Many word problems involving percents fit the above model—that is, a quantity being increased or decreased. Often you can solve these problems using one of the following formats:
Examples
A $100
jacket will be reduced by 15% for a sale. What will the sale price be?
The sale
price is $85.
More often, the sale price will be known and the original price is not known.
The sale price for a computer is $1200, which represents a 20%
markdown. What is the original price?
Let x represent
original price. Then the sale price is x – 0.20x = 0.80x.
The sale
price is also 1200. This gives us the equation 0.80x = 1200.
The original
price is $1500.
In the first example, the percent was multiplied by the number given;
and in the second, the percent was multiplied by the unknown. You must be very careful
in deciding of which quantity you take the percent. Suppose the first problem
was worded, ‘‘An item is marked down 20% for a sale. The sale price is $80,
what is the original price?’’ The equation to solve would be x – 0.20x ¼ 80x,
where x represents the original price. A common mistake is to take 20% of $80
and not 20% of the original price. The data used in the following examples and
practice problems are taken from the 117th edition of Statistical Abstract of
the United States. Many quantities and percentages are approximate.
Examples
The average
number of hours of television watched by U.S. adults during 1980 was 1470. By
1995, the number of hours of television viewing increased by about 7.1%. What
was the average number of hours of television viewing in 1995?
1980 hours +7.1% of 1980 hours = 1995
hours
1470 + (0.071)(1470) = 1470 + 104.37 =
1574.37
The average
number of hours U.S. adults spent watching television in 1995 was about 1574.
On January 1, 1995, the price of a first
class postage stamp was $0.32, which is a 60% increase from the cost on
November 1, 1981. What was the cost of a first class stamp on November 1, 1981?
Let x = 1st class price on November 1, 1981
$0.32 is 60%
more than this quantity.
The price of
a first-class stamp on November 1, 1981 was $0.20.
Practice
1. A
local cable television company currently charges $36 per month. It plans an
increase in its monthly charge by 15%. What will the new rate be?
2. In
1980, the median age of U.S. residents was 30 years. By 1996, the median age
had increased by about 15.3%. What was the median age in 1996?
3. In
1995, the death rate per 100,000 U.S. residents from major cardiovascular disease
was 174.4, which is about a 313 4% decrease from 1980. What was the 1980 death
rate per 100,000 from major cardiovascular disease?
4. A
worker’s take-home pay was $480 after deductions totaling 40%. What is the
worker’s gross pay?
5. A
cereal company advertises that its 16-ounce cereal represents 25% more than
before. What was the original amount?
6. A
couple does not wish to spend more than $45 for dinner at their favorite
restaurant. If a sales tax of 71 2%is added to the bill and they plan to tip
15% after the tax is added, what is the most they can spend for the meal?
7. A
discount store prices its blank videotapes by raising the wholesale price by
40% and adding $0.20. What must the tape’s wholesale price be if the tape sells
for $3.00?
Solutions
1. $36
will be increased by 15%:
36
+ (0.15)(36) = 36 + 5.4 = 41.4
The
new rate will be $41.40.
2. 30
is increased by 15.3%:
30
+ (0.153)(30) = 30 + 4.59 = 34.59
The
median age of U.S. residents in 1996 was about 34.6 years.
3. The
1980 rate is decreased by 
. Let x = 1980 rate.
The
1980 death rate per 100,000 from major cardiovascular disease was about 255.5.
4. The
gross pay is reduced by 40%. Let x = gross pay.
The
worker’s gross pay is $800.
5. The
original amount is increased by 25%. Let x = original amount.
The
original box of cereal contained 12.8 ounces.
6. The
total bill is the cost of the meal plus the tax on the meal plus the tip.
Let
x = cost of the meal. The tax, then, is 0.075x.
The
tip is 15% of the meal plus tax: (x + 0.075x = 1.075x is the price of the
meal), so the tip is 0.15(1.075x) = 0.16125x.
The
total bill is x + 0:075x + 0.16125x. We want this to equal 45:
The
couple can spend $36.40 on their meal.
7. Let
x ¼ wholesale price. 40% of the wholesale price is 0.40x. The retail price is
the wholesale price plus 40% of the wholesale price plus $0.20:
The wholesale price is $2.
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