Section 5.1
Converting between Percents,
PRE-ACTIVITY
PREPARATION
Decimals, and Fractions
Think about how often you have heard, read, or used the term percent (%) in its many everyday applications:
•
The sales tax in Michigan is 6%.
•
A 15% gratuity will be automatically added for parties of six
or more.
•
Hurry in! End-of-season clearance! 80% savings on all redticketed items!
•
Full-time student enrollment increased this year by 2.5
percent.
•
Last season our quarterback had a pass completion rate of
61.9 %.
•
My score on the English quiz? 85%.
You might use an equivalent fraction or decimal number instead of the percent in each example:
•
The sales tax in Michigan is 3/50 (or 0.06).
•
A gratuity of 3/20 (or 0.15) will be automatically added for parties of six or more.
•
Hurry in! End-of-season clearance! 4/5 (or 0.80) savings on all red-ticketed items!
•
Full-time student enrollment increased this year by 35/1400 (or 0.025).
•
Last season our quarterback had a pass completion rate of 289/467 (or 0.619).
•
My score on the English quiz? 17/20 (or 0.85).
However, the percent is probably the most familiar and effective way to communicate the numbers.
In this section, you will learn what the term percent represents. You will also learn the practical skill of
converting between fractions, decimals, and percents, so that you can choose the appropriate form for
presenting specific information.
LEARNING OBJECTIVES
• Master the mathematical language of percents.
• Convert between equivalent decimals, fractions, and percents.
• Order a mixed set of fractions, decimals, and percents.
453
Chapter 5 — Percents
454
TERMINOLOGY
PREVIOUSLY USED
NEW TERMS
TO
LEARN
build up
decimal fraction
convert
non-terminating repeating decimal
factor
percent
order
percent number
ratio
percent sign (%)
reduce
terminating decimal
simple fraction
BUILDING MATHEMATICAL LANGUAGE
A decimal fraction is a simple fraction with a whole number numerator and a denominator of 10
or 100 or 1000, and so on.
You can write any decimal number as a decimal fraction by first translating into words, then writing the
fraction from the words.
35
as a decimal fraction.
For example, the decimal number 0.035 is “thirty-five thousandths,” or
1000
A percent is a special type of ratio, comparing any number in the numerator to 100 in the denominator.
The numerator is referred to as the percent number.
The percent sign (%) means “per 100.” For example, by the definition of percent,
22
= 22%
100
100
= 100%
100
198
= 198%
100
1
12
2 = 12 1 % or 12.5%
100
2
0.7
7
= 0.7% or
%
100
10
"twenty-two percent"
22 is the percent number.
"one hundred percent"
100 is the percent number.
"one hundred ninety-eight percent"
198 is the percent number.
"twelve and one-half percent"
"seven tenths percent"
or "seven tenths of a percent"
12
1
is the percent number.
2
0.7 is the percent number.
Section 5.1 — Converting between Percents, Decimals, and Fractions
455
When the percent number is less than 1, as in ¼ %, .02%, or .82%, the percent is between 0% and 1%.
When the percent number is greater than 1, but not a whole number, the percent is between two whole
number percents. For example:
3¼ % is greater than 3% and less than 4%.
16.5% is halfway between 16% and 17%.
You can use decimals, fractions, and percents interchangeably. To present information, you can choose the
form that will convey the information most effectively.
On the other hand, when you solve problems involving percents, you cannot calculate using a number in
its percent form. Your only options are to use its fraction equivalent or its decimal equivalent. (Even a
calculator with a % key converts the percent to its decimal form when that key is used.)
Converting between the Decimal, Fraction, and Percent Form of a Number
TECHNIQUES
The following six techniques demonstrate how you can convert any decimal number, fraction, or percent to
one of its other two equivalent forms. The six conversions are:
•
Decimal Number to a Fraction
•
Decimal Number to a Percent
•
Fraction to a Percent
•
Fraction to a Decimal Number
•
Percent to a Decimal Number
•
Percent to a Fraction
Converting a Decimal Number to a Fraction
Technique
Write the decimal number as a decimal fraction and reduce the fraction to lowest terms.
Models
Convert the following decimals to their equivalent fractions or mixed numbers.
Reduce fractions to lowest terms.
A
►
0.022
22
1000
22 ÷ 2
11
=
Answer
1000 ÷ 2 500
0.022 =
B
►
4.48
48
100
48 ÷ 4
12
4
=4
Answer
100 ÷ 4
25
4.48 = 4
Chapter 5 — Percents
456
Converting a Fraction to a Decimal Number
Technique
Divide the numerator of the fraction by its denominator.
Special
Case:
The fraction is already (or can be built up to)
a decimal fraction (see page 457, Model 2)
???
Why do you do this?
In its broadest sense, a fraction represents the quotient of two quantities. The division process will always result
in a decimal number.
When you divide a whole number by another whole number, the quotient will always be either a terminating
decimal or a non-terminating repeating decimal.
Terminating Decimal
A terminating decimal is
one for which there is a zero
remainder at some point in
the long division process.
For example, if you divide 3
by 8, the result will be .375,
a terminating decimal.
0.375
8 3.000
−2 4
)
60
−56
40
−40
0
Non-terminating Repeating Decimal
5.333....
3 16.000
−15
10
−9
10
)
When the division results in a digit or in a
sequence of digits that repeats, the quotient
is a non-terminating repeating decimal. For
example, when you divide 16 by 3, the result
is 5.333...
−9
10
−9
The notation for this is 5.3, with the bar over the
3 indicating that it is the digit 3 that repeats.
To indicate a sequence of digits that repeats, use
a bar over the sequence.
For example, the result of dividing 14 by 11 is
1.272727.... or 1.27.
Model 1
Convert the following fractions and mixed numbers to decimals.
Round to the ten-thousandth place for non-terminating decimals.
►
A
0.625
8 5.000
−4 8
)
5
8
20
−16
Divide 5 by 8.
The quotient is a terminating decimal.
5
= 0.6
625 Answer
8
40
−40
0
Section 5.1 — Converting between Percents, Decimals, and Fractions
B
►
3
1
16
3
1
1
=3+
16
16
457
0.0625
16 1.0000
−9 6
)
The whole number 3 does not change.
40
−32
1
Convert the fraction
to a decimal.
16
80
−80
0
The quotient is a terminating decimal. 3 1 = 3.0625 Answer
16
C
►
5
12
0.41666
12 5.00000
−48
)
The quotient is a non-terminating repeating decimal.
5
Rounded to the nearest ten-thousandth,
≈ 0.4167 Answer
12
20
−12
80
−72
80
−72
80
−72
8
Special
Case:
Model 2
The Fraction is Already (or Can be Built Up to)
a Decimal Fraction
Convert the following to decimals.
►
A
9
10
THINK
B
►
If the fraction is already a decimal fraction, or can easily be built up
to a decimal fraction, simply write it in standard decimal notation.
It is not necessary to divide.
9
10
8
4
25
8
4×4
16
=8
25 × 4
100
is “nine tenths” or 0.9 Answer
THINK
“eight and sixteen hundredths”
8.16 Answer
Chapter 5 — Percents
458
Converting a Decimal Number to a Percent
Technique
Multiply the decimal by 100%.
That is, move the decimal point two places to the right and attach the percent sign (%).
???
Why can you do this?
Starting with a decimal number, for example 0.285, the Technique says that 0.285 × 100% = 28.5%.
The Identity Property of Multiplication states that multiplying any number by 1 does not change the value of
the number. For this Technique, multiplying by 100% is multiplying by 1:
100% means
100
100
= 1.
and, from your knowledge of fractions,
100
100
That is why the value of 0.285 does not change. Only its form changes.
Now consider the second part of the Technique. Why is it that, to multiply by 100%, you can simply move the
decimal point two places to the right and add a percent sign? Think about the process this way:
0.285 × 100% = 28.5%
THINK
The shortcut for multiplying by 100 is to move the decimal point two places to the right.
Models
Convert the following decimals to their equivalent percents.
A
►
0.8225
0.8225 × 100% = 82.25%
B
►
OR
0.8225% = 82.25%
OR
1.723% = 172.3%
Answer
1.723
1.723 × 100% = 172.3%
Answer
Section 5.1 — Converting between Percents, Decimals, and Fractions
C
►
0.04
0.04% = 4%
D
►
0.009
0.009% = 0.9%
Answer
E
►
4
4.00% = 400%
Answer
459
Answer
Converting a Percent to a Decimal Number
Technique
Drop the percent sign and divide the percent number by 100.
That is, drop the percent sign (%) and move the decimal point two places to the left.
Special Case: The percent number contains a fraction (see page 460, Model 2)
???
Why can do you do this?
This Technique goes back to the definition of a percent as the ratio of the percent number to 100.
For example, 28.5% =
THINK
28.5
= 28.5 ÷ 100 or 0.285
100
The shortcut for dividing by 100 is to move the decimal point two places to the left.
Model 1
Convert the following percents to their equivalent decimals.
A
►
85.4%
85.4% = 85.4 ÷ 100 = 0.854
OR
85.4% = 0.854
Answer
B
►
258%
258% = 258 ÷ 100 = 2.58
OR
258% = 2.58
Answer
C
►
2.3%
2.3% = 02.3%
D
►
0.75%
0.023
Answer
Zero place digit needed.
0.75% = 00.75%
0.0075
Answer
Two zero place digits needed.
Chapter 5 — Percents
460
Model 2
Special Case: The Percent Number Contains a Fraction
Convert each percent to its equivalent decimal. Round to the nearest ten-thousandths place, if necessary.
A
►
62
1
%
4
When the conversion is from a percent to a decimal, and the percent number
contains a fraction, rewrite the fractional portion of the percent as a decimal.
1
= 1 ÷ 4 = 0.25
4
B
►
so 62
1
% = 62.25% = 0.6225 Answer
4
1
14 %
3
1
= 1 ÷ 3 = 0.3 ≈ 0.33
3
1
14 % ≈ 14.33%
3
14.33% = 0.1433 Answer
Before looking at the remaining two techniques for converting between fractions and percents, think about the
first four conversion techniques, as summarized in the table below.
Fraction
13
40
Decimal
0.325
40 13.000
−120
)
100
−80
200
−200
Percent
0.325
and
32.5%
0.325
× 100%
0
32.5%
0.325
13
40
and
0.325
reduce
32.5
100
325
1000
Notice that the decimal form is the connecting conversion between fraction and percent, and between percent
and fraction in the opposite direction. This bridge is the basis for the final two conversion techniques.
Section 5.1 — Converting between Percents, Decimals, and Fractions
461
Converting a Fraction or a Mixed Number to a Percent
Technique
Step 1 — Convert the fraction or mixed number to a decimal.
Step 2 — Convert the decimal to a percent.
Special Case: The denominator is 100 (see below, Model 2)
Special Case: The denominator is a factor of 100 (see page 462, Model 3)
Special Case: The result of Step 1 is a repeating decimal (see page 462, Model 4)
Model 1
Convert the following to percents.
0.875
8 7.000
)
A
►
7
8
Step 1
7
= 0.875
8
Step 2
0.875% = 87.5% Answer
−64
60
−56
40
−40
0
B
►
2
1
2
Step 1
Step 2
Model 2
2
1
= 2.5
2
2.50% = 250% Answer
0.5
2 1.0
)
−10
0
Special Case: The Denominator of the Fraction is 100
Convert to a percent:
37
100
If the denominator of the fraction is 100,
the numerator IS the percent number.
37
= 37% Answer
100
Chapter 5 — Percents
462
Model 3
Special Case: The Denominator is a Factor of 100
9
Convert to a percent:
20
0.45
20 9.00
)
9
= 0.45
20
Step 1
0.45% = 45% Answer
Step 2
−8 0
100
−100
0
If the denominator of the fraction is a factor of 100, instead of
dividing to determine the decimal, build the fraction to have 100
as its denominator. The numerator will be the percent number.
9
9 ×5
45
=
=
= 45% by the definition of a percent.
20 20 × 5 100
Model 4
Special Case: The Result of Step 1 is a Repeating Decimal
5
6
Round to the nearest hundredth percent, if necessary.
Convert the fraction to a percent.
Step 1
5
≈ 0.8333
6
0.83333
6 5.00000
−48
)
If the directions say to round the percent
to the nearest hundredth percent, round
the decimal to the ten-thousandth place in
the decimal form so that there will be two
decimal places in the percent form to indicate
accuracy to the hundredth percent place.
20
−18
20
−18
20
−18
20
−18
2
Step 2
0.8333% = 83.33% Answer
Note: If the directions do not specify a rounding place, one option is to present the repeating
decimal part using the bar notation.
That is, 5 = 83.3%
6
Another is to present the repeating decimal part as a fraction.
That is, 5 = 83 1 %
6
3
2
1
0.83 = 0.83
6
3
6 5.00
−48
)
20
−18
2
Section 5.1 — Converting between Percents, Decimals, and Fractions
463
Converting a Percent to a Fraction or Mixed Number
Technique
Step 1 — Convert the percent to its equivalent decimal.
Step 2 — Convert the decimal to its equivalent fraction or mixed number and reduce.
Special Case: The percent number is a whole number (see below, Model 2)
Special Case: The percent number contains a fraction (see page 464, Model 3)
Model 1
Convert the following to fractions or mixed numbers. Reduce the fractions to lowest terms.
A
►
0.06%
Step 1
Step 2
B
►
3.2%
Step 1
Step 2
Model 2
00.06% = 0.0006
0.0006 =
6
6÷2
3
=
=
Answer
10, 000 10, 000 ÷ 2 5000
03.2% = 0.032
0.032 =
32
32 ÷ 4
8÷2
4
=
=
=
Answer
1, 000 1, 000 ÷ 4 250 ÷ 2 125
Special Case: The Percent Number is a Whole Number
Convert the following to fractions or mixed numbers. Reduce the fractions to lowest terms.
A
►
54%
54% = 0.54 =
54 ÷ 2
27
=
Answer
100 ÷ 2 50
When the percent number is a whole number, there is no advantage to using its decimal
equivalent. Simply write the percent number over 100 (definition of percent) and reduce.
54% =
B
►
215%
54 ÷ 2
27
=
Answer
100 ÷ 2 50
215
215% =
=2
100
3
20
15
100
=2
3
Answer
20
Chapter 5 — Percents
464
Model 3
Special case: The Percent Number Contains a Fraction
Convert the following to fractions. Reduce to lowest terms.
►
A
23
1
%
6
When the percent number contains a fraction, drop the percent
sign and multiply the percent number by 1/100 (same as dividing
the percent number by 100). Reduce the fraction.
Note: This procedure assures that the resulting fractions will be the exact
equivalent of the percent.
1
23
1
6 = 23 1 × 1 = 139 × 1 = 139 Answer (reduced fully)
23 % =
6
100
6 100
6
100
600
B
►
2
1
%
2
1
1
1
1
5
2 %=2 ×
=
×
2
2 100
2
1
20
100
=
1
Answer
40
Note: Since the ½ in the percent number can be converted to a terminating decimal,
0.5, you can choose to work through the decimal and convert as follows:
2
C
►
1
25 ÷ 25
1
% = 02.5% = 0.025 =
=
2
1000 ÷ 25
40
1
33 %
3
1
1
1
33 % = 33 ×
=
3
3 100
1
100
1
1
×1
=
Answer
3
3
100
Section 5.1 — Converting between Percents, Decimals, and Fractions
465
Validating Conversions
In general, unless you rounded a decimal in the process, you can validate your
answer by converting it back to the original form.
Following are examples of validation for each of the six conversion techniques.
Example 1: Decimal to Fraction
Conversion:
Validation:
11
Answer
50
11
(Convert
to a decimal)
50
0.22 =
0.22 9
50 11.00
−1 0 0
)
10 0
−100
0
Example 2: Fraction to Decimal
A.
Conversion:
Validation:
5
= 0.625 Answer
8
(Convert 0.625 to a fraction)
0.625 =
B.
Conversion:
Validation:
625 ÷ 25
25 ÷ 5
5
9
=
=
1000 ÷ 25
40 ÷ 5
8
5
≈ 0.417 Answer (rounded)
12
Since 0.417 is rounded, it is not possible to convert it back
to the exact original fraction. You can however, validate the
equality of the decimal fraction form of your answer and the
original fraction. The cross-products will be close but not
exact because of the rounded decimal.
417 ? 5
=
1000 12
?
417 × 12 = 5 × 1000
5004 ≈ 5000 9
Example 3: Decimal to Percent
Conversion: 1.723 = 172.3% Answer
Validation:
(Convert 172.3% to a decimal.)
172.3% = 1.723
continues on the next page
Chapter 5 — Percents
466
Example 4: Percent to Decimal
A.
Conversion:
2.3% = 0.023 Answer
Validation:
(Convert 0.023 to a percent)
0.023% = 2.3%
B.
1
% = 0.6225 Answer
4
Conversion:
62
Validation:
(Convert 0.6225 to a percent)
0.6225% = 62.25% = 62
25
1
% = 62 % 9
100
4
Example 5: Fraction to Percent
A.
Conversion:
9
= 45% Answer
20
Validation:
(Convert 45% to a fraction)
45% = 0.45 =
B.
45 ÷ 5
9
=
9
100 ÷ 5 20
5
≈ 83.33% (rounded)
6
Conversion:
Validation:
Change the rounded 83.33% to its decimal equivalent. Then
validate the equality of the decimal fraction and the original
fraction.
83.33% = 0.8333
8, 333 ? 5
=
6
10, 000
?
8, 333 × 6 = 5 × 10, 000
49, 998 ≈ 50, 000 9
Example 6: Percent to Fraction
Conversion:
Validation:
3
Answer
500
3
(Convert
to a percent.)
500
0.6% =
0.006
500 3.000
−3000
)
0
0.006% = 0.6% 9
Section 5.1 — Converting between Percents, Decimals, and Fractions
467
How Estimation/Prediction Can Help
As a mental math check to assure that your conversion answer is reasonable, keep the
following comparison chart in mind.
Fraction form
Decimal form
Percent form
proper fraction (< 1)
2
Example:
5
< 1.0
< 100%
improper fraction or
mixed number (> 1)
3
Example: 1
4
Example:
> 1.0
Example:
1.75
Example:
40%
> 100%
Example:
175%
Since 0.22 < 1, the fraction will be proper (< 1).
THINK
Actual answer:
Prediction:
0.4
Convert 0.22 to a fraction.
Prediction:
Example:
Example:
11
50
Convert
7
to a percent.
8
THINK
Since
7
< 1, the percent will be < 100%.
8
Actual answer: 87.5%
Example:
Prediction:
Convert 1.6 to a percent.
THINK
Since 1.6 > 1.0, the percent will be > 100%.
Actual answer: 160%
You can, in fact, refine your predictions even more so; the following chart presents just
a few of the comparisons you can use to predict or estimate your answer.
Fraction form
Decimal form
Percent form
1
2
1
<
10
< 0.5
< 50%
< 0.1
< 10%
1
2
3
>
4
> 0.5
> 50%
> 0.75
> 75%
> 2.0
> 200%
<
>
>2
Chapter 5 — Percents
468
Example:
Convert 0.22 to a fraction.
Prediction:
Convert
Prediction:
⎛ 1
⎞
⎜⎜< or 25 ⎟⎟
⎜⎝ 2
50 ⎟⎠
11
50
Actual answer:
Example:
Since 0.22 < 0.5, the fraction will be < ½ .
THINK
7
to a percent.
8
THINK
Since
7 3
>
8 4
⎛ 6 ⎞⎟
⎜⎜ ⎟ , the percent will be > 75%
⎜⎝ 8 ⎟⎠
Actual answer: 87.5%
Example:
Convert 2.8 to a percent.
Prediction:
THINK
Since 2.8 > 2.0, the percent will be > 200%
and since 2.8 > 2.5, the percent will be > 250%.
Actual answer: 280%
Example:
Convert 240% to a fraction.
Prediction:
THINK
Since 240% > 200%, the fraction will be a mixed number > 2
and
1
2
Since 40% < 50%, the fraction part of the mixed number will be < .
Actual answer: 2
Example:
Convert 0.07 to a fraction.
Prediction:
THINK
Actual answer:
Example:
Prediction:
2
5
7
100
Since 0.07 < 0.1, the fraction will be <
1
.
10
⎛ 1
10 ⎞⎟
⎜⎜<
or
⎟
⎜⎝ 10
100 ⎟⎠
Convert 0.07 to a percent.
THINK
Actual answer: 7%
Since 0.07 < 0.1, the percent will be < 10%.
Section 5.1 — Converting between Percents, Decimals, and Fractions
469
Ordering a Mixed Set of Decimals, Fractions, and Percents
You know how to order a set of decimal numbers (see Chapter 2) and a set of fractions (see Chapter 3).
Now with your knowledge of percents and of how all three forms are interchangeable, you can order a mixed
set of decimals, fractions, and percents. Since the process of converting from percents to decimals and from
fractions to decimals is simple, using the decimal forms for comparison is usually most efficient.
TECHNIQUE
Ordering a Mixed Set of Decimals, Fractions, and Percents
Technique
Step 1 — Convert all numbers to the same equivalent form.
Step 2 — Order them as you would for your chosen form.
Model
A
►
List the following in order of their values, from lowest to highest:
Step 1: Write as decimals.
0.111
9 1.000
−9
)
10
−9
10
−9
1
B
►
1
, 0.115, 11%
9
Step 2: Order the decimals
using trailing zeros.
Rank
1
= 0.1 ≈ 0.111
9
0.111
2
0.115 is in decimal form
0.115
3
11% = 0.11
0.110
1
Answer : 11%,
List the following in order of their values, from lowest to highest: 0.75,
Step 1: Write as decimals.
0.75 = 0.75
3
% = 00.75% = 0.0075
4
7.5% = 07.5%
1
, 0.115
9
3
%, 7.5%
4
Step 2: Order the decimals
using trailing zeros.
Rank
0.7500
3
0.0075
1
0.0750
2
Answer :
3
% , 7.5% , 0.75
4
Chapter 5 — Percents
470
ADDRESSING COMMON ERRORS
Issue
Losing track
of the percent
sign (%) while
converting a
percent that
contains a
decimal or
fraction
Incorrect
Process
3
Convert 1 %
5
decimal.
to a
0.6
5 3.0
−3 0
)
0
Convert 2.35% to a
fraction.
7
2.35
5% = 2
Answer: 2
Making an error
in the shift of
the decimal
point when
converting
between
decimals and
percents
20
2
Resolution
Converting the fraction
in the percent number to
a decimal or the decimal
in the percent number to
a fraction only changes
the form of the percent
number.
Drop the percent sign
(%) only when you
divide the percent
number by 100.
100
7
20
. = 0.015%
0
1.5
Convert 2.27% to a
decimal.
% = 227
2
2.27%
Convert 1 3 %
5
decimal.
to a
3
1 % = 1.6%
5
01.6% = 0.016
Convert 2.35% to a
fraction.
2.35% = 0.0235
235 ÷ 5
=
10, 000 ÷ 5
47
=
2, 000
35
Convert 1.5 to a
percent.
Correct Process
When converting from a
decimal to a percent, the
resulting percent number
will always be 100 times
as large as the decimal
number.
When converting from
a percent to a decimal,
the resulting decimal
number will always be
100 times smaller than
the percent number.
Think about the
decimal point shift in a
conversion you know.
25 converts to 25% and
25% converts to 0.25
0.25
25%
Decimal
Percent
Decimal
Percent
point shifts right
point shifts left
Convert 1.5 to a percent.
THINK
1.5 > 1 so the percent will
be greater than 100%
1.5 × 100% = 150%
Convert 2.27% to a
decimal.
THINK
2.27% < 100%, so the
decimal will be less than 1.
02.27% = 0.0227
Section 5.1 — Converting between Percents, Decimals, and Fractions
Issue
Not presenting
the required
fraction form
when converting
from a percent
to a fraction
Incorrect
Process
Resolution
Convert 0.8% to a
fraction.
0.8% =
First convert the percent
to a decimal. Then write
the decimal as a decimal
fraction and reduce to
lowest terms.
0.8 ÷ 4
100
0÷4
The fraction
equivalent of a
decimal or percent
should not contain a
decimal point.
0.2
Answer =
25
471
Correct Process
Convert 0.8% to a
fraction.
0.8% = 00.8% = 0.008
1
0.008 =
=
It must be in the form
of a simple fraction—a
whole number over a
whole number.
Misinterpreting
the denominator
for the decimal
fraction when
converting from
a decimal to a
fraction
Convert 0.125 to its
fraction equivalent.
0.125 =
125
125
12
25 ÷ 12
10
0,, 000
00 ÷ 125
1
=
80
5
400
=
1
80
Review Section 2.1 on
how to read a decimal
number.
The place value of
the last digit to the
right of the decimal
point determines the
denominator for the
decimal fraction.
125
8
1000
1
125
Convert 0.125 to its
fraction equivalent.
0.125 =
125 ÷ 125
1
=
1000 ÷ 125
8
Chapter 5 — Percents
472
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the terminology and notation associated with percents
how to convert between fractions and decimals
how to convert between percents and decimals
how to convert between fractions and percents
why you can apply a Technique without changing the value of the original number
how, in general, to validate conversions
how to order a mixed set of fractions, decimals, and percents
Section 5.1
Converting between Percents,
ACTIVITY
Decimals, and Fractions
PERFORMANCE CRITERIA
• Converting between percents, decimals, and fractions
– accuracy
– required presentation of the answer
• Ordering a mixed set of percents, decimals, and fractions
– accurate conversions to the same form
– correct order
CRITICAL THINKING QUESTIONS
1. When writing a decimal fraction for a given decimal number, how do you determine its denominator?
2. What are two ways to convert a fraction to decimal? For each procedure, when and why would you choose
to use it?
3. What are the two required characteristics of the final answer for a decimal to fraction conversion?
473
474
Chapter 5 — Percents
4. Why does 100% = 100 = 1?
100
5. Why is a percent called a special type of ratio?
6. What are four ways to convert fractions to percents? For each procedure, when and why would you choose
to use it?
•
•
•
•
Section 5.1 — Converting between Percents, Decimals, and Fractions
475
7. What are three possible ways to convert a percent to a fraction? For each procedure, when and why would
you choose to use it?
•
•
•
8. How can you be sure that you have moved the decimal point in the proper direction when converting
between decimals and percents?
9. What does a decimal or a fraction within a percent mean?
Chapter 5 — Percents
476
10. How can you go about ordering a mixed set of fractions, decimals, and percents?
TIPS
FOR
SUCCESS
• Before you begin a conversion, identify what kind of number you have to start with (Fraction, Decimal, or
Percent) and the kind of number you want to end with (F, D, or P).
• For most conversions, use decimals as a “bridge” between fractions and percents, as indicated in the
individual rows of this chart.
Fraction
Decimal
Percent
0.75
75%
Begin with a Fraction
3
4
1
2
50
←
100
1
4
25
←
100
Begin with a Decimal
50%
0.5
Begin with a Percent
0.25
25%
• Use estimation/prediction to assure that your answer “makes sense.”
• Know some of the most common conversions:
1
= 10%,
10
1
= 25%,
4
1
= 50%,
2
3
= 75%,
4
1
1
= 33 %,
3
3
2
2
= 66 %
3
3
Section 5.1 — Converting between Percents, Decimals, and Fractions
477
DEMONSTRATE YOUR UNDERSTANDING
1. Convert each of the following to a simple fraction or mixed number. Reduce to lowest terms.
Convert to a
Fraction or
Mixed Number
Process
Answer
Validation (optional)
a) 0.03
b) 0.156
c) 8.42
2. Convert each of the following to a decimal. Round to the nearest thousandth for non-terminating decimals.
Convert to a
Decimal
a)
17
40
b)
1
500
Process
Answer
Validation (optional)
Chapter 5 — Percents
478
Convert to a
Decimal
c)
2
7
d) 1
13
20
e) 4
4
5
f)
7
16
Process
Answer
Validation (optional)
Section 5.1 — Converting between Percents, Decimals, and Fractions
479
3. Convert each of the following to a percent.
Convert to a
Percent
Process
Answer
Validation
(optional)
a) 1.25
b) 0.3
c) 0.0785
d) 0.0005
e) 3
f) 10.25
4. Convert each of the following to a decimal. Round to the nearest ten-thousandth, if necessary.
Convert to a
Decimal
a) 34%
b) 2.57%
c) 0.74%
d) 382%
e) 3%
f) 24
g) 6
1
%
2
2
%
3
Process
Answer
Validation
(optional)
Chapter 5 — Percents
480
5. Convert each of the following to a percent. Round to the nearest hundredth percent, if necessary.
Convert to a
Percent
a)
3
16
b)
7
50
c)
3
d)
1
12
3
8
Process
Answer
Validation (optional)
Section 5.1 — Converting between Percents, Decimals, and Fractions
Convert to a
Percent
e) 1
f)
Process
481
Answer
Validation (optional)
3
4
1
2
100
15
6. Convert each of the following to a simple fraction or mixed number. Reduce to lowest terms.
Convert to a
Fraction or
Mixed Number
a) 125%
b) 38%
Process
Answer
Validation (optional)
Chapter 5 — Percents
482
Convert to a
Fraction or
Mixed Number
2
c) 61 %
3
d) 3
3
%
4
e) 0.74%
f) 0.8%
g) 62.5%
Process
Answer
Validation (optional)
Section 5.1 — Converting between Percents, Decimals, and Fractions
483
7. Order the following from lowest to highest value: 3 , 38%, 0.34, 37%
8
8. Order the following from highest to lowest value: 5.3%,
1
1
1
, 0.555, 5 %,
20
2
2
TEAM EXERCISES
1. In the grids below, fill in the correct number of rectangles to represent the following:
a) 4%
Use a pencil.
c) 140% Use a highlighter.
b) 30%
Use a pen.
d) 0.6% Use a different color highlighter.
484
2. Fill in 17% of this grid.
3. What percent of this figure is unshaded?
4. Refer to the photograph of recyclable containers at
right. Round each percent to the nearest tenth of a
percent.
a) What percent of the containers have a spray
trigger ?
b) What percent of the containers are red?
c) What percent of the containers are yellow?
d) What percent of the containers are not yellow?
Chapter 5 — Percents
Section 5.1 — Converting between Percents, Decimals, and Fractions
IDENTIFY
AND
CORRECT
THE
485
ERRORS
Identify the error(s) in the following worked solutions. If the worked solution is correct, write “Correct” in the
second column. If the worked solution is incorrect, solve the problem correctly in the third column.
Worked Solution
What is Wrong Here?
1) Write the equivalent fraction
for 0.72%.
Identify the Errors
The fraction form should
not contain a decimal
point.
Convert the percent to
a decimal, then to a
fraction.
Reduce the fraction.
Correct Process
00.72% = .0072
=
72 ÷ 4
18
=
10, 000 ÷ 4
2,500
18 ÷ 2
9
=
2,500 ÷ 2 1, 250
Answer:
2) Change 5.5% to a fraction.
3) Write 1.12% as a fraction.
4) Write 0.005 as a percent.
9
1, 250
Chapter 5 — Percents
486
Worked Solution
What is Wrong Here?
5) Convert 17.3 to a percent.
6) Convert 2
3
to a percent.
5
Identify the Errors
Correct Process
Section 5.1 — Converting between Percents, Decimals, and Fractions
ADDITIONAL EXERCISES
Convert each of the following to a decimal. Round to the nearest ten-thousandth, if necessary.
1.
3
5
5.
2. 126%
3.
4
9
4.
3
2 %
4
4
5
8
6. 48%
7.
5
11
1
8. 12 %
6
Convert each of the following to a percent. Round to the nearest hundredth percent, if necessary.
9.
0.325
13.
0.006
10.
5
16
14.
11.
11
25
15.
5
1
4
12. 0.23
16.
1
3
5
1
3
Convert each of the following to a simple fraction or mixed number. Reduce to the lowest term.
17.
0.275
21.
4.8%
18.
2.65
22.
264%
19.
17.5%
23.
96%
20.
1
5 %
4
24.
Order the following from smallest to largest.
25.
5
, 0.632,
8
26.
1.4,
29
,
20
62.9%,
11
16
3
1 , 142.5%
8
2
66 %
3
487