WORD PROBLEMS AND APPLICATIONS
 
 

Objectives:  To continue to build on the techniques for solving word problems.

Recall 1:  (Starting a word problem.)

  1. Read the problem quickly to get a rough idea of the    nature of the problem.
  2. If there is something to sketch, then attempt to sketch it.     Label all the knowns and unknowns on your sketch.     This generally helps to identify what is known and what    is not known.
  3. Isolate the question in the problem.  This usually comes    at the end of the word problem.
  4. Each problem should begin with Let x = something.     The x should equal the quantity that you wish to find.     That quantity is usually determined by the question.
  5. Translate English phrases into mathematical symbols    and find any relations between the known quantities and    the unknown quantities.
  6. Interpret your answer and check if it makes sense.


 
Examples

Example 1.    The length of a rectangle is equal to four times the width.  If the perimeter of the rectangle is 490 meters, what is the dimensions of the rectangle?

 Solution:    Make a sketch of the rectangle and label the sides.


 We know that the perimeter of the rectangle (the distance around the rectangle) is 490 meters.  Let P = the perimeter.  Then

P = x + 4x + x + 4x = 490
So,   10x = 490,
and
        x = 49.

 Therefore, one side is 49 meters and the other is 4*49 meters.  The dimensions of the rectangle are therefore 49 meters  196 meters.


 
Example 2. (Percent Problem)
In an experiment, it is calculated that in administering a new drug for the treatment of ulcers, 85.5% of patients who received the drug had had significant improvements of their ulcer condition.  If 1881 ulcer patients had significant improvements in their condition, how many patients participated in the experiment?

 Solution: Remember that percent means hundredths.  So, if you say 85.5 percent you mean 85.5 hundredths or .85.5/100   This means that if you say

85.5 percent of x    you mean    85.5/100 times x .

  Let x be the number of patients participating in the experiment.

  We know that (85.5/100)x = 1881.

  Solving for x we find that

    x = 1881/ (85.5/100)

    x = 2200.


 
Note
Remember to check on your answer.  Is it true that 85.5% of 2200 = 1881?
To find out, multiply  85.5/100 times 2200.  The result should be 1881.  Is it?

 
Example 3. (percentage)
A division of the Transylvania Company manufactures light bulbs.  Each month 20% of the bulbs it manufactures do not pass inspection.  Another 12% are damaged on route to stores.  If the company can sell 18,020 bulbs, how many bulbs should it manufacture?

 Solution: Let x = the number of bulbs that should be manufactured.

  20% of x is the number of bulbs that do not pass.  Hence,
  (20/1 00)x = the number of bulbs that do not pass.

  12% of x is the number of damaged bulbs.

  (12/100)x = the number of damaged bulbs.
 

  Use the fact that

 number of bulbs to be manufactured
                minus
    number of bulbs that do not pass
                minus
        number of damaged bulbs
                equals
               18,020

  This translate into: x - (20/100)x - (12/100)x = 18020

  Solving for x, we get

    x - .20x - .12x = 18020,
    or x - .32x = 18020,
    x(.68) = 18020,
    x = 26,500 light should be manufactured.


 
Example 4. (Mixture problem)
A supermarket contracts an ice cream manufacturer to mix 2000 gallons of Haag ice cream that contains 18 % fat with 1,800 gallons of Dass ice cream that contains 12% fat to produce an ice cream called Hagg and Dass.  What is the amount of fat content in the Hagg and Dass mixture?

 Solution: Let x be the content of fat in Hagg and Dass.

  We use the fact that

amound of fat in one cream + amound of fat in second cream =amound of fat in micture
      18% of 2000               +   12% of 1800                        = x% of (2000 + 1800)  .
  
   The quantity (2000 + 1800)  comes from the fact that  the new mixture will contain (2000 + 1800) gallons of ice cream.
 
      (18/100)2000            +      (12/100)18000                    =     (x/100)3800

  Or,
        360 + 216 = 38x,
        576 = 38x,
        x = 15.157894 percent,
  or approximately 15 percent.


 
Note
Check that this makes sense.  If the answer is correct, then 15% of the total amount of the mixture should be approximately equal 18% of 2000 gallons plus 12% of 1800 gallons.

18% of 2000 = 360,
12% of 1800 = 216,
15% of 3800 = 570.            check that 570 = 360 + 216.


 
Note
In the example above, two batches of ice cream are mixed to get a new batch of ice cream, the mixture.  The amount of substance in the mixture is the amount of substance in the first batch plus the amount of substance in the second batch.  The substance in the above example is ice cream.  This means that the amount of ice cream in the first mixture plus the amount of ice cream in the second mixture is equal to the amount of ice cream in the mixture.  We used this fact when grouping the 2000 gallons with the 1800 gallons to get the 3800 gallons of mixture.

     
Example 5.(Geometry problems)
A high rise apartment building is planned for a 20,000 square foot lot.  Because of strict zoning laws, the number of stories permitted for an apartment building is tied to the number of tenants in the building.  Each tenant must have a parking space; there are 20 tenants to a floor; and, each parking space requires 40 square feet of space.  If the area of the base of the building is 10 times the square of the number of floors, what is the largest number of floors that the zoning law will permit?

 Solution: Let x = number of floors that the zoning law will permit.

   Then the size of the base of the building is 10x^2.
   The number of tenants is 20x.
   The amount of space required for the parking lot is (20x)40

   We therefore know that

 the building space      +     the parking space  <=   the size of the lot
 
           10x^2            +       (20x)40             <=         20,000.

   The maximum x will be achieved when the inequality is an equality; in other words, when

            10x^2 + (20x)40 = 20,000.
 
   Solving for x, we get 10x^2 + 800x = 20,000,
   or                               x^2 + 80x - 2,000 = 0,
   This quadratic equation factors as

    (x + 100)(x - 20) = 0.
   The solution to this quadratic is x = -100 and x = 20.  Since x  is the number of floors of the high rise, it must be a positive  number.  We must, therefore, reject the negative solution    x = -100.  The answer is x=20 floors.


 
Example 6. A horse race track is 1.5 miles long.  The winner came in only five seconds before the second horse.  If the winner averaged 80 miles/hour during the race, how much faster (on average) should the second horse have gone in order to keep up with the first?

 Solution: Let v2 = the velocity of the second horse.

We use the relation between distance, velocity and time as follows:

    distance = velocity x time.

We must not forget, however, that all the units must match up.  Therefore, we must convert the 80 miles/hour to miles/second or convert the 5 seconds to hours.

1 hour = 60 minutes  and  1 minute = 60 seconds.  So, 1 hour = 60(60) seconds.
 
Therefore, the velocity of the winner--converted into miles per second--is

 80 miles/hour= (1/45) miles/second

The total distance of the race is 1.5 miles.  Therefore, using the

velocity x  time = distance

relationship, we get

   (1/45) miles/second*t = 1.5 miles

where t is winner's time in finishing the race (in seconds).

We know that the second horse came in 5 seconds after the winner.  Hence, we know that the velocity of the second horse times (t + 5) equals 1.5.  In other words, if v2 = the velocity of the second horse, then

    v2 *(t + 5) = 1.5.

We may find t by solving the equation

    (1/45) miles/second*t = 1.5 miles

for t.  We find that t = 67.5 seconds.  Therefore,

    v2 *(67.5 + 5) = 1.5.

Solving for v2, we get v2 = 0.021 miles/second or 21/1000 miles/second.

The difference between the winners velocity and this velocity is

1/45  - 21/1000  = 55/45000 miles/second.

The answer could be considered to be 55/45000 miles/second, or converted back to a velocity in miles per hour by multiplying by 3600, since there are 3600 seconds to an hour.  We get  (55/45000)*3600 miles/hour.


 
EXERCISES

 
1. Two numbers have a relationship where one minus eight times the other equals 57.  The average of the two numbers is 6.  Find the numbers.
 

 
2. In a certain far away land, a duke was given on his twentieth birthday a large number of gold ingots.  When he reached 30, he fell in love with a beautiful princess whose fancy he could not easily attract.  After many attempts, he fell to drink and gambling.  At 31, he gambled and lost 1/6 of his gold ingots, at 32 he lost 1/5 of the remaining ingots.  At age 34 he had only 10 ingots left.  How many gold ingots did the duke receive for his twentieth birthday?
 

 
3  The editor of a large publishing company reviews algebra manuscripts for possible publication.  If she rejects 98.5% of the manuscripts that she reads per year, how many manuscripts does she review for publication, if she accepts 15 new manuscripts per year?
 

 
4 A chemical plant discharges diluted waste that contains 0.04% H2S (hydrogen sulfide) into a 15,000 gallon holding container that contains 10,000 gallons of liquid containing 0.01% H2S.  The mixture is released into a stream when the concentration reaches 0.02% H2S.  How much waste is needed before the valves of the holding container are opened to allow the mixture at 0.02% to flow into the stream?
 

 
5 The English Channel tunnel was dug by an English team and a French team.  The French team drilled toward England at the rate of 130 feet per week, while the English team, having poorer drilling equipment drilled at a rate of 90 feet per week.  If it took two years to complete the drilling, how long is the tunnel?
 

 
6 A group of young teenagers assemble computers in a garage.  They decide to go into business by selling three different computers.  One contains a single floppy drive and costs $740.00.  Another has one floppy drive and one hard drive and costs $1080.00.  A third contains two floppy drives plus a hard drive and costs $1200.00.  If the costs are entirely due to the drives, what does a floppy drive cost?  What does a hard drive cost?
 

 
7 A health food store offers three mixtures of Colombian coffee.  The first contains 5% chicory and costs $4.50 a pound.  The second has 5% chicory and 5% cocoa bean and costs $4.55 a pound.  The third has 5% cocoa bean and costs $4.75 per pound.  If the price of the coffee is entirely due to the chicory and cocoa bean, what is the price per pound of the cocoa bean?
 

 
8 During the Ming Dynasty in China cricket fighting was so much in vogue in the Imperial Palace that the people were permitted to pay taxes by sending crickets.  Taxes were based on the amount of crops that were grown and brought to market.  A bushel of rice could be sold at the market for 40 yuan.  The taxes were assessed at 10% of market value; but one good fighting cricket could be offered in place of 55 yuan worth of taxes.  If a peasant sold 500 bushels of rice in the market during the year, how many crickets would need if he wished to pay 22% of his taxes in good fighting crickets?
 

 
9 Suppose that the value of a good fighting cricket is as it was in exercise 8.  Also suppose that a bushel of rice sells for 40 yaun but requires 10% tax, a bushel of maize for 25 yaun requiring 12% tax and a bushel of potatoes for 12 yaun with 15% tax.  If a peasant sells 200 bushels of rice, 100 bushels of maize and 50 bushels of potatoes, how many good fighting crickets will he need to pay 50% of the taxes in good fighting crickets?  (According to Ming tax laws any fractional value of a cricket is equivalent to the next larger whole number of crickets.)
 

 
10. The Emperor's cricket was entered in a cricket match.  Tickets were sold at 20 yaun, 40 yaun and 80 yaun each.  Ch'ang-an collected tickets and noticed that the number of 40 yaun tickets was three times the number of least expensive tickets and 50 less than the number of most expensive tickets.   The total amount collected was 8,000 yaun.  How many of each class of tickets was sold.
 
 

 
11 Two 19-th Century fictional characters are to fight a dual over a beautiful woman.  They choose their weapons, stand back-to-back and walk in opposite directions.  The taller of the two walks at a rate of one feet per second faster than the shorter.  At the end of 8 seconds they are 56 feet apart.  Find the speed of each person.
 

 
12 A troop of soldiers is marching in a line that is 400 feet long.  A messenger at the end of the line must deliver a message to the leader of the troop in the front of the line.  If the troops are marching at a rate of 8 feet per second and the messenger is running at a rate of 10 feet per second, how long will it take for the messenger to deliver his message?
 

 
13 In a foot race between Achilles and a tortoise the tortoise was given a head start of 400 feet.  If Achilles runs at a rate of 20 feet per second and the tortoise runs at a rate of 2 feet per second, how long will it take for Achilles to pass the tortoise?   


ANSWERS
 
1. 17 and -5

 
2.  x =15

 
3   1,000 manuscripts

 
4  15,000 gals. of mix
 
5  22,880 feet

 
6  $120.00 for the floppy drive;
   $340.00 for the hard drive.
 
7  $0.80/pound

 
8  8 crickets

 
9  11 crickets

10 7 tickets at 20 yaun, 21 tickets at 40 yaun and 71 tickets at 80 yaun
 

11. One walks at a rate of 3 feet/sec, the other at 4 feet/sec.
 

12 200 seconds

13 400/18 seconds or  or 5,000 gals of waste  approx. 22.23 secs