the region. The area, in square meters, of the remaining region can be expressed in the standard polynomial form $Ax^2 + Bx + C$ , where $A$ , $B$ , and $C$ are integers. What is the value of the coefficient $B$ ?

Show answer & explanation

Answer: -3

Answer

The coefficient of the linear term,

B

, is

-3

.

Subtracting the area of the square,

x^2 - 4x + 4

, from the area of the rectangle,

6x^2 - 7x - 3

, yields

5x^2 - 3x - 7

. Thus, the coefficient

B

of the

x

term is

-3

.

Step-by-Step Solution

1

Calculate the area of the rectangle.

Area =

6x^2 - 7x - 3

The area of a rectangle is found by multiplying its length and width:

(2x - 3)(3x + 1) = 6x^2 + 2x - 9x - 3 = 6x^2 - 7x - 3

.

2

Calculate the area of the square.

Area =

x^2 - 4x + 4

The area of a square is the square of its side length:

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

.

3

Subtract the square's area from the rectangle's area.

Remaining Area =

5x^2 - 3x - 7

Subtracting the area of the removed square from the total area requires distributing the negative sign to each term of the square's polynomial:

(6x^2 - 7x - 3) - (x^2 - 4x + 4) = 6x^2 - 7x - 3 - x^2 + 4x - 4 = 5x^2 - 3x - 7

.

4

Identify the coefficient

B

.

B = -3

In the standard quadratic form

Ax^2 + Bx + C

, the coefficient of the linear term

x

is

B

, which corresponds to

-3

in the polynomial

5x^2 - 3x - 7

.

Key Concept

Operations on Polynomials (multiplication, squaring binomials, and subtraction with negative sign distribution)

Question 109Question →

For what value of $y$ is the equation $3(y - 4) = 5y + 2$ true?

Show answer & explanation

Answer: -7

Answer

The value of

y

that satisfies the equation is

-7

.

Distributing the 3 yields

3y - 12 = 5y + 2

. Subtracting

3y

from both sides gives

-12 = 2y + 2

. Subtracting 2 from both sides gives

-14 = 2y

. Dividing by 2 results in

y = -7

.

Step-by-Step Solution

1

Distribute the 3 on the left side of the equation.

3y - 12 = 5y + 2

To simplify the expression by expanding the parentheses.

2

Subtract

3y

from both sides of the equation.

-12 = 2y + 2

To collect the variable terms on the right side of the equation.

3

Subtract 2 from both sides of the equation.

-14 = 2y

To isolate the variable term.

4

Divide both sides by 2.

y = -7

To solve for

y

.

Key Concept

Solving linear equations by distributing and isolating the variable.

Estimated Time:45s

Question 110Question →

A commercial bakery has two bread-kneading machines, Machine X and Machine Y. Machine X can knead dough at a constant rate of $p$ pounds per hour. Machine Y's kneading rate is $14$ pounds per hour more than half the kneading rate of Machine X. On a busy morning, Machine X starts kneading and operates for exactly $5$ hours. Machine Y starts operating $1$ hour after Machine X starts and operates for the next $4$ hours. Together, the two machines knead a total of $336$ pounds of dough. What is the total number of pounds of dough kneaded by Machine Y?

34

116

136

148

200

Show answer & explanation

Answer: 136

Answer

The total number of pounds of dough kneaded by Machine Y is 136.

To find the total dough kneaded by Machine Y, we first define the rates of both machines. Let Machine X's rate be

p

pounds per hour. Machine Y's rate is

14

more than half of Machine X's rate, which is written as

0.5p + 14

pounds per hour. Machine X runs for

5

hours, producing

5p

pounds of dough. Machine Y runs for

4

hours, producing

4(0.5p + 14) = 2p + 56

pounds of dough. Setting their sum equal to the total of

336

gives the equation

7p + 56 = 336

, which simplifies to

7p = 280

and yields

p = 40

. Machine Y's total work is then

4(0.5(40) + 14) = 136

pounds.

Step-by-Step Solution

1

Define the variable for Machine X's rate and write the expression for Machine Y's rate based on the text.

Machine X's rate is

p

pounds per hour. Machine Y's rate is

0.5p + 14

pounds per hour.

We need to express both rates in terms of a single variable to set up the equation.

2

Determine the operating duration for each machine.

Machine X operates for

5

hours. Machine Y operates for

4

hours.

Machine X starts first and runs for

5

hours. Machine Y starts

1

hour later and operates for the remaining

4

hours.

3

Write the equation for the total pounds of dough kneaded by both machines combined.

5p + 4(0.5p + 14) = 336

Total work is the sum of the work done by Machine X (rate times time) and Machine Y (rate times time).

4

Solve the equation for the variable

p

.

5p + 2p + 56 = 336 \implies 7p + 56 = 336 \implies 7p = 280 \implies p = 40

.

Distribute the

4

and combine like terms to isolate

p

.

5

Calculate the total work done specifically by Machine Y.

Machine Y's total work =

4(0.5(40) + 14) = 4(20 + 14) = 4(34) = 136

pounds.

The question asks for the total pounds of dough kneaded by Machine Y, which is its rate multiplied by its operating time.

Key Concept

Translating verbal descriptions of rates and times into algebraic equations and solving them.

Alternative Method

Instead of solving for

p

first, we can write the equation directly in terms of Machine Y's work. Let

y

be the work done by Machine Y. Since Machine Y worked for

4

hours, its rate is

\frac{y}{4}

. This rate is

14

more than half of Machine X's rate, so

\frac{y}{4} = 0.5R_X + 14 \implies R_X = 2(\frac{y}{4} - 14) = \frac{y}{2} - 28

. Machine X's work is

5 R_X = 5(\frac{y}{2} - 28) = 2.5y - 140

. Since the total work is

336

, we have

(2.5y - 140) + y = 336 \implies 3.5y = 476 \implies y = 136

.

Estimated Time:3m 0s

Question 111Question →

What is the complete set of real numbers $x$ that satisfy the inequality $\frac{2 - x}{3} - \frac{3x - 1}{4} \geq 2$ ?

x \geq -1

x \leq -\frac{19}{13}

x \geq \frac{5}{4}

x \leq -1

x \leq \frac{9}{13}

Show answer & explanation

Answer: $x \leq -1$

Answer

The complete set of real numbers satisfying the inequality is

x \leq -1

.

To solve the inequality

\frac{2 - x}{3} - \frac{3x - 1}{4} \geq 2

, we first eliminate the denominators by multiplying both sides by their least common multiple, which is

12

. This yields

4(2 - x) - 3(3x - 1) \geq 24

. Distributing the terms gives

8 - 4x - 9x + 3 \geq 24

. Combining like terms results in

11 - 13x \geq 24

. Subtracting

11

from both sides yields

-13x \geq 13

. Finally, dividing by

-13

requires reversing the inequality sign, leading to the solution

x \leq -1

.

Step-by-Step Solution

1

Multiply all terms on both sides of the inequality by

12

, which is the least common multiple of the denominators

3

and

4

.

4(2 - x) - 3(3x - 1) \geq 24

Multiplying by a positive number preserves the inequality direction while eliminating the fractions to simplify the equation.

2

Distribute the coefficients

4

and

-3

to the terms inside the parentheses.

8 - 4x - 9x + 3 \geq 24

Applying the distributive property expands the expression. Note that multiplying

-3

by

-1

results in

+3

.

3

Combine like terms on the left side of the inequality.

11 - 13x \geq 24

Grouping the constants (

8 + 3 = 11

) and the variable terms (

-4x - 9x = -13x

) simplifies the inequality.

4

Subtract

11

from both sides of the inequality.

-13x \geq 13

Isolating the variable term on the left side by moving the constant term to the right side.

5

Divide both sides of the inequality by

-13

and reverse the direction of the inequality sign.

x \leq -1

Dividing by a negative number requires reversing the inequality sign from

\geq

to

\leq

.

Key Concept

Solving linear inequalities by clearing fractions, distributing terms correctly, and reversing the inequality sign when multiplying or dividing by a negative number.

Estimated Time:2m 0s

Question 112Question →

If $x$ and $y$ are positive real numbers, the expression

\frac{(x^{1/2} + y^{1/2})^2 - (x^{1/2} - y^{1/2})^2}{(2x^{-1/4} y^{3/4})^{-2}}

can be simplified to the form $k y^p$ . What is the value of the sum $k + p$ ?

1

10

15

17

18

Show answer & explanation

Answer: 18

Answer

18

Expanding the numerator yields

4x^{1/2}y^{1/2}

. Simplifying the denominator using exponent rules yields

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

. Dividing the numerator by the denominator gives

\frac{4}{1/4} \cdot x^{1/2 - 1/2} \cdot y^{1/2 - (-3/2)} = 16y^2

, which corresponds to

k = 16

and

p = 2

. The sum is

16 + 2 = 18

.

Step-by-Step Solution

1

Expand and simplify the numerator.

4x^{1/2}y^{1/2}

Expand both squared binomials:

(x^{1/2} + y^{1/2})^2 = x + 2x^{1/2}y^{1/2} + y

and

(x^{1/2} - y^{1/2})^2 = x - 2x^{1/2}y^{1/2} + y

. Subtracting the second expression from the first yields

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

.

2

Simplify the denominator using exponent rules.

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

Apply the power of a product rule

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

to distribute the exponent of

-2

:

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

. This simplifies to

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

.

3

Divide the simplified numerator by the simplified denominator.

16y^2

Divide the coefficients and subtract the exponents of like bases:

\frac{4x^{1/2}y^{1/2}}{\frac{1}{4}x^{1/2}y^{-3/2}} = \left(\frac{4}{1/4}\right) x^{1/2 - 1/2} y^{1/2 - (-3/2)} = 16 x^0 y^{1/2 + 3/2} = 16y^2

.

4

Identify the values of

k

and

p

and calculate their sum.

18

Comparing

16y^2

to the form

k y^p

gives

k = 16

and

p = 2

. Therefore, the sum is

k + p = 16 + 2 = 18

.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 113Question →

Two courier drones, Drone A and Drone B, fly in opposite directions along a straight path from the same distribution center. Drone A departs at 10:00 AM and flies at a constant speed of $40\text{ miles per hour}$ . Drone B departs from the same location at 10:30 AM and flies in the opposite direction at a constant speed of $60\text{ miles per hour}$ . At what time will the two drones be exactly $220\text{ miles}$ apart?

12:00 PM

12:12 PM

12:30 PM

12:42 PM

12:54 PM

Show answer & explanation

Answer: 12:30 PM

Answer

12:30 PM

The correct answer is 12:30 PM. To find this, define

t

as the travel time of Drone B in hours. Since Drone A departs 30 minutes (

0.5

hours) earlier, its travel time is

t + 0.5

hours. Using the relationship that the sum of the distances traveled in opposite directions equals the total distance, we set up the equation

40(t + 0.5) + 60t = 220

. Solving this yields

100t + 20 = 220

, which simplifies to

100t = 200

and

t = 2\text{ hours}

. Adding 2 hours to Drone B's departure time of 10:30 AM results in 12:30 PM.

Step-by-Step Solution

1

Define variables for the travel times of both drones relative to their departure times.

Let

t

be the number of hours Drone B travels after departing at 10:30 AM. Since Drone A departs 30 minutes (

0.5

hours) earlier at 10:00 AM, Drone A's travel time is

t + 0.5

hours.

Establishing a single variable for time allows us to write a single-variable linear equation for the total distance covered.

2

Express the distance traveled by each drone using the formula

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

and sum them to equal the total separation distance.

The distance traveled by Drone A is

40(t + 0.5)

miles, and the distance traveled by Drone B is

60t

miles. The equation is:

40(t + 0.5) + 60t = 220

.

Since the drones fly in opposite directions from the same starting point, the total distance between them is the sum of their individual distances.

3

Solve the equation for

t

.

40t + 20 + 60t = 220 \implies 100t + 20 = 220 \implies 100t = 200 \implies t = 2\text{ hours}

.

Solving for

t

gives the number of hours Drone B travels before they are

220\text{ miles}

apart.

4

Determine the final time by adding the calculated travel time to the corresponding departure time.

Drone B departs at 10:30 AM and travels for

2\text{ hours}

.

10:30\text{ AM} + 2\text{ hours} = 12:30\text{ PM}

. (Alternatively, Drone A departs at 10:00 AM and travels for

2 + 0.5 = 2.5\text{ hours}

, which also yields 12:30 PM).

Adding the elapsed time to the initial departure time gives the clock time when the condition is met.

Key Concept

Translating relative motion and time-offset word problems into linear equations

Question 114Question →

A food truck charges a flat fee of $6 for a custom lunch box container, plus$ 8 for each scoop of specialty salad added to the box. If a customer paid a total of $38 for one custom lunch box filled with specialty salad, how many scoops of specialty salad did they receive?

14

32

4

5

8

Show answer & explanation

Answer: 4

Answer

4 scoops of specialty salad

The correct answer is 4. The total cost is represented by the equation

6 + 8s = 38

, where

s

represents the number of scoops. Subtracting the flat container fee of

6 from the total of

38 leaves

32. Dividing

32 by the per-scoop cost of $8 yields 4 scoops.

Step-by-Step Solution

1

Set up the linear equation representing the total cost.

6 + 8s = 38

, where

s

is the number of scoops of specialty salad.

The total cost consists of a one-time flat fee of

6 plus

8 per scoop of salad.

2

Subtract the flat fee from both sides of the equation.

8s = 32

To isolate the variable term, we subtract the constant cost of the container from the total cost.

3

Divide both sides of the equation by the cost per scoop.

s = 4

Dividing the remaining cost by the price per scoop yields the number of scoops purchased.

Key Concept

Translating a verbal description of a linear cost scenario into a one-variable linear equation and solving it.

Question 115Question →

If $\frac{3}{5}x - 7 = 8$ , what is the value of $2x - 3$ ?

15

25

44

47

\frac{1}{3}

Show answer & explanation

Answer: 47

Answer

47

To find the value of

2x - 3

, first solve the equation

\frac{3}{5}x - 7 = 8

for

x

. Adding 7 to both sides of the equation gives

\frac{3}{5}x = 15

. Multiplying both sides by the reciprocal

\frac{5}{3}

yields

x = 25

. Finally, substitute

25

for

x

in the expression

2x - 3

to get

2(25) - 3 = 50 - 3 = 47

.

Step-by-Step Solution

1

Add 7 to both sides of the equation

\frac{3}{5}x - 7 = 8

to isolate the variable term.

\frac{3}{5}x = 15

Adding 7 to both sides eliminates the constant on the left side of the equation.

2

Multiply both sides of the equation by

\frac{5}{3}

to solve for

x

.

x = 25

Multiplying by the reciprocal of

\frac{3}{5}

isolates

x

on the left side of the equation.

3

Substitute

x = 25

into the expression

2x - 3

.

2(25) - 3 = 50 - 3 = 47

The question asks for the value of the expression

2x - 3

rather than just the variable

x

.

Key Concept

Solving two-step linear equations involving fractions, and evaluating algebraic expressions.

Alternative Method

Instead of solving for

x

first, you can express the target expression in terms of

\frac{3}{5}x

. Specifically, note that

2x - 3 = \frac{10}{3}(\frac{3}{5}x) - 3

. Since

\frac{3}{5}x = 15

, substituting this directly gives

\frac{10}{3}(15) - 3 = 50 - 3 = 47

.

Estimated Time:45s

Question 116Question →

Which of the following expressions is equivalent to the expression below?

(3x^3 - 2y^2)^2 - 2(x^3 - 4y^2)(4x^3 - 5y^2)

x^9 + 30x^3y^2 - 36y^4

x^6 - 54x^3y^2 + 44y^4

x^6 + 30x^3y^2 - 36y^4

x^6 + 42x^3y^2 - 36y^4

-7x^6 + 72x^3y^2 - 76y^4

Show answer & explanation

Answer: $x^6 + 30x^3y^2 - 36y^4$

Answer

The expression

x^6 + 30x^3y^2 - 36y^4

.

To find the correct expression, we first square the binomial to obtain

9x^6 - 12x^3y^2 + 4y^4

. Next, we multiply the two binomials

(x^3 - 4y^2)(4x^3 - 5y^2)

to get

4x^6 - 21x^3y^2 + 20y^4

, and distribute the factor of 2 to obtain

8x^6 - 42x^3y^2 + 40y^4

. Finally, we subtract this from the squared binomial by distributing the negative sign to all terms:

9x^6 - 12x^3y^2 + 4y^4 - 8x^6 + 42x^3y^2 - 40y^4

. Combining like terms yields

x^6 + 30x^3y^2 - 36y^4

.

Step-by-Step Solution

1

Expand the squared binomial

(3x^3 - 2y^2)^2

.

9x^6 - 12x^3y^2 + 4y^4

Use the binomial squaring formula

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

, where

a = 3x^3

and

b = 2y^2

.

2

Multiply the two binomials

(x^3 - 4y^2)(4x^3 - 5y^2)

.

4x^6 - 21x^3y^2 + 20y^4

Apply the FOIL method to expand the product:

4x^6 - 5x^3y^2 - 16x^3y^2 + 20y^4

, and combine the like middle terms.

3

Multiply the resulting trinomial from Step 2 by the constant factor of 2.

8x^6 - 42x^3y^2 + 40y^4

Distribute the factor of 2 to each term of the simplified trinomial expression.

4

Subtract the expression in Step 3 from the expression in Step 1.

x^6 + 30x^3y^2 - 36y^4

Distribute the negative sign to all terms of the subtracted polynomial:

9x^6 - 12x^3y^2 + 4y^4 - 8x^6 + 42x^3y^2 - 40y^4

, and combine the remaining like terms.

Key Concept

Operations on Polynomials

Question 117Question →

Which of the following is the completely factored form of the expression $3x^3 - 12x^2 + 12x$ ?

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

3x^2(x - 2)^2

3x(x - 2)(x + 2)

3x(x - 2)^2

3x(x + 2)^2

Show answer & explanation

Answer: $3x(x - 2)^2$

Answer

The completely factored form of the expression is

3x(x - 2)^2

.

To factor the expression

3x^3 - 12x^2 + 12x

completely, we first look for the greatest common factor (GCF) of the three terms. The GCF of

3x^3

,

-12x^2

, and

12x

is

3x

. Factoring out

3x

yields

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

. Next, we factor the quadratic trinomial inside the parentheses. The expression

x^2 - 4x + 4

is a perfect square trinomial that factors into

(x - 2)^2

. Putting it all together, the completely factored form is

3x(x - 2)^2

.

Step-by-Step Solution

1

Identify and factor out the Greatest Common Factor (GCF) from all three terms of the polynomial

3x^3 - 12x^2 + 12x

.

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

Each term in the polynomial is divisible by

3

, and the lowest power of

x

common to all terms is

x^1

. Thus, the GCF is

3x

.

2

Factor the remaining quadratic trinomial

x^2 - 4x + 4

inside the parentheses.

(x - 2)^2

The trinomial

x^2 - 4x + 4

is a perfect square trinomial matching the form

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

, where

a = x

and

b = 2

.

3

Combine the factored parts to write the final completely factored expression.

3x(x - 2)^2

Bringing together the GCF and the factored trinomial yields the simplest, fully factored form.

Key Concept

Factoring a polynomial completely by first extracting the greatest common factor (GCF) and then factoring the remaining perfect square trinomial.

Alternative Method

Instead of factoring directly, you can expand the answer choices to see which one is equivalent to the original polynomial. For example, expanding the correct option:

3x(x-2)^2 = 3x(x^2 - 4x + 4) = 3x^3 - 12x^2 + 12x

. This matches the original expression.

Estimated Time:1m 0s

Question 118Question →

A certain value $x$ satisfies the relationship where the square of the difference between $x$ and $4$ is equal to $16$ decreased by $3$ times $x$ . If $x_1$ and $x_2$ are the two real solutions to this relationship, with $x_1 > x_2$ , what is the value of $3x_1 - 2x_2$ ?

3

6

10

15

33

Show answer & explanation

Answer: 15

Answer

The value of the expression is 15.

The value 15 is correct. First, translate the relationship into the algebraic equation

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

. Expanding the left side yields

x^2 - 8x + 16 = 16 - 3x

. To solve the quadratic equation by factoring, rearrange the terms to set one side to zero: subtract 16 and add

3x

to both sides, which simplifies to

x^2 - 5x = 0

. Factoring the left-hand side gives

x(x - 5) = 0

. The solutions are

x = 5

and

x = 0

. Since the problem defines

x_1 > x_2

, the larger solution is

x_1 = 5

and the smaller solution is

x_2 = 0

. Substituting these values into the expression

3x_1 - 2x_2

yields

3(5) - 2(0) = 15

.

Step-by-Step Solution

1

Translate the verbal description into a mathematical equation.

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

The 'square of the difference between

x

and

4

' is represented as

(x-4)^2

, and '16 decreased by 3 times

x

' is represented as

16 - 3x

.

2

Expand the squared binomial on the left side of the equation.

x^2 - 8x + 16 = 16 - 3x

Using the binomial expansion formula

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

, expand

(x-4)^2

to

x^2 - 2(4)(x) + 4^2 = x^2 - 8x + 16

.

3

Rearrange the equation to set one side to zero.

x^2 - 5x = 0

Add

3x

and subtract

16

from both sides of the equation to gather all terms on one side:

x^2 - 8x + 3x + 16 - 16 = 0

, which simplifies to

x^2 - 5x = 0

.

4

Factor the quadratic equation.

x(x - 5) = 0

Factor out the greatest common factor, which is

x

.

5

Solve for the roots of the equation.

x = 0

or

x = 5

Set each factor to zero:

x = 0

or

x - 5 = 0

.

6

Identify the values of

x_1

and

x_2

and calculate the required expression.

x_1 = 5

,

x_2 = 0

, so

3x_1 - 2x_2 = 15

Since

x_1 > x_2

, assign

x_1 = 5

and

x_2 = 0

. Calculate

3(5) - 2(0) = 15

.

Key Concept

Solving quadratic equations by rearranging terms, expanding binomials, factoring out the greatest common factor, and solving for roots.

Question 119Question →

When the polynomial $12x^2 + 11x - 15$ is factored completely into the form $(ax + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are integers such that $a > c > 0$ , what is the value of the constant term $d$ ?

Show answer & explanation

Answer: 5

Answer

The value of the constant term

d

is

5

.

Factoring the trinomial

12x^2 + 11x - 15

completely gives

(4x - 3)(3x + 5)

. Applying the constraint

a > c > 0

means the factor with the larger

x

-coefficient must be written first in the template

(ax + b)(cx + d)

. This yields

a = 4

,

b = -3

,

c = 3

, and

d = 5

. Thus, the constant term

d

is

5

.

Step-by-Step Solution

1

Find the factor pair for the AC method.

We need two numbers that multiply to

12 \times (-15) = -180

and add up to

11

. The numbers are

20

and

-9

.

This allows us to split the linear middle term to factor by grouping.

2

Rewrite the polynomial and factor by grouping.

12x^2 + 20x - 9x - 15 = 4x(3x + 5) - 3(3x + 5) = (4x - 3)(3x + 5)

.

Grouping the first two terms and the last two terms reveals a common binomial factor of

(3x + 5)

.

3

Apply the given inequality constraints to match the template.

Comparing

(4x - 3)(3x + 5)

to

(ax + b)(cx + d)

with

a > c > 0

yields

a = 4

,

b = -3

,

c = 3

, and

d = 5

.

Since the lead coefficient

4

is greater than

3

, the factor

(4x - 3)

must correspond to

(ax + b)

.

Key Concept

Factoring quadratic trinomials of the form

Ax^2 + Bx + C

using the grouping (AC) method.

Question 120Question →

A chemist wants to create $100$ milliliters of a $42.5\%$ acid solution by mixing three different acid solutions: a $10\%$ acid solution, a $20\%$ acid solution, and an $80\%$ acid solution. She decides that the volume of the $20\%$ acid solution used must be exactly $3$ times the volume of the $10\%$ acid solution used. What is the difference, in milliliters, between the volume of the $80\%$ acid solution and the volume of the $10\%$ acid solution used in the final mixture?

Show answer & explanation

Answer: 25

Answer

The difference between the volume of the

80\%

acid solution and the volume of the

10\%

acid solution is

25

milliliters.

Solving the system of equations yields that

15

milliliters of the

10\%

solution,

45

milliliters of the

20\%

solution, and

40

milliliters of the

80\%

solution are needed. The difference between the volume of the

80\%

solution and the

10\%

solution is

40 - 15 = 25

milliliters.

Step-by-Step Solution

1

Define variables for the volume of each acid solution.

Let

x

be the volume of the

10\%

solution,

y

be the volume of the

20\%

solution, and

z

be the volume of the

80\%

solution.

This establishes algebraic representations for the unknowns.

2

Set up a system of linear equations based on the relationships given in the problem statement.

x + y + z = 100

(total volume)

y = 3x

(relationship between the

20\%

and

10\%

solutions)

0.10x + 0.20y + 0.80z = 42.5

(total acid content)

Translating word problems to mathematical equations allows us to solve for the variables systematically.

3

Reduce the system to a single equation in terms of

x

by substituting

y = 3x

and expressing

z

in terms of

x

.

4x + z = 100 \implies z = 100 - 4x

Substitute both into the acid equation:

0.10x + 0.20(3x) + 0.80(100 - 4x) = 42.5

Substitution simplifies the system of equations to a single linear equation with one variable.

4

Solve the simplified linear equation for

x

.

0.70x + 80 - 3.20x = 42.5 \implies -2.50x = -37.5 \implies x = 15

This determines the volume of the

10\%

acid solution.

5

Calculate the volume of the

80\%

solution,

z

.

z = 100 - 4(15) = 40

This determines the volume of the

80\%

acid solution.

6

Find the difference between

z

and

x

.

z - x = 40 - 15 = 25

The question asks for the difference between these two volumes.

Key Concept

Translating and solving systems of linear equations from verbal descriptions (mixture problems).

Estimated Time:2m 30s

Elementary Algebra

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Step-by-Step Solution

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