Solving Linear Equations Practice Questions — ACT — Page 3

Question 41Question →

A marathon runner plans to run a total of $d$ miles this week. The runner schedules $\frac{2}{3}(d - 6)$ miles of the run on soft dirt trails and $0.4(d + 15)$ miles on asphalt roads. If the runner completes only these two segments for a total of $34$ miles, what is the value of $d$ ?

Show answer & explanation

Answer: 30

Answer

The value of

d

is 30.

The value of

d

that satisfies the equation is

30

because substituting

30

back into the original equation yields a true statement:

\frac{2}{3}(30 - 6) + 0.4(30 + 15) = 16 + 18 = 34

.

Step-by-Step Solution

1

Set up the equation based on the given information: the trail running distance plus the road running distance equals the total distance.

\frac{2}{3}(d - 6) + 0.4(d + 15) = 34

This relates the individual segment distances to the total planned mileage of

34

miles.

2

Distribute the coefficients to eliminate the parentheses.

\frac{2}{3}d - 4 + 0.4d + 6 = 34

Distributing

\frac{2}{3}

to

(d-6)

gives

\frac{2}{3}d - 4

, and distributing

0.4

to

(d+15)

gives

0.4d + 6

.

3

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

\frac{2}{3}d + 0.4d + 2 = 34

Combining the constants

-4

and

6

yields

2

.

4

Subtract 2 from both sides of the equation to isolate the variable terms.

\frac{2}{3}d + 0.4d = 32

Subtracting

2

from both sides simplifies the equation to have variable terms on one side and constant terms on the other.

5

Convert the decimal

0.4

to a fraction to combine the coefficients of

d

.

\frac{2}{3}d + \frac{2}{5}d = 32

Converting

0.4

to

\frac{4}{10} = \frac{2}{5}

allows us to work with a common denominator.

6

Find a common denominator to add the fractions.

\frac{10}{15}d + \frac{6}{15}d = 32 \Rightarrow \frac{16}{15}d = 32

The least common multiple of

3

and

5

is

15

.

7

Multiply both sides of the equation by the reciprocal of the coefficient of

d

to solve for

d

.

d = 32 \times \frac{15}{16} \Rightarrow d = 2 \times 15 \Rightarrow d = 30

Multiplying by

\frac{15}{16}

isolates

d

on the left side of the equation.

Key Concept

Solving linear equations involving fractions and decimals by distributing, combining like terms, and isolating the variable.

Alternative Method

Instead of converting the decimal

0.4

to a fraction, you can multiply the entire equation by a common multiple like

15

to eliminate both the fraction and the decimal:

15 \left[ \frac{2}{3}(d - 6) + 0.4(d + 15) \right] = 15(34) \Rightarrow 10(d - 6) + 6(d + 15) = 510 \Rightarrow 10d - 60 + 6d + 90 = 510 \Rightarrow 16d + 30 = 510 \Rightarrow 16d = 480 \Rightarrow d = 30

. This method avoids working with fractional coefficients.

Estimated Time:1m 30s

Question 42Question →

A botanist models the growth of a rare seedling. The number of weeks $w$ that the seedling has been growing satisfies the linear equation:

\frac{1}{4}(3w - 8) + 0.6 = \frac{1}{5}(2w + 7)

If the seedling's growth continues to follow this model, what is the value of 3 less than 5 times the number of weeks the seedling has been growing?

-37

-17

22

25

37

Show answer & explanation

Answer: 37

Answer

37

Solving the given equation for

w

yields

w = 8

. The question asks for the value of 3 less than 5 times the number of weeks, which translates to the expression

5w - 3

. Substituting

w = 8

into the expression results in

5(8) - 3 = 37

.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common multiple of the denominators, which is 20.

5(3w - 8) + 12 = 4(2w + 7)

This clears the fractions and simplifies the equation to integer coefficients.

2

Distribute the coefficients across the terms inside the parentheses.

15w - 40 + 12 = 8w + 28

This removes the parentheses so terms can be grouped.

3

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

15w - 28 = 8w + 28

Combining

-40

and

12

simplifies the expression on the left.

4

Isolate the variable

w

by moving the variable terms to the left side and constant terms to the right side.

7w = 56

Subtracting

8w

and adding

28

to both sides groups like terms together.

5

Solve for

w

by dividing both sides by 7.

w = 8

This isolates the variable

w

to find the number of weeks.

6

Translate '3 less than 5 times the number of weeks' into an algebraic expression and evaluate it for

w = 8

.

5w - 3 = 5(8) - 3 = 37

This translates the verbal question into mathematical terms and calculates the final value.

Key Concept

Solving multi-step linear equations containing fractions and decimals, and translating verbal expressions into algebraic terms.

Alternative Method

Instead of clearing the fractions first, one could convert the fractions to decimals:

0.25(3w - 8) + 0.6 = 0.2(2w + 7)

. Distribute to get

0.75w - 2 + 0.6 = 0.4w + 1.4

, which simplifies to

0.75w - 1.4 = 0.4w + 1.4

. Subtracting

0.4w

and adding

1.4

to both sides yields

0.35w = 2.8

, which gives

w = \frac{2.8}{0.35} = 8

. Then, compute

5(8) - 3 = 37

.

Estimated Time:1m 30s

Question 43Question →

A business owner registers $4$ identical cell phone lines under a group plan. The service provider charges a monthly base fee of $\$15.00$ per line, plus $\$0.05$ per minute of call time. If the total monthly bill for all $4$ lines was $\$96.00$ and each line used the exact same number of minutes, how many minutes did each line use?

180

405

720

31.5

1,620

Show answer & explanation

Answer: $180$

Answer

Each cell phone line used

180

minutes.

The correct answer is

180

minutes. The total bill for the

4

lines is represented by

4(15.00 + 0.05m) = 96.00

. Dividing both sides by

4

gives the monthly charge per line:

15.00 + 0.05m = 24.00

. Subtracting the base fee of

15.00

from both sides leaves the total per-minute cost of

0.05m = 9.00

. Dividing by the per-minute rate of

0.05

gives

m = 180

.

Step-by-Step Solution

1

Set up the linear equation representing the total monthly bill.

4(15.00 + 0.05m) = 96.00

, where

m

represents the number of minutes used per line.

The total bill is the sum of the charges for all

4

lines, where each line costs

\$15.00

plus

\$0.05

per minute.

2

Divide both sides of the equation by

4

to isolate the single-line cost expression.

15.00 + 0.05m = 24.00

Dividing both sides by

4

simplifies the equation and isolates the cost per line.

3

Subtract the base fee of

15.00

from both sides of the equation.

0.05m = 9.00

This isolates the variable charge term on the left side of the equation.

4

Divide both sides by

0.05

to solve for

m

.

m = 180

Dividing the remaining total of

9.00

by the rate of

0.05

per minute yields the total number of minutes used.

Key Concept

Solving Linear Equations

Alternative Method

Instead of dividing by

4

first, distribute the

4

to both terms inside the parentheses:

4(15.00) + 4(0.05m) = 96.00

. This simplifies to

60.00 + 0.20m = 96.00

. Subtract

60.00

from both sides to get

0.20m = 36.00

. Finally, divide by

0.20

to find

m = 180

.

Estimated Time:1m 15s

Solving Linear Equations

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method