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The Humongous Book of Algebra Problems by W. Michael Kelley

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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 dened 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.

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