MTH 030 -- Elementary Algebra -- Exercise Solutions
Section:

Percent Problems

All percent problems are solved by referring to the basic formula that relates percentage, percent, and the base of the percentage.

     That formula is Percentage = (Percent)(Base)

Notice that the formula contains just three quantities (percentage, percent, and base). If any two of these three quantities are known, their value may be substituted into the basic formula and it may then be solved to determine the third.

There are only three percent related problems:
   (1) Given Percent and Base, calculate Percentage
   (2) Given Percent and Percentage, calculate Base
   (3) Given Percentage and Base, calculate Percent

If we let A represent Percentage, let P represent percent and B represent the Base, the the basic formula may be written as
           A = PB
which is read as follows:    The percentage A is P percent of the base B.

Use this verbal statement and the basic formula to help translate percent problems into an equation.

Finally remember the meaning of percent is "per 100" and convert all percents to decimals.

15. Translate the following statement into an equation:  12 is 40% of what number?
Solution: Clearly 40% is the percent P. Compare 12 is 40% of what number with percentage A is P percent of the base B and it is pretty clear that in this problem the percentage is 12 and the unknown quantity is the base.
This yields the equation   12 = (40%)(B)  and converting percent to a decimal gives the equation 12 = 0.40B

16. Translate the following statement into an equation:   99 is what percent of 200?
Solution: Clearly the missing part is percent. Compare 99 is what percent of 200 with
percentage A is P percent of the base B and it is pretty clear that in this problem 99 is the percentage and 200 is the base.
This yields the equation 99 = P(200)

17: Change the following to decimals:
     a. 35% means 35 per 100 which may be written as the ratio 35/100 from which we get the decimal 0.35.
         Therefore 35% = 0.35   (or you can just move the decimal point two places to the left -- that's what division by 100 does)
     b. 3.5% means 3.5 per hundred whcih may be written as the ratio 3.5/100 from which we get the decimal 0.035
          Therefore 3.5% = 0.035
(or you can just move the decimal point two places to the left -- that's what division by 100 does)
     c.  350% means 350 per hundred which may be written as the ration 350/100 from which we get 3.50
          
Therefore 350% = 03.5 (or you can just move the decimal point two places to the left -- that's what division by 100 does)
      d. 1/2 % is best understood by changing 1/2% to 0.5% which means 0.5 per hundred which may be written as the ratio .5/100
           from which we get the decimal 0.005
           Therfore 1/2 % = 0.005
(or you can just move the decimal point two places to the left -- that's what division by 100 does)
           

18. Change each decimal to a percent

   a. 0.9
Solution:  
 

b.    0.09
Solution:

c.   9
Solution:

d.   0.999
Solution:

27. An existing 1000 foot water line had to be extended to a length of 1525 feet to reach a new restroom facility. Describe a quantity in two ways. Write the resulting equation.
Solution:
The new length is 1525 feet.
The new length is also the old length plus the extension.
Let x be the length of the extension, then the new length is 1000 + x.
We now have the new length expressed in two ways and since these two expressions represent the same thing they must be equal.
Therefore 1000 + x = 1525.

28. Because of overgrazing, state agriculture officials determined that the 4500 head of cattle currently on the ranch had to be reduced to 2750.
Solution:
The current population is 4500.
The future population is 2750.
The future population is also the current population minus the reduction.
We don't know the reduction. Let x represent the reduction. Then we can express the future population as 4500 - x.

We now have future population expressed in two ways and since these two expressions represent the same quantity, they must be equal.
Therefore 4500 - x = 2750.

29. A length of gold chain, cut into 12-inch-long pieces makes five bracelets
Solution:
The length of the chain is unknown.  Let it be represented by the variable x.

A bracelet is 12 inches long. the total length of chain produces five bracelets.
The total length of chain is therefore (5)(12) = 60 inches.
We now have the total length of chain represented as 60 and as x.
Therefore x = 60.

30. The 24 ounces of walnuts used to make a fruitcake were twice what was called for in the recipe.
Solution: The amount called for in the recipe is unknown.  Let it be reperesented by the variable x.
The amount used was 24 ounces, but it was also twice the amount called for or 2x.

Therefore 24 = 2x.

The following percent problems are presented in an unconventional manner.  Diagrams are used to help relate the statement of the problem with the fundamental formula A = PB by which ALL percent problems are solved.

31.  

32.

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34.

35.

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38.

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40.

41.

42.

Some of the same percent problems are shown below in a more conventional manner.