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Problem Solving and Rational
Equations
MATH 018
Combined Algebra
S. Rook
Overview
• Section 7.6 in the textbook:
– Solving proportions
– Using proportions to solve word problems
– Solving word problems about numbers
– Solving word problems about work
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Solving Proportions
Ratios & Proportions
• Ratio – a quotient of two quantities
e.g. 1/5 or x/(x – 1)
– An expression
• Proportion – a mathematical equation of two
equal ratios
Ex: 2/3 = 6/9 or x/(x – 1) = 7/8
– Only one rational expression on each side of the =
• We can solve a proportion as a rational equation
• We can also cross-multiply to solve a
proportion
e.g. x/4 = 1/2
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Solving Proportions (Example)
Ex 1: Solve:
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3
a)

x x 1
b)
5
3

y2 y4
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Using Proportions to Solve
Word Problems
Using Proportions to Solve Word
Problems
• Key is to extract information from the word
problem to set up proportions
• Proportions compare two units
– e.g. cups of sugar to batches, number of
cards to people, etc
• Align the units
– e.g. Put cups of sugar is in the numerator and
batches in the denominator for both sides of
the proportion
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– DO NOT mix up the units on each side
Using Proportions to Solve
Word Problems (Example)
Ex 2: Solve:
a) If 2 boxes of staples can fill 3 staplers,
how many boxes of staples are needed
to fill 9 staplers?
b) On a certain map 1.5 inches
represents 30 miles. How many
inches on this same map would be
used to represent 75 miles?
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Solving Word Problems about
Numbers
Solving Word Problems about
Numbers
• Look for key words to help set up the
problem
• Solve the resulting rational equation
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Solving Word Problems about
Numbers (Example)
Ex 3: Solve:
a) The quotient of three more than a number
AND two times the number plus five is equal
to three-fifths. Set up an equation and find
the number.
b) The quotient of twelve and a number equals
two-thirds. Set up an equation and find the
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number.
Solving Word Problems about
Work
Solving Word Problems about
Work
• Allows us to consider situations where two
people or objects perform some task
– Work problems answer one of the following
two questions:
• How long does it take a person or object to finish
the task alone
• How long does it take both people or objects to
finish the task together
• Key is to identify how much of a task can
be completed in 1 hour
– If a job can be finished in 2 hours, what part of
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the job has been completed in 1 hour?
Solving Word Problems about
Work (Example)
Ex 4: Solve:
a) If pump A can fill a swimming pool in 3
hours and pump B can fill the same swimming
pool in 2 hours, how long would it take pumps
A and B to fill the pool if they worked together?
b) A bricklayer’s apprentice can finish a job in 8
hours whereas the bricklayer and his
apprentice can finish the same job together in
2 hours. How long would it take the bricklayer
to finish the same job alone?
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Summary
• After studying these slides, you should know how to do
the following:
– Understand the concept of a proportion
– Solve a proportion
– Solve word problems involving proportions
– Solve these types of word problems that involve
rational equations:
• Numbers
• Work
• Additional Practice
– See the list of suggested problems for 7.6
• Next lesson
– Simplifying Complex Fractions (Section 7.7)
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