Mastery Check Solutions

Show What You Know

Two tanker trucks are needed to fill an Olympic-sized swimming pool at Sandy’s Swim Club. The first truck can fill the pool in x hours. The second truck takes twice as long to fill the pool alone. When the trucks fill the pool together it takes $3 \frac{1}{2}$ hours.

Make a chart to show time and rate of work for Truck A, Truck B, and Together.

	Truck A		Truck B		Together
time:	x		2x		$3 \frac{1}{3}$
rate of work:	$\frac{1}{3}$	+	$\frac{1}{2 x}$	=	$\frac{1}{3 \frac{1}{3}}$

Use part A to write and solve an equation to find the time it takes each truck to fill the pool alone.

$\begin{array}{rcl} \frac{1}{x} + \frac{1}{2 x} & = & \frac{1}{3 \frac{1}{3}} \\ \frac{1}{x} + \frac{1}{2 x} & = & \frac{1}{\frac{10}{3}} \\ \frac{1}{x} + \frac{1}{2 x} & = & \frac{3}{10} \end{array}$

$\begin{array}{rcl} LCD : 10 x \\ \frac{1 (10)}{(10) x} + \frac{1 (5)}{(2 x) (5)} & = & \frac{3 x}{10 (x)} \\ 10 + 5 & = & 3 x \\ 15 & = & 3 x \\ x & = & 5 \end{array}$

It would take Truck A 5 hours and Truck B 10 hours to fill the pool individually.

Note

Q: What are the restrictions for your equation? Explain.
A: The value of x cannot equal zero because then the denominator would be undefined (equal to zero).

The average person at Sandy’s Swim Club can swim the length of the pool, 50 meters, in 70 seconds. The club’s Olympic swimmer can swim the length of the pool in $2 p - 3$ seconds. It takes three times longer for the average swimmer to swim the length of the pool.

Determine how long it takes an Olympic swimmer to swim 50 meters. Show your work.

$\begin{array}{rcl} \frac{average}{olympic}, \frac{3}{1} & = & \frac{70}{2 p - 3} \\ 3 (2 p - 3) & = & 70 \\ 6 p - 9 & = & 70 \\ 6 p & = & 79 \\ p & = & \frac{79}{6} \end{array}$

$\begin{array}{rcl} 2 p - 3, p & = & \frac{79}{6} \\ 2 (\frac{79}{6}) - 3 \\ \frac{79}{3} - \frac{9}{3} & = & \frac{70}{3} \\ \frac{70}{3} & = & 23.3 \end{array}$

It will take an Olympic swimmer 23.3 seconds to swim the 50 meter pool.

Note

Write your answer as a mixed number or decimal because time is not written as an improper fraction.

Finding the value of p does not complete the problem because this is not the time of the Olympic swimmer. The step of substituting p into the expression 2p – 3 gives the time of the Olympic swimmer.

Q: What is the excluded value for this proportion?
A: $\begin{array}{rcl} p & \neq & \frac{3}{2} \end{array}$

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Determine the excluded values for the rational equation.
Solve rational equations that are proportions.
Solve rational equations by finding the least common denominator (LCD).
Apply rational equations to word problems.