WORD PROBLEMS INVOLVING RATIONAL EQUATIONS
 

NOTE:    This batch of word problems is the culmination of a more leisurely introduction.  For the quiz and homework you should expect the questions to be easier, in the main, than the ones here.  Check out the earlier topics here and here.     
 

Objective:  To learn how to solve word problems that reduce to rational equations.


 
Recall 1. The examples and exercises below exhibit word problems   that are solvable through translation to a rational equation.    Be sure to check that the solutions to the rational equations   make sense for the problem.  Take care to check for poles   and to eliminate any solutions that do not fit the problem.

 
Example 1.    A food distribution company applies to the local authorities for a permit to build a 900,000 cubic foot warehouse.  The local zoning board does not approve the plan, but does suggest that the plan would pass if the warehouse could be designed to be 50 feet longer and 40 feet narrower.  If the company wishes to preserve the 30 foot height and the volume, what were the dimensions of the warehouse that the zoning board rejected?

 Solution:    Make a sketch of the warehouse and label the dimensions.

 Let x be the length of the warehouse under the old plan.  The volume of the warehouse under the old plan and new one is the same 900,000 cubic feet.  Sketch a picture of the second plan for the warehouse

Since the volumes and heights are the same in both plans, we have

    (x + 50)(y - 40)(30) = 900,000,
and
    xy(30) = 900,000.

From the second equation, we find that y = 30,000/x.  Substituting this value in for y in the first equation, we find that

    

This reduces to the quadratic equation

    4x^2 + 200x - 150000 = 0,

which, by the quadratic formula, has solutions

    x = 

So, the length of the warehouse according to the old plan was approximately 170.26 feet.

You should check that this makes sense by finding the width of the warehouse according to the old plan.  Since

    y =30,000/x ,

we know that 30,000/170.26, or that y is approximately 176.2.  Now check that xy(30) is approximately 900,000.

Check:  (170.26)(176.2) = 899994.36, which is surely close enough to approve the check.


 
Example 2. Two honey bees take off from the same hive in pursuit of a honeysuckle bush that is 1000 feet from the hive.  If the second takes off 10 seconds after the first, reaches the bush 10 seconds before the first and averages 30 feet per second faster than the first, what was the first bee's average speed?

 Solution: Let x = the average speed of the first bee in feet per second.   Then the average speed of the second bee is x - 30.  Using the fact that
 
    time = distance / speed

and that both bees travel the same distance of 1000 feet, we get

 first bee's time    =    second bee's time      - 20 seconds.
 
    1000/x            =          1000/ (x-20)        -        20.

This last equation may be solved by multiplying by the L.C.D., which is
x(x - 30).  We get

     x(x - 30)(1000/x) = x(x - 30)(1000/(x - 30))  - x(x - 30)(20).

This reduces to the quadratic equation

    2x^2 - 600x - 3000 = 0.

By the quadratic formula, the solutions are

    x = ,

which is approximately x = 305 and x = -5.  The average speed cannot be negative; so we may treat the x = -5 as being extraneous.  The first bee's averaged speed was approximately 305 feet/second.


 
Example 3. Huckleberry Finn and Tom Sawyer have a raft race down the Mississippi River.  They start from a dock, race to a buoy that is 300 feet away and return.  Huckleberry's average speed was 1.4 miles/hour, relative to land.  If Tom makes the round trip in 5 minutes and if the river current is 25 feet/minute downstream, who won the race?

 Solution: Let x = Tom's speed relative to the land.  We are told that the total time it takes Tom to make the round-trip is 5 minutes; so,

 time going downstream + time going upstream = 5.

Using this fact, together with the fact that
 
    time = distance / speed

gives
     300/ (x + 25) + 300/ (x - 25) = 5.

This is the equation we must solve for x.

The L.C.D. is (x + 25)(x - 25).  Multiplying both sides of the equation by this L.C.D. gives
    (x - 25) (x + 25)(300/ (x + 25)) + (x - 25) (x + 25)(300/ (x - 25)) = 5.,

which reduces to the quadratic polynomial equation

    x^2 - 120x - 625 = 0.

This quadratic factors as

    (x + 5)(x - 125) = 0.

Therefore, the solutions are x = -5 and x = 125.  If Tom's average velocity had been negative, he would never have returned.  Therefore, the only sensible solution is x = 125 feet/minute.  Huckleberry's speed was 1.4 miles/hour.  We must convert Tom's 125 feet/minute speed into a mile/hour speed.
 

Tom's average speed converts to slightly more than 1.42 miles per hour.  Huckleberry's speed is short by about  2/100 of a mile per hour.  Therefore Tom wins the race.


     
EXERCISES

1. One Halloween evening a wizard and a witch counted the number of Snickers Bars they had each received as a result their trick-or-treat expedition.  The witch simply counted that he had received 7 Snicker Bars.  The wizard, being a math genius, noticed that if he divided 6 times the number of Snickers Bars that he received by five less than that number, he would have the square of the number of Snickers Bars that he had received.  Who had more Snickers Bars?
 

 
2. An aging king decided that it was time to divide his vast land among his three daughters--Goneril, Regan and Cordelia.   He decided to have them race for the amount of land that they would acquire.  Each would get a square piece that was as large as the square of the distance they ran in 1 hour.   Goneril's average speed was 2 miles/hour less than Regan's, but 3 miles/hour faster than Cordelia's.  How much land did the king leave to Cordelia?
 

 
3  A harried parent leaves his house during rush-hour each weekday to fetch his wife at the railroad station and return to pick up his child from school.  His house, station and school are equal distance of 1 mile from each other.  (On an imaginary triangle.)  He usually makes the whole trip in 15 minutes.  Because of heavy traffic at rush-hour, he averages 10 miles/hour slower between the station and the school.  How fast is his speed between the house and the station?
 

 
4 A sorcerer's love potion boils over and leaves a huge pool of potion in his laboratory.  He knows that if he asks his apprentice to clean it up, it would take 20 minutes longer than if he did it himself.  So he decides to have the apprentice help him do the job.  Together, they clean the mess in 30 minutes.  How long would it have taken if the sorcerer did the job alone ?

 
5 A Patriot missile is launched at the same time as a Scud missile and is aimed at destroying the Scud  .  The Scud is traveling at the rate of 1000 miles/hour, while the Patriot is traveling at 800 miles/hour.  If it takes two minutes for them to be within 1000 feet from each other, What is the distance between their launching sites?
 

 
6 A company manufactures computer terminals and receives a contract for designing a new model terminal.  It specifies that the new model should have a screen that is 2 inches narrower and 5 inches taller than the old model.  If the total screen area of the old model was 36 square inches and new model is 10 square inches larger than the old, what are the dimensions of the old model?
 

 
7 Two speed boats depart from a harbor and head in opposite directions on a river.  The river speed, relative to the ground is 14 miles per hour.  After 8 minutes they are 20 miles apart.  Find the speed of each boat.
 

 
8 A police car arrives at the scene of a robbery of The Fifth National Bank of Hometown, just as the thief in a get-away car rounds a corner 1000 yards away.  The police car chases after the thief at an average speed of 70 miles per hour. If the get-away car averages 55 miles per hour, how long will it be before the cop catches the thief?

 ANSWERS 
 
1 The witch (The wizard had only 6 Bars).
 

 
2  16 square miles
 

 
3   approximately 17.4 miles per hour.

 
4 approximately 51.62 minutes.

 
5  Approximately 60 miles.
 

 
6  approx. 6.29 by 5.22.
 

 
7  82 miles/hour and 68 miles/hour
 

 
8  approximately 2.27 minutes or 136 seconds