Chapter Twenty-Three — Word Problems

The Humongous Book of Algebra Problems

538

23.36 A collection of 44 coins containing pennies, dimes, and quarters is worth

$5.90. There are twice as many dimes as quarters. How many pennies are in

the collection?

Let q represent the number of quarters, d = 2q represent the number of dimes,

and p = 44 – 3q represent the number of pennies. Multiply the total number of

each coin by its value and set the sum equal to 5.90.

Solve for q.

There are q = 13 quarters, 2(13) = 26 dimes, and 44 – 3(13) = 44 – 39 = 5

pennies.

Work

How much time does it save to work together?

23.37 Two computer programmers are instructed to write code for an application.

Working by herself, Donna could complete the task in ﬁve days, but when

working with Chris, she is able to complete the program in three days. How

long would it take Chris to complete the task by himself?

To set up a work problem, create one fraction for each of the individuals

involved. The numerator of each fraction is the length of time the individual

worked, and the denominator is the length of time it would take that individual

to complete the task working alone. Add the fractions and set the sum equal to

one.

Chris and Donna worked together for the same length of time, three days.

Donna can complete the task alone in ﬁve days. Let x represent the length of

time it would take Chris to complete the task alone.

There are

q quarters and

2q dimes. The

combined number of

quarters and dimes

is q + 2q = 3q. The

total number of coins

is 44, so subtract the

combined number of

quarters and dimes

from 44 to calculate

the number of

pennies.

This is the

expression for

pennies dened in

the beginning of the

problem: p = 44 – 3q.

The sum

of the fractions

always equals 1.

Chapter Twenty-Three — Word Problems

The Humongous Book of Algebra Problems

539

Multiply the entire equation by 5x, the least common denominator, to eliminate

fractions.

Solve for x.

It would take Chris days to complete the task by himself.

Note: Problems 23.38–23.39 refer to a pair of carpenters named Lisa and Karen. Lisa can

build a bookshelf in 30 minutes, and Karen can build the same bookshelf in 45 minutes.

23.38 How long will it take Lisa and Karen to build the bookshelf together?

Let x represent the length of time Lisa and Karen work together to complete

the bookshelf. Create a rational expression for each carpenter, dividing x (the

length of time worked by both) by the length of time it would take each to

complete the bookshelf alone. Add the fractions and set the sum equal to one.

Multiply the entire equation by 90, the least common denominator, to eliminate

fractions.

Working together, Lisa and Karen complete the bookshelf in 18 minutes.

“1” represents “1

complete bookshelf.”

The fractions on

the left side of the

equation represent

the portions of that

complete job that each

person is responsible for.

Chapter Twenty-Three — Word Problems

The Humongous Book of Algebra Problems

540

Note: Problems 23.38–23.39 refer to a pair of carpenters named Lisa and Karen. Lisa can

build a bookshelf in 30 minutes.

23.39 Karen is injured and can no longer complete the bookshelf alone in 45

minutes. It now takes Lisa and Karen 28 minutes 48 seconds to complete the

bookshelf working together. How long would it take Karen to complete the

bookshelf working alone?

Divide 48 seconds by 60 to convert into minutes: . It takes the

pair of carpenters 28.8 minutes to build the bookshelf together, and Lisa can

complete the bookshelf in 30 minutes by herself. Let x represent the length of

time it would take Karen to build the bookshelf.

Multiply the entire equation by 30x, the least common denominator, to

eliminate fractions.

Solve for x.

It would take Karen 720 minutes, or hours, to complete the task alone.

Note: Problems 23.40–23.41 refer to Rob and Matt, who go into business together mowing

lawns during the summer. Rob works twice as fast as Matt.

23.40 If it takes Rob and Matt 90 minutes to mow one lawn together, how long would

it take each of them to mow it separately?

Let x be the length of time it takes Rob to mow the lawn alone, and let 2x be

the time Matt needs to complete the same task. They spend 90 minutes working

together.

Start Free Trial

No credit card required