Applications of Rational Equations

When am I ever going to use this?

Example #1 – Beach SweepIt will take Hannah 6 hours to clean up her stretch of beach.  Kendrick will take 10 hours to clean up his stretch of the beach.  If they work together, how long will it take to clean up: a) One stretch of beach?  b)  Both stretches of the beach?

Unit Rate: Hannah’s Rate: Kendrick’s Rate: Their combined rate to clean up 1 stretch of beach is the addition of their rates.

Find the LCM: Multiply each part of the equation by the LCM:

Simplify each part:

Solve the remaining equation:

This means that Hannah and Kendrick have a combined rate of 3.75 hours to clean up 1 stretch of beach. To clean up two stretches of beach it would take 3.75 + 3.75 = 7.50 hours.Be sure to TEST your solution! Use your calculator if needed. It works!

Example #2 – Sailing to OcracokeTony is bound for Ocracoke from harbor in Oriental.  He will return in five days.  The current of the waters along the way have an average rate of 20 miles per hour.  The round trip is 90 miles and will take a total of 10 hours.  If he is traveling against the current on his trip to Ocracoke and with the current on his return to Oriental, what is his speed in still water? 

It is important for us to remember that We adjust the equation to get: . Since the round trip is 90 miles, it is 45 miles each way. We will let , our unknown.When Tony is traveling to Ocracoke, his speed will be the and when he is traveling to Oriental his speed will be the These combined parts must equal the total time of 10 hours.

To solve the rational equation we will find the LCM. Multiply each part by the LCM. Simplify the parts, and then solve the remaining equation.LCM =

It would be appropriate to either factor or use quadratic formula to solve this.

It would be appropriate to either factor or use quadratic formula to solve this.By factoring: This gives us two rates: r = 25 and r = -16. Since we are solving for a rate of speed, a negative speed does not make sense in this application, so we only use the rate of 25 miles per hour.We need to TEST this solution:

Tony’s speed in still water is 25 miles per hour.

Example #3 – Kayak ToursDown East Kayaks & Tackle Co. offers Kayak Tours of the Down East waters.  In still water, they paddle at a rate of 6 miles per hour.  The tour takes 3 hours round trip and covers 10 miles.  What is the rate of the current?

Similar to example #2, this will solve for the rate of the current, not the rate in still water.

Multiply by the LCM of

 

Since we are talking about a rate of current, the negative solution does not apply. Our rate of the current is 4 miles per hour.Test the solution:

Example #4 – Cleaning SolutionsTo save money, while still supporting the environment, Joyce mixes her own Citric Acid based cleaning products.  She has 12 milliliters of a 15% citric acid solution and wants to make a 60% Citric Acid solution.  How much of a 70% Citric Acid solution does she need to add?

 The percent of acid found in the solution is calculated as:

  Original Added Solution Final Solution

Amount of Acid .15(12) .70x .15(12) + .70x

Total Amount 12 X 12 + x

The percentage of the final solution needs to be 60 %, so we get:

LCM =

 

This means that 54 milliliters of a 70% solution must be added to the original.Test the solution!