Free Pre-Algebra
Lesson 40 ! page 1
Lesson 40
Percent
We see percents everywhere. The sales tax is 8%. The sale offers 25% off, shop now! 51% of those surveyed support a
change in policy. Percents are a common and useful way to talk about ratios. In the upcoming sections we learn to interpret
and calculate with percents in a variety of common applications.
Per One Hundred
Percent means “per one hundred,” because cent is one hundred in Latin. It sounds
like a rate, because of the word “per,” but it actually means a ratio in which the
denominator is one hundred. When the word percent or symbol % follows a
number, it indicates that the number is a ratio of two quantities.
There are One Hundred
Cents in a dollar…
The percent symbol, %, is not really a mathematical operation symbol the way + or
x are. The percent symbol is really a stylized version of a fraction bar followed by a
denominator of 100. You can see how the symbol evolved as a kind of shorthand.
%
means
/100
Years in a Century…
Percent to Fraction
If you interpret the percent symbol as above, it becomes very easy to change a
number written as a percent to a fraction.
Men commanded by a Centurion.
Example: Write the percents as fractions.
25%
8%
51%
25/100
Continue to lowest
terms:
8/100
Continue to lowest
terms:
51/100
Already in lowest
terms.
1/4
2/25
Let’s read the example sentences at the top of the page with the fractions in place of percents.
The sale offers 25% off, shop now!
becomes
The sale offers 1/4 off, shop now!
The sales tax is 8%.
becomes
The sales tax is 2/25ths.
51% of those surveyed support a change in policy.
becomes
51/100 of those surveyed support a change in policy.
Note that in the last example both the percent and the fraction are followed with the word “of.” The word “of” tells us what we
are taking a fraction of, that is, what is the whole? 51/100 of those surveyed tells us that the whole behind the fraction is
the total number of people surveyed. The other examples assume that we know what the whole is. The sale offers 1/4 off of
what? 1/4 off of the original price. The sales tax is 2/25ths of what? 2/25ths of the purchase price. Just as it’s very
important to know what we are taking a fraction of, it’s very important to know what a percent is of. The whole behind a
percent is called the base.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 40 ! page 2
Example: Re-write the sentence with a fraction in place of the percent. Underline the words that indicate the base.
Only 20% of students at the college used the
tutoring services offered.
20% =
Only 1/5 of students at the college used
the tutoring services offered.
20/100 = 1/5
Example: Re-write the percent as a fraction. Rewrite the sentence with the word “of” and supply the words that
indicate the base.
They shipped twenty thousand televisions,
but over 5% were defective.
He saved 30% by shopping at Discount$mart.
5% =
Over 1/20 of the twenty thousand
televisions shipped were defective.
5/100 = 1/20
30% =
30/100 = 3/10
He saved 3/10 of the price he would
have paid elsewhere by shopping at
Discount$mart.
Percent to Decimal
Just as we can interpret the % symbol as a fraction bar with 100 in the denominator, we can also interpret it as a division
symbol followed by 100. To write a percent as a decimal, re-write the % sign as ÷ 100.
Example: Write the percents in decimal form.
25%
8%
51%
25 ÷ 100
8 ÷ 100
51 ÷ 100
= 0.25
= 0.08
= 0.51
Again, let’s read the example sentences with decimals in place of percents.
The sale offers 25% off, shop now!
becomes
The sale offers 0.25 times the item price taken off the
item price, shop now!
The sales tax is 8%.
becomes
The sales tax is 0.08 times the purchase price.
51% of those surveyed support a change in
policy.
becomes
0.51 times the number of those surveyed support a
change in policy.
The decimal examples read very awkwardly. They are more like word translations of mathematical equations than real
English sentences. Notice that we have to supply the base to make the sentence understandable – no one would ever say
“The sale price is 0.25 off.” The word “of” is changed to “times.” The word or symbol for percent is a much more elegant way
to speak or write. But the percent symbol is not used in mathematical expressions – in equations we use the decimal
representation of the percent for calculations. That’s why we practice translating the percents to decimals here.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 40 ! page 3
The Shortcut for Changing Percent to Decimal
If you need to change a percent to a decimal, you can always whip out your trusty calculator and quickly divide by 100. But
when you look at an example like 51% = 0.51, it seems so straightforward that there ought to be a way to do the
conversion more quickly in your head.
And there is. Since the percent symbol means to divide by 100, what we are doing is shifting the digits in the number two
place values lower. The digit that used to be in the ones place is moved to the hundredths place, and the others all keep
their relative positions.
Some people prefer to think of this as moving the decimal point to the left.
51% contains an invisible decimal point after the 1s place:
51.0%
Drop the percent, and shift the decimal point two digits right:
.51
Supply the zero in the ones place for good form:
0.51
Use whatever method makes the most sense to you. If you prefer to do the division on the calculator, that is a reliable
method that doesn’t require much thought. Or, you can use a shortcut and move the digits down in place value or the
decimal point to the right if either of those seems more simple and clear.
The examples below show how some percents translate to fraction and decimal form. The fractions will need to be put in
lowest terms eventually, but you can see how the % symbol works.
37% =
37
= 0.37
100
75% =
75
= 0.75
100
06% =
06
= 0.06
100
50% =
50
= 0.50
100
Note the placeholder zero.
The decimal does not need
the trailing 0 and should
really be written 0.5.
100%
0.5%
Below are some examples that are slightly tougher.
Example: Write the percents as decimals.
137%
7.5%
137 ÷ 100
7.5 ÷ 100
100 ÷ 100
0.5 ÷ 100
= 1.37
= 0.075
=1
= 0.005
It’s really important to remember that 100% = 1.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 40 ! page 4
Changing Percents with a Decimal Point to Fractions
If a percent has a decimal point, as in the example 7.5% above, you can still change it to a fraction. You have two choices
for how to proceed.
METHOD 1: First write 100 in the denominator.
7.5% =
METHOD 2: Change to a decimal first.
7.5
100
7.5% = 7.5 ÷ 100 = 0.075
This is a ratio, but not really a fraction, because there is a
decimal point in the numerator. Since there is one decimal
place, tenths, to get rid of it multiply top and bottom by 10:
7.5 10
75
3• 5 • 5
3
•
=
=
=
100 10 1000 2 • 2 • 2 • 5 • 5 • 5 40
Then write the decimal as a fraction (as we did in Lesson 31)
by using the place value of the digit furthest to the right as
denominator.
0.075 =
75
3• 5 • 5
3
=
=
1000 2 • 2 • 2 • 5 • 5 • 5 40
Example: Write the percents as fractions.
3.7%
0.5%
0.01%
205%
3.7 ÷ 100 = 0.037
0.5 ÷ 100 = 0.005
0.01 ÷ 100 = 0.0001
205 ÷ 100 = 2.05
= 37/1,000
= 5/1,000
= 1/10,000
= 2 and 5/100
= 1/200
= 2 and 1/2
From Decimal to Percent
To change a percent to a decimal, we interpret the percent symbol as division by 100. To go from a decimal to a percent, we
do the opposite of dividing by 100, that is, we multiply by 100. (One way to think about this is that we are multiplying by
100%, which is equal to 1, and so we’re not really changing the number, just its form. But don’t enter the % symbol when
multiplying if you are using a calculator.) You can also use the reverse shortcut of moving the digits two place values higher,
or the decimal point two places to the right. If you use a shortcut don’t forget the percent symbol in the answer.
Example: Write the decimals as percents.
0.43
0.09
0.4
1.1
0.43 x 100%
0.09 x 100 %
0.4 x 100 %
1.1 x 100 %
= (0.43 x 100)%
= 9%
= 40%
= 110%
= 43%
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 40 ! page 5
From Fraction to Percent
There are two ways to change a fraction to a percent. Here we change the fraction 2/5 to a percent.
METHOD 1: Write a proportion to change the fraction to a
denominator of 100.
METHOD 2: Change the fraction to a decimal by dividing
numerator by denominator.
2
x
=
5 100
Solve by cross-multiplying.
5x = 200
x = 40
2
= 2 ÷ 5 = 0.4
5
Change the decimal to a percent by multiplying by 100%.
0.4 ! 100% = 40%
5x / 5 = 200 / 5
The value for x goes before the percent symbol.
2
= 40%
5
If the fraction has a repeating decimal form, it will need to be rounded or the percent will have to be written with a repeat bar
over the decimal part.
Example: Write the fractions as percents.
3
8
3
= 3 ÷ 8 = 0.375
8
1
3
0.375 ! 100% = 37.5%
1
= 1÷ 3 = 0.33333...
3
0.33333... ! 100% = 33.3333....%
= 33.3%
The skill of moving between the different representations of percent, fraction, and decimal is fundamental to practical work
with percents.
!
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 40 ! page 6
Lesson 40: Percent
Worksheet
Name__________________________________
1. Fill in the missing values in the chart to create a PERCENT-DECIMAL-FRACTION Reference Sheet.
PERCENT
DECIMAL
RATIO (/100)
FRACTION
0.5%
0.01
2/100
1/20
3/40
8/100
0.1
15%
16%
0.2
25/100
3/10
2/5
50/100
0.6
70%
75%
0.8
83/100
49/50
99/100
1
101%
102%
1.5
2
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 40 ! page 7
2. Underline or supply the words that tell the base of the percent.
a. 30% of the class felt the test was too easy.
b. Alicia scored 90% on the test.
c. 28% of the test-takers studied more than
8 hours for the test.
d. Davis was paid a commission that was 20%
of the band’s income.
e. This jacket was 40% off.
f. The copy is 90% of the original size.
g. 60% of the crowd refused to dance at all.
h. The interest on the loan was 18%.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 40 ! page 8
Lesson 40: Percent
Homework 40A
Name_________________________________________
1. The litter your cat prefers comes in two sizes. There is a
4 lb box for $5.99 or a 10 lb box for $12.99. Find the price
per pound for each of the two sizes.
2. The shadow of a flagpole is 11 feet long at the same time
the shadow of a yardstick (3 feet tall) is 2.4 feet. What is the
height of the flagpole?
3. The Nutrition Facts label shows
4. A floor plan for a house indicates that 1 inch on the plan is
equal to 8 feet in the house. If the living room is 11 feet wide,
how many inches wide is it on the plans?
Serving Size: about 14 crackers (15 g)
Servings Per Container: about 16
About how many crackers are in the box?
About how many grams does 22 crackers weigh?
What is the true scale of the plans?
5. Compare the fractions with > , <, or =. Show your work.
a.
1
6
1
10
b.
3
8
1
3
c.
6
7
7
6
© 2010 Cheryl Wilcox
6. Compare the decimals with > , <, or =.
a. 0.34
0.4
b. 1.109
1.111
c. 0.005
0.05
d. 1/9
0.1
Free Pre-Algebra
7. Write the fractions as decimals.
Lesson 40 ! page 9
8. Write the decimals as fractions in lowest terms.
a.
5
6
a. 0.44
b.
9
1000
b. 0.005
9. Write in scientific notation.
10. Write in standard form.
a. 96,988,000,000,000,000
a. 9.7 trillion
b. 21.4 million
b. 7.15 x 1019
11. Write with a decimal and place value name (as in #9b).
12. Underline or supply the words that tell the base of the
percent.
a. 67,500,000,000,000
b. 4.399 x 108
a. 95% of the dentists chose sugarless gum.
b. Tomas scored 83% on his test.
13. Re-write the percents as decimals.
14. Re-write the decimals as percents.
a. 67%
a. 0.015
b. 5%
b. 0.29
15. Re-write the percents as fractions in lowest terms.
16. Re-write the fractions as percents.
a. 85%
a. 3/4
b. 5%
b. 3/5
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 40 ! page 10
Lesson 40: Percent
Homework 40A Answers
1. The litter your cat prefers comes in two sizes. There is a
4 lb box for $5.99 or a 10 lb box for $12.99. Find the price
per pound for each of the two sizes.
2. The shadow of a flagpole is 11 feet long at the same time
the shadow of a yardstick (3 feet tall) is 2.4 feet. What is the
height of the flagpole?
$5.99 $1.4975
=
! $1.50 per lb.
4 lb
1 lb
$12.99 $1.299
=
! $1.30 per lb.
10 lb
1 lb
height
shadow
3
h
=
2.4 11
2.4h = 33
2.4h / 2.4 = 33 / 2.4
h = 13.75
The flagpole is 13 and 3/4 feet high.
3. The Nutrition Facts label shows
Serving Size: about 14 crackers (15 g)
Servings Per Container: about 16
4. A floor plan for a house indicates that 1 inch on the plan is
equal to 8 feet in the house. If the living room is 11 feet wide,
how many inches wide is it on the plans?
1 inch x inches
=
8 feet
11 feet
About how many crackers are in the box?
14 crackers 16 servings
•
1 serving
1 container
= 224 crackers per container
About how many grams does 22 crackers weigh?
14 crackers 22 crackers
=
15 grams
x grams
14x = 330
x ! 23.6
8x = 11
3
x =1
8
8x / 8 = 11/ 8
It is 1 and 3/8 inches on the plans.
What is the true scale of the plans?
14x / 14 = 330 / 14
1 inch
1 inch
1
=
=
8 feet 96 inches 96
22 crackers weigh about 23.6 g.
5. Compare the fractions with > , <, or =. Show your work.
a.
1 1
>
6 10
5
3
>
30 30
6. Compare the decimals with > , <, or =.
a. 0.34 < 0.4
b. 1.109 < 1.111
b.
3 1
>
8 3
9
8
>
24 24
c. 0.005 < 0.05
c.
6 7
<
7 6
proper < improper fraction
d. 1/9 > 0.1
© 2010 Cheryl Wilcox
1/9 = 0.11111…
Free Pre-Algebra
Lesson 40 ! page 11
7. Write the fractions as decimals.
a.
5
= 0.83
6
b.
9
= 0.009
1000
8. Write the decimals as fractions in lowest terms.
a. 0.44 =
44 11
=
100 25
b. 0.005 =
5
1
=
1000 200
9. Write in scientific notation.
10. Write in standard form.
a. 96,988,000,000,000,000
a. 9.7 trillion
9,700,000,000,000
9.6988 x 1016
b. 7.15 x 10
b. 21.4 million
71,500,000,000,000,000,000
21,400,000 = 2.14 x 107
11. Write with a decimal and place value name (as in #9b).
a. 67,500,000,000,000
67.5 trillion
b. 4.399 x 108
439,900,000 = 439.9 million
19
12. Underline or supply the words that tell the base of the
percent.
a. 95% of the dentists chose sugarless gum.
b. Tomas scored 83% on his test.
83% of the possible points
13. Re-write the percents as decimals.
14. Re-write the decimals as percents.
a. 67%
a. 0.015
67 ÷ 100 = 0.67
b. 5%
b. 0.29
5 ÷ 100 = 0.05
15. Re-write the percents as fractions in lowest terms.
a. 85% =
b. 5% =
0.015 x 100% = 1.5%
85 17
=
100 20
5
1
=
100 20
© 2010 Cheryl Wilcox
0.29 x 100% = 29%
16. Re-write the fractions as percents.
a. 3/4 = 0.75 = 75%
b. 3/5 = 0.6 = 60%
Free Pre-Algebra
Lesson 40 ! page 12
Lesson 40: Percent
Homework 40B
Name_________________________________________
1. A case of paper towels that contains 30 rolls is on sale for
$30.99. You can get the same brand in a smaller package of
12 rolls for $14.99. Find the price per roll for each size
package.
2. The shadow of a flagpole is 15 feet long at the same time
the shadow of a yardstick (3 feet tall) is 3.8 feet. What is the
height of the flagpole?
3. The Nutrition Facts label shows
4. A floor plan for a house indicates that 1 inch on the plan is
equal to 6 feet in the house. If the living room is 14 feet wide,
how many inches wide is it on the plans?
Serving Size: about 4 pieces (25 g)
Servings Per Container: about 12
About how many pieces are in the container?
About how many grams does 10 pieces weigh?
What is the true scale of the plans?
5. Compare the fractions with > , <, or =. Show your work.
a.
1
5
1
4
6. Compare the decimals with > , <, or =.
a. 0.55
0.9
b. 5.109
b.
1
4
2
9
c.
11 9
9 11
© 2010 Cheryl Wilcox
1.101
c. 0.0015
d. 1/8
0.0005
0.125
Free Pre-Algebra
7. Write the fractions as decimals.
a.
5
12
b.
9
10,000
Lesson 40 ! page 13
8. Write the decimals as fractions in lowest terms.
a. 0.88
b. 0.6
9. Write in scientific notation.
10. Write in standard form.
a. 986,900,000,000,000
a. 29.7 million
b. 1.4 billion
b. 7.99 x 1011
11. Write with a decimal and place value name (as in #9b).
12. Underline or supply the words that tell the base of the
percent.
a. 13,800,000
b. 9.9 x 109
a. 50% of the pizza contained pineapple.
b. The sale price was 10% off.
13. Re-write the percents as decimals.
14. Re-write the decimals as percents.
a. 45%
a. 0.15
b. 1%
b. 0.029
15. Re-write the percents as fractions in lowest terms.
16. Re-write the fractions as percents.
a. 80%
a. 1/2
b. 32%
b. 19/25
© 2010 Cheryl Wilcox