Section 9.6: Applications of Rational Expressions
Rational equations often arise in real-life problems involving rates, proportions, and work. To solve these problems, set up an equation using rational expressions, then solve as usual, being mindful of restrictions.
Example 1: Work Problem
It takes Alice 4 hours to paint a room and Bob 6 hours to paint the same room. How long will it take them if they work together?
Alice’s rate: \( \tfrac{1}{4} \) room/hour.
Bob’s rate: \( \tfrac{1}{6} \) room/hour.
Together: \( \tfrac{1}{4} + \tfrac{1}{6} = \tfrac{5}{12} \) room/hour.
Time = reciprocal = \( \tfrac{12}{5} = 2.4 \) hours.
Example 2: Distance/Rate Problem
A boat travels 30 miles downstream in 2 hours and returns upstream in 3 hours. Find the speed of the boat in still water and the current.
Downstream speed: \( \tfrac{30}{2} = 15 \) mph.
Upstream speed: \( \tfrac{30}{3} = 10 \) mph.
Let \( b \) = boat speed, \( c \) = current.
\( b+c=15 \), \( b-c=10 \).
Solve: Add equations → \( 2b=25 \) → \( b=12.5 \). Then \( c=2.5 \).
Example 3: Mixing Problem
A tank is filled by Pipe A in 5 hours and emptied by Pipe B in 8 hours. How long will it take to fill the tank if both are open?
Pipe A: \( \tfrac{1}{5} \) tank/hour (fills).
Pipe B: \( -\tfrac{1}{8} \) tank/hour (empties).
Net rate: \( \tfrac{1}{5} - \tfrac{1}{8} = \tfrac{3}{40} \) tank/hour.
Time = reciprocal = \( \tfrac{40}{3} \approx 13.33 \) hours.
Practice Problems
- Tom can complete a task in 6 hours. Jane can do the same in 4 hours. How long if they work together?
- A cyclist rides 60 miles downstream in 3 hours and back upstream in 4 hours. Find the cyclist’s speed in still water and the current.
- A pump can fill a pool in 7 hours. A drain can empty it in 10 hours. How long will it take to fill the pool if both are open?
- A car travels 180 miles at 60 mph. How long would the trip take at 45 mph?
- If one pipe fills a tank in 3 hours and another in 5 hours, how long if both work together?