Lesson 50: Applications of Rational Equations (Rates, Proportions)
Rational equations are commonly used to solve real-world problems involving rates, work, distance, and proportions. These applications often require setting up an equation where the variable represents time, speed, or another quantity.
Steps to Solve Application Problems
- Identify the variable and what it represents.
- Write expressions for each quantity in terms of the variable.
- Set up a rational equation relating the quantities.
- Find a common denominator if necessary, and solve for the variable.
- Check that your solution is reasonable and does not make any denominator zero.
Examples
Example 1: Work Problem
Two pumps fill a tank. Pump A can fill the tank in 4 hours, Pump B in 6 hours. How long to fill the tank working together?
- Rate of A: 1/4 tank/hour
- Rate of B: 1/6 tank/hour
- Together: 1/4 + 1/6 = 5/12 tank/hour
- Time to fill tank: t = 1 ÷ (5/12) = 12/5 = 2.4 hours
Example 2: Distance/Rate Problem
A car travels 150 miles. If it travels x mph, time = distance/rate = 150/x. If speed increases by 10 mph, time decreases by 0.5 hours. Find original speed.
- Equation: 150/x − 150/(x + 10) = 0.5
- Multiply through by x(x + 10) and solve quadratic: x² + 10x − 3000 = 0
- Solution: x = 50 mph (discard negative solution)
Practice Problems
- Two pipes fill a pool. Pipe A fills in 5 hours, pipe B in 3 hours. How long to fill together?
- A boat travels 60 miles downstream in 2 hours and returns upstream in 3 hours. Find speed of boat in still water.
- A man can paint a house in 8 hours, another in 6 hours. How long together?
- A car travels 120 miles. Increasing speed by 20 mph saves 1 hour. Find original speed.
- Two taps fill a tank in 4 and 6 hours. A leak empties 1/12 tank per hour. Time to fill tank with both taps and leak?