WARM UP 1.Solve 1.If a computer’s printer can print 12 pages in 3 minutes, how many pages can it print in 1 minute? Multiply through by x – x(x – 1) = 4(x – 1) 2 + x – x = 4x – 4 or 4 pages in 1 minute.

USING RATIONAL EXPRESSIONS

OBJECTIVES Solve work problems using rational equations. Solve motion problems using rational equations. Apply algebraic models to real world situations.

WORK PROBLEMS Tom knows that he can mow a golf course in 4 hours. He also knows that Perry takes 5 hours to mow the same course. Tom must complete the job in 2-1/2 hours. Can he and Perry get the job done in time? How long will it take them to complete the job together? Solving Work Problems If a job can be done in t hours, then of it can be done in one hour. (The above condition holds for any unit of time.)

SOLVING THE PROBLEM  UNDERSTAND the problem. Question: How long will it take the two of them to mow the lawn together? Data: Tom takes 4 hours to mow the lawn. Perry takes 5 hours to mow the lawn.  Develop and carry out a PLAN. Let r represent the total number of hours it takes them working together. Then they can mow of it in 1 hour.

TRANSLATION We can now translate to an equation. Translating to an equation We solve the equation. Multiplying on both sides by the LCD to clear fraction. hours

PROBLEM CONTINUED  Find the ANSWER and CHECK. Tom can do the entire job in 4 hours, so he can do just over half in 2 hours. Perry can do the entire job in 5 hours, so he can do just under half the job in 2 hours. It is reasonable that working together they can do the job in 2 hours. It will take them 2 hours together, so they will finish in time.

TRY THIS… Carlos can do a typing job in 6 hours. Lynn can do the same job in 4 hours. How long would it take them to do the job working together with two typewriters.

EXAMPLE 2 At a factory, smokestack A pollutes the air twice as fast as smokestack B. When the stacks operate together, they yield a certain amount of pollution in 15 hours. Find the time it would take each to yield that same amount of pollution operating alone. Let x represent the number of hours it takes A to yield the pollution. Then 2x is the number of hours it takes B to yield the same amount of pollution. is the fraction of the pollution produced by A in 1 hour. is the fraction of the pollution produced by B in 1 hour. Together the stacks yield of the total pollution in 1 hour. They also yield of it in 1 hour. We now have an equation: Solving for x, we get x = 22 ½ hours for smokestack A and 45 hours for B

TRY THIS… Pipe A can fill a tank three times as fast as pipe B. Together they can fill the tank in 24 hours. Find the time it takes each pipe to fill the tank.

MOTION PROBLEMS Recall from Chapter 4 the formula for distance, d = rt. From this we can easily obtain rational equations for time and for the rate of speed. and We can use these equations to solve motion problems.

EXAMPLE 3  UNDERSTAND the problem. Question: What is the speed of the wind? Data: An airplane has a speed of 200 km/h. The airplane flies 1062 km with the wind and 738 km against the wind in the same amount of time. An airplane flies 1062 km with the wind. In the same amount of time it can fly 738 km against the wind. The speed of the plane in still air is 200 km/h. Find the speed of wind.

EXAMPLE 2 CONT.  Develop and carry out a PLAN. First draw a diagram. Let r represent the speed of the wind and organize the facts in a chart km t hours r (The wind increases the speed) t hours 738 km r (The wind decreases the speed) DISTANCERATETIME With wind rt Against wind – 4t

EXAMPLE 3 CONT. The times are the same, so we write the equation in the form: This is the translation. and Using substitution, we obtain Solving for r, we get 36.

EXAMPLE 3 CONTINUED  Find the ANSWER and CHECK. The number 36 checks in t he equation. A 36 km/h wind also makes sense in the problem. With the wind, the plane has a speed of 236 km/h. If it travels 1062 km, it must fly for 4.5 hours. Against the wind, the plane travels at 164 km/h. It will travel 738 km, also in 4.5 hours. Thus the speed of the wind is is 36 km/h.

TRY THIS… A boat travels 246 mi downstream in the same time it takes to travel 180 mi upstream. The speed of the current in the stream is 5.5 mi/h. Find the speed of the boat in still water.