PERCENT PROBLEMS
Remember,
"percent" means "per hundred" so that 50% means 50 per
hundred or 
which is equal to 
.
To change from
decimals to percents move the decimal point two places to the right, and to
change from percents to decimals move the decimal point two places to the left.
There are three
good methods with which to solve percent problems.
METHOD 1: Proportion Method
There are three
numbers involved in percent problems.
For example, if I get 18 out of 20 problems correct on a test, my score
is 90%. In other words, I get
"part" of the problems right, the "whole" test has 20
questions, and the "rate" at which I answer questions correctly is
90%. My grade is "based" on
20 questions. For percent problems we
use the formula
In this case, 
In any percent
problem you might be looking for "part" of something, the whole or "base" amount, or a
"rate." The base amount might
also be the original amount, which we'll see in problems that are looking for a
percent change.
Trick: If you don't know the percent then you are
looking for the rate. But if you know
the percent, then which is the part and which is the whole amount? You can't be sure that the smaller number is
the "part" because what if I not only answered all 20 questions
correctly but answered the bonus question correctly as well? (I didn't mention the bonus question, did
I?) In that case my score might be
105%! So look for the words
"is" and "of." For
example, 18 "is" what percent "of" 20?
METHOD 2: Algebraic Method.
Since 18 is 90%
of 20 we can "translate" from English to algebra and write
18 = 0.90 x 20
The formula is
now
Part = Rate x
Base
Note that if we
were to solve for the rate our answer will be a decimal rather than a
percent. We need to change our answer
to a percent. See Example 3.
METHOD 3: Fraction Shortcut Method
Sometimes it can
be easier to use fractions when solving percent problems. For example, if I wanted to know my grade on
the test, I could simply reduce
18/20 to 9/10
and then convert 9/10 to a percent.
9/10 = 0.9 = 90%
Similarly, if I
knew my score and wanted to know how many problems I answered correctly, I could
find 90% (or 9/10) of 20.
Example 1: Find 15% of 90.
Proportion
Method
Now using cross
multiplication we have
100 p = 90 x 15
p = 13.5
Algebraic Method
p = 15% x 90
p = 0.15 x 90
p = 13.5
Fraction Method
Since 15% = 3/20
we can multiply
Check: Does the answer make sense?
Yes, since 10%
of 90 is 9, 15% is a little bit more.
Example 2: 21 is 7% of what number?
Proportion
Method
Now using cross
multiplication we have
7b = 21x100
b = 300
Algebraic Method
21 = 7% x b
21 = 0.07 x b
21/0.07 = b
300 = b
Fraction
Shortcut
Note that if we
use the proportion method we can use equivalent fractions.
Then b = 100 x
3 = 300
Check: Does the answer make sense?
Yes, since 21 is
only 7% of something the number we are looking for is pretty big. If 21 were 10% of a number the number would
be 210. But 7% is less than 10% so 21
represents an even smaller part of a number, and the number must be larger than
210.
Example 3: There are 45 men in a group of 150
people. What percent are men?
Here we must be
looking for rate! There "is"
45 men in a group "of" 150 (forgive the poor grammar) lets us know
that p = 45 and b = 150.
Proportion
Method
Now using cross
multiplication we have
150 r = 45 x 100
r = 30
and 30 /100 =
30%
Algebraic Method
45 = r x 150
45/150 = r
0.30 = r
Changing 0.30 to
a percent we have 30%.
Fraction
Shortcut
Note that we can
reduce 45/150 to 3/10 and then use equivalent fractions to solve:
Does the answer
make sense? If half of the people were
men there would be 75 men. However,
there are 45 men, which is about half of 75 or about 25%.
PERCENT CHANGE
PROBLEMS
Often we wish to
know the percent increase or decrease.
For example, if my salary goes from $20 an hour to $25 an hour then it
will increase by 25%.
That is
In this case,
percent change = $5/$20 = 1/4 = 25%.
Note that my
salary increased $5, or 25%.
If my salary is
later lowered by $5 to the old $20 per hour, is the percent change again 25%?
Percent change =
$5/$25 = 1/5 = 20%
Note that in the
first case the base pay or the original amount was $20 so that the raise was a
larger percent than in the second case where the base pay was $25.
Example: Two out of 1000 people wear calculator
watches when they take standardized tests.
What percent of the people wear calculator watches when they take
standardized tests?
Answer: 0.02%
Only part of the
people wear calculator watches, so let 2=p and 1000=b.
Then 2/1000 =
0.002 = 0.02%
Example:
On December 1st
a store owner tells the manager to increase the sales force by 25% in order to
accompany the expected surge of holiday shoppers. Then on December 31 the store owner tells the manager to fire 25%
of the sales force.
Will there
be
A. more people working in January than in
November?
B. fewer people working in January than in
November?
C. the same amount of people working in January
than in November?
D. impossible to tell without knowing how many
people the store employed in November.
Answer: B
Solution: Choose a number for the sales force and see
what happens. Assume there are 100
people working in the store in November.
When the sales force is increased by 25%, 25 people are hired, bringing
the total sales force to 125. At the
end of December, 25% of the people are fired.
This is 25% of 125 or 31.25 people.
This leaves 93.75 employees (125 less 31.25), so the sales force has
fallen.
Algebraic
solution: Let N be the number of people
working in the store on November 30th.
Increasing the sales force by 25% of N brings the sales force to 125%N
during December. At the end of December
the manager fires 25% of the people working in the store leaving 75% of the
people who worked in December still employed in January. In January the number of people in the sales
force is:
75%(125%N)
=0.75(1.25N)
=0.9375N.
Since
0.9375N<N, the answer is B.
Example:
Edna withdrew
$45.00 from her savings account, which was 9% of her total savings. How much was in her savings account before
her withdrawal?
A. $4.05
B. $50.00
C. 405.00
D. $494.50
E. $500.00
Answer: E
Let S be the
amount of money in Edna's savings account.
If $45 is 9% of S, then
$45 = .09S
$45/.09 = S
$500 = S
Example:
On a math test
there are 25 questions. If Aaron
answers 8 of them correctly and if each question is worth the same amount, what
grade will he get on the test?
Answer: 32%
Use a ratio of
part/whole.
8/25 = 0.32 =
32%
This problem can
also be set up as a proportion:
part/whole =
rate/100
8/32 = r/100
8 x 100 = 25 x r
800 = 25r
800/25 = r
32 = r
Therefore,
Aaron's grade is 32%.
Important: Do not
change 32 to a percent (32=3200%) since r is the percent number and is already
divided by 100.
Example:
If 3 red marbles
can be exchanged for 4 blue marbles and if 5 blue marbles can be exchanged for
6 white marbles, how many red marbles can be traded for 600 white marbles?
Let r, b, and
represent the number or red, white, and blue marbles, respectively.
Then 3r = 4b and
5b = 6w
Answer: 375
Ratio
approach:
6w = 5b 
600w = 500b
4b = 3r 
500b = 375r
Therefore, 600 w
= 375 r.
Algebraic
approach:
4b = 3r 
b = (3/4)r
6w = 5b 
w = (5/6)b =
(5/6)[(3/4)r] = (15/24)r = (5/8)r
Then 600w =
600(5/8)r = 375r
Example:
Courtney was
making $18.00 per hour. When she got a
better job offer her boss offered her a 20% raise, which she accepted. What is her new salary?
A. $3.60
B. $5.00
C. $18.20
D. $20.00
E. $21.60
Answer: E
The raise is 20%
of $18.00 so the raise is 0.2 x $18.00 = $3.60. Added to her original salary she is now making $18.00 + $3.60
which is $21.60 an hour.
Shortcut: If she was making 100% of her salary and now
she is making 20% more, she is making 120% of $18.00 or 1.2 x $18.00 = $21.60.
Example:
If a pair of
boots that normally cost $260.00 are marked down 40% and if the tax rate is
6.25%, what will a customer pay at the register for the boots?
A. $110.50
B. $146.25
C. $156.00
D. $165.75
E. $276.25
Answer: D
If the discount
is 40% multiply the original price by 40% to find the discount:
0.4 x $260.00 =
$104.00.
Then the
discount subtracted from the price gives the sale price:
$260.00 -
$104.00 = $156.00
Now the tax on
$156 is .0625 x $156.00 = $9.75. This
must be added to the sale price:
$156.00 + $9.75
= $165.75.
Super
Shortcut: If the discount is 40% then
the sale price is 60% of $260.00:
0.6 x $260.00 =
$156.00.
If the tax rate
is 6.25% the final price is 106.25% of $156.00:
1.0625 x $156.00
= $165.75.
This problem can
be done with one calculation on the calculator:
260 x 0.60 x
1.0625 = 165.75.