Applying Rational Equations

Rational expressionsA fraction with a polynomial in the numerator and/or denominator. and rational equationsAn equation that contains one or more rational expressions. can be useful tools for representing real-life situations and for finding answers to real problems. In particular, they are quite good for describing distance-speed-time questions, and modeling multi-person work problems.

Work problems often ask us to calculate how long it will take different people working at different speeds to finish a task. The algebraic models of such situations often involve rational equations derived from the work formula, `W = rt`. The amount of work done (`W`) is the product of the rate of work (`r`) and the time spent working (`t`). The work formula has `3` versions:

Some work problems have multiple machines or people working on a project together for the same amount of time but at different rates. In that case, we can add their individual work rates together to get a total work rate. Let’s look at an example:

Example
Problem	Myra takes `2` hours to plant `500` flower bulbs. Francis takes `3` hours to plant `450` flower bulbs. Working together, how long should it take them to plant `1500` bulbs?
	Myra: `500` bulbs in `2` hours `=250` bulbs per hour Francis: `450` bulbs in `3` hours `=150` bulbs per hour	Think about how many bulbs each person can plant in one hour. This is their planting rate.
	Myra and Francis together: `250+150` bulbs per hour `=400` bulbs per hour	Combine their hourly rates to determine the rate they work together.
	`400/1=1500/t`	Use one of the work formulas to write a rational equation, for example `r=W/t`. We know `r`, the combined work rate, and we know `W`, the amount of work that must be done. What we don't know is how much time it will take to do the required work at the designated rate.
	`400/11t=1500/t1t` `400t=1500` `t=1500/400=15/4` `t = 3 3/4 " hours"`	Solve the equation by multiplying both sides by the common denominator, then isolating `t`.
Answer	It takes `3` hours `45` minutes for Myra and Francis to plant `1500` bulbs together.

Other work problems go the other way. We'll calculate how long it will take one person to do a job alone when we know how long it takes a group to get it done:

Example
Problem	Jamie, Pria and Paul can paint a room together in `2` hours. If Pria does the job alone she can paint the room in `5` hours. If Paul works alone, he can paint the room in `6` hours. If Jamie works alone, how long would it take her to paint the room?
Pria `+` Paul `+` Jamie `=1/2` room/hour Pria `=1/5` room/hour Paul `=1/6` room/hour Jamie `=1/t` room/hour		Determine the hourly rates for each person and for the whole group using the formula `r=W/t`. Work is painting `1` room, so `W=1`. We don’t know how long Jamie will take, so we need to keep the variable `t`.
`1/5+1/6+1/t=1/2`		Write an equation to show that the sum of their individual rates equals the group rate. (Think of it this way: Pria works for one hour and paints `1/5` of the room. Paul works for an hour and paints `1/6` of the room. Jamie works for an hour and paints `1/t` of the room. Together they have painted half the room in one hour.
`1/5(6t)/(6t)+1/6(5t)/(5t)+1/t*30/30=1/2` `(6t)/(30t)+(5t)/(30t)+(30)/(30t)=1/2` `(11t+30)/(30t)=1/2`		Solve the rational equation. First find the least common denominator of the individual rates. It is `5 * 6 * t = 30t`. Then multiply each term on the left by a fractional form of `1` so that all rates have the same denominator and can be added. (Note: we could also have found the common denominator of the entire equation, which is also `30t`, and multiplied both sides by it.)
`(11t+30)/(30t)30t=1/230t` `11t+30=30/2t` `11t + 30=15t` `11t - 11t + 30 = 15t - 11t` `30 = 4t` `7.5 = t`		Now multiply both sides of the equation by the common denominator, then simplify.
Answer	It would take Jamie `7.5` hours to paint the room by herself.

Sometimes work problems describe rates in a relative way: someone works `3` times as fast as someone else or a machine takes `2` fewer hours to finish a job than another model of machine. In these instances, we express one rate using information about another rate. Let’s look at an example:

Example
Problem	One pipe can fill a pool `1.5` times faster than a second pipe. If both pipes are open, the pool can be filled in `6` hours. If only the slower pipe is open, how long would it take to fill the pool?
	`r=W/t` fast pipe `=1/p` slow pipe `=1/(1.5p)` fast pipe `+` slow pipe `=1/6`	Find the rates of each pipe alone and the two working together. Work `=1` pool filled. Hours needed for fast pipe to fill pool: `p` Hours needed for slow pipe to fill pool alone: `1.5p` Hours needed for both pipes together: `6`
	`1/p+1/(1.5p)=1/6`	Write an equation that shows that the amount of work completed by both pipes in one hour is equal to the sum of the work of each pipe.
	`1/p6/6+1/(1.5p)4/4=1/6*p/p` `6/(6p)+4/(6p)=p/(6p)` `6 + 4 = p` `p=10` hours	Solve for `p`. One way to do this is to rewrite the rational expressions using a common denominator. Common denominator of `p`, `1.5p`, and `6` is `6p`.
	`10 * 1.5 = 15` hours	Slow pipe takes `1.5p` hours to do the work alone.
Answer	The slower pipe will take `15` hours to fill the pool alone.

Rational equations can be used to solve a variety of problems that involve rates, times, and work. Using rational expressions and equations can help us answer questions about how to combine workers or machines to complete a job on schedule.

Learning Objective

Introduction

Solving Work Problems

Summary