The set-up for the applications in this chapter will include rational equations. The keyword, reciprocal, appears often. Remember that the reciprocal of a number is 1 divided by that number.
Example: One positive integer is 5 more than the other. When the reciprocal of the larger number is subtracted from the reciprocal of the smaller the result is 5/14. Find the two integers.
Example: The difference between the reciprocals of two consecutive positive odd integers is 2/15. Find the integers.
There are two equivalent formulas to choose from when solving work-rate problems. If two people are working together on a job then their work rates add and they can perform the job working together in a shorter amount of time. If we let x = the time it takes a person to complete a task then his work rate is 1/x. In other words, he can complete the 1 job in x number of hours. If we let y = the time it takes a second person to complete the task and t = the time it takes for both people working together we get the following work-rate formulas:
We have set up uniform motion problems using the formula D = r * t. For the following motion problems, we will need the equivalent formula D/r = t to set up the equations.
Example: The first leg of Mary’s road trip consisted of 120 miles of traffic. When the traffic cleared, she was able to drive twice as fast for 300 miles. If the total trip took 9 hours, then for how much time was she stuck in traffic?
Example: Brett lives on the river 45 miles upstream from town. When the current is 2 miles per hour he can row his boat downstream to town for supplies and back in 14 hours. What is his average rowing speed in still water?