# Linear Equation Word Problems Worksheet

Linear Equation Word Problems Worksheet
• Page 1
1.
Plant A is 25 ft tall and plant B is 20 ft tall. Plant A grows 2 ft per week and Plant B grows $2\frac{1}{5}$ ft per week. In how many weeks will the heights of the two plants become equal?
 a. 20 b. 30 c. 25 d. 50

#### Solution:

Height of plant A is 25 ft, whose rate of growth is 2 ft per week.

Height of the plant B is 20 ft, whose rate of growth is 21 / 5ft per week.

Let both the plants will have the same height after c weeks.

So, height of plant A after c weeks = height of plant B after c weeks.

25 + 2(c) = 20 + (21 / 5 ) (c)
[Algebraic model.]

1 / 5 c = 5
[Simplify.]

c = 25.
[Solve for c.]

The plants will have the same height in 25 weeks.

2.
The front page of a newspaper is $12\frac{1}{2}$ inches wide. The left and right margins are each $x$ inches wide. The news is in four columns each of width 2 inches and the space between two columns is $\frac{1}{2}$ inch. Find the value of $x$.
 a. 2 b. 3 c. 1.5 d. 1

#### Solution:

Width of the newspaper = 121 / 2 inches.

Left, right margins are each of x inches

The news is in four columns each of width 2 inches and the space between the two columns is 1 / 2 inch.

Now the front page of the newspaper can be seen as shown.

From the figure, we have x + 2 + 1 / 2 + 2 + 1 / 2 + 2 + 1 / 2 + 2 + x = 121 / 2
[Algebraic model.]

2x + 8 + 3 / 2 = 121 / 2

2x = 3 x = 1.5
[Solve for x.]

The value of x = 1.5.

3.
Ethan has two cars A and B. He used car A on the first day of the week and car B on the next day. He drove car A at 45 miles per hour for 3 hours and car B at 49 miles per hour for 4 hours. What is the total distance traveled by Ethan in these two days?
 a. 135 miles b. 61 miles c. 331 miles d. 196 miles

#### Solution:

Ethan drove the car A at 45 miles per hour for 3 hours.

Distance traveled by Ethan by the car A = (45)(3) = 135 miles.
[Distance = speed × time.]

Ethan drove the car B at 49 miles per hour for 4 hours.

Distance traveled by Ethan by the car B = (49)(4) = 196 miles.
[Distance = speed × time.]

So, the total distance traveled by Ethan = 135 + 196 = 331 miles.

4.
Brian has $1,500 and his sister Quincy has$1,920. Each gets $10 as an allowance per week. Brian saves his allowance and Quincy spends her allowance plus$20 each week. In how many weeks will they have equal amount with them?
 a. 342 weeks b. 42 weeks c. 14 weeks d. 390 weeks

#### Solution:

The initial amount with Brian is $1,500 and the initial amount with Quincy is$1,920.

Brian adds $10 each week to his amount and Quincy spends$20 (She gets $10 every week and spends it along with another$20) each week from her amount.

Let after x number of weeks the amount with Brian will be equal to the amount with Quincy.

1500 + 10x = 1920 - 20x
[Algebraic model.]

30x = 420 x = 14
[Solve for x.]

Brian and Quincy will have the same amount with them after 14 weeks.

5.
The number of students opting for Japanese and French in a school are 60 and 281 respectively. The number of students opting for Japanese has been increasing at the rate of 6 students per year and that for French has been decreasing at the rate of 11 students per year. How long will it take for the number of students taking Japanese to equal the number of students taking French?
 a. 2 years b. 24 years c. 26 years d. 13 years

#### Solution:

The number of students opted for Japanese = 60 and the rate of increase of students who opted for Japanese = 6 per year

The number of students who opted for French = 281 and the rate of decrease of students who opted French = 11 per year

Let after n number of years the number of students who opted for Japanese will be equal to the number of students who opted for French.

The number of students who opted for Japanese after n years = 60 + 6n
[Increases at 6 students per year.]

The number of students who opted for French after n years = 281 - 11n
[Decreases at 11 students per year.]

60 + 6n = 281 - 11n
[Algebraic model.]

17n = 221 n = 13
[Solve for n.]

After 13 years, the number of students opting for Japanese will be equal to the number of students opting for French.

6.
Brian wants to mail a packet. He packed it in a rectangular box as shown, which just meets the postal regulations. According to the postal regulations the combined length and girth of the package should not exceed 72 cm. If the length($l$) of the box is 28 cm, then what is the width of the box?
[Out of the three dimensions of a box the longest dimension is considered to be the length. The other two dimensions are each doubled and then added together to obtain the girth.]

 a. 22 cm b. 11 cm c. 44 cm d. 3 cm

#### Solution:

The rectangular box has square ends each of side x cm. So, the width of the box is x cm, and height is x cm.

Length of the required box = l = 28 cm

Girth of the box + Length of the box = 72 cm
[The box just meets the regulations.]

4x + 28 = 72
[The girth is the distance around the box.]

4x = 72 - 28, x = 11 cm

So, the width of the box is 11 cm.

7.
At West High University, 1430 students major in arts and 910 major in science. Arts students have been increasing at the rate of 40 students per year and the science students have been decreasing at the rate of 30 students per year. At these rates, after how many years the number of students who major in arts will be 3 times the number of students who major in science?
 a. 11 years b. 9 years c. 12 years d. 10 years

#### Solution:

Number of students who major in Arts = 1430
and rate of increase of students who major in Arts = 40 students per year

Number of students who major in Science = 910
and the rate of decrease of students who major in Science = 30 students per year

Let x be the number of years after which the number of students major in Arts is 3 times the number of students major in Science.

1430 + 40x = 3(910 - 30x)
[Algebraic model.]

130x = 1300 x = 10
[Solve for x.]

After 10 years the number of students major in Arts will be 3 times the number of students major in Science.

8.
Andrew traveled on a bike at a speed of 70 miles per hour and traveled in a car at a speed of 300 miles per hour. His time of traveling on the bike is twice as the time of traveling in the car. If the total distance traveled was 1760 miles, then how many hours did Andrew travel on the bike?
 a. 4 hours b. 510 hours c. 2.63 hours d. 8 hours

#### Solution:

The speed of the bike on which Andrew traveled is 70 miles per hour.

The speed of the car in which Andrew traveled is 300 miles per hour.

Let x hours be the time he traveled by car, then the time he traveled on the bike is 2x hours.

So, the distance traveled on the bike = (70)(2x) and

The distance traveled by the car = (300)x

So, the total distance traveled = (70)2x + (300)x

(70)2x + (300)x = 1760
[Total distance = 1760 miles.]

440x = 1760 x = 4
[Solve for x.]

So, Andrew traveled on the bike for 8 hours.
[2x = 8.]

9.
Matt has $3,000 and adds$10 each week to it. Tommy has $4,040 and spends$30 from it each week. After how many weeks will both Matt and Tommy have the same amount with them?
 a. 26 weeks b. 28 weeks c. 33 weeks d. 24 weeks

#### Solution:

The initial amount with Matt = $3,000 and his savings per week =$10

The initial amount with Tommy = $4,040 and he spends$30 from it each week.

Let both Matt, Tommy will have the same amount after n number of weeks.

After n weeks, the amount with Matt = 3000 + 10n and the amount with Tommy = 4040 - 30n
[Matt increases his amount by $10 every week and Tommy decreases his amount by$30 every week.]

3000 + 10n = 4040 - 30n
[After n weeks they will have equal amount.]

40n = 1040 n = 26
[Solve for n.]

So, after 26 weeks both Matt, Tommy will have the same amount with them.

10.
The girth of a rectangular box of length 40 in., height 30 in. is 100 in.. What is the width of the box?
[Out of the three dimensions of a box the longest dimension is considered to be the length. The other two dimensions are each doubled and then added together to obtain the girth.]
 a. 10 in. b. 40 in. c. 20 in. d. 100 in.

#### Solution:

The girth of a rectangular box of length l, width x and height y is 2x + 2y.
[Definition.]

Height of the box is y = 30 in. and the girth is 100 in..

2x + 2y = 100

2x + 2(30) = 100
[Substitute y = 30.]

x = 20 in.
[Solve for x.]

So, the width of the box is 20 in..