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453
PRE-ACTIVITY
PREPARATION
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 red-
ticketed 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 specifi c information.
• Master the mathematical language of percents.• Convert between equivalent decimals, fractions, and percents.• Order a mixed set of fractions, decimals, and percents.
Converting between Percents,
Decimals, and Fractions
LLEARNINGEARNING OOBJECTIVESBJECTIVES
Section 5.1
454 Chapter 5 — Percents
TTERMINOLOGYERMINOLOGY
NEW TERMS TO LEARN
decimal fraction
non-terminating repeating decimal
percent
percent number
percent sign (%)
terminating decimal
PREVIOUSLY USED
build up
convert
factor
order
ratio
reduce
simple fraction
BBUILDING UILDING MMATHEMATICAL ATHEMATICAL LLANGUAGEANGUAGE
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.
For example, the decimal number 0.035 is “thirty-five thousandths,” or
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,
35
1000 as a decimal fraction.
22
10022= % "twenty-two percent" 22 is the percent number.
100
1100100= % "one hundred percent" 100 is the percent number.
198
100==198% "one hundred ninety-eight ppercent" 198 is the
or
percent number.
1212
10012
1
2= % "twelve and one-half percent" 112 5. % 22
1
2 is the
or %
percent number.
0 7
1000 7
7
10
.. %= "seven tenths percent" 0.7 is the percent number.
or "seven tenths of a percent"
455Section 5.1 — Converting between Percents, Decimals, and Fractions
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.)
TTECHNIQUESECHNIQUES
Technique
Write the decimal number as a decimal fraction and reduce the fraction to lowest terms.
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
►►A 0.022 4.48►►B0 022
221000
22 21000 2
. =
÷÷= 11
500 Answer
4 48 448
100
448 4
100 4
. =
÷÷
= 41225
Answer
Models
Convert the following decimals to their equivalent fractions or mixed numbers. Reduce fractions to lowest terms.
Converting between the Decimal, Fraction, and Percent Form of a Number
456 Chapter 5 — Percents
Technique
Divide the numerator of the fraction by its denominator.
Converting a Fraction to a Decimal Number
Model 1
? ? ? 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.
)8 3 000
2 4
60
56
40
40
0
0 375..
−
−
−
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.
)3 16 000
15
10
9
10
9
10
9
5 333..
−
−
−
−
....Non-terminating Repeating Decimal
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...
The notation for this is , with the bar over the 3 indicating that it is the digit 3 that repeats.
5 3.
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. .
►►A
Convert the following fractions and mixed numbers to decimals. Round to the ten-thousandth place for non-terminating decimals.
58
Divide by .
The quotient is a terminating decimal.
5 858= 0.6625 Answer
)8 5 00048
2016
4040
0
0 625..
−
−
−
The fraction is already (or can be built up to) a decimal fraction (see page 457, Model 2)
Special Case:
457Section 5.1 — Converting between Percents, Decimals, and Fractions
►►B 31
16
31
163
116
= + The whole number 3 does not change.
Convert the fraction to a decimal. 116
)16 1 000096
4032
8080
0
0 0625..
−
−
−
The quotient is a terminating decimal. 31
16= 3.0625 Answer
►►C 512
)12 5 0000048
20128072807280
0 41666
728
.
.
−
−
−
−
−
The quotient is a non-terminating repeating decimal.
Rounded to the nearest ten-thousandth, 512
≈ 0.4167 Answer
Model 2 The Fraction is Already (or Can be Built Up to) a Decimal Fraction
Convert the following to decimals.
►►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.
►►A 910
THINK 910
is “nine tenths” or 0.9 Answer
8425
84 425 4
816100
××= THINK “eight and sixteen hundredths” 8.16 Answer
Special Case:
458 Chapter 5 — Percents
Technique
Multiply the decimal by 100%.
That is, move the decimal point two places to the right and attach the percent sign (%).
Converting a Decimal Number to a Percent
? ? ? 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:
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%
100100
1% means and, from your knowledge of fractions,
100 000
1001= .
The shortcut for multiplying by 100 is to move the decimal point two places to the right.THINK
Convert the following decimals to their equivalent percents.
0.8225
0.8225 × 100% = 82.25% OR 0.8225% = 82.25% Answer
1.723
1.723 × 100% = 172.3% OR 1.723% = 172.3% Answer
Models
►►A
►►B
459Section 5.1 — Converting between Percents, Decimals, and Fractions
0.04 0.04% = 4% Answer
0.009 0.009% = 0.9% Answer
4 4.00% = 400% Answer
►►C
►►D
►►E
Model 1
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.
Converting a Percent to a Decimal Number
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 defi nition of a percent as the ratio of the percent number to 100.
For example, = 28.5 ÷ 100 or 0.28528 528 5
100. %
.=
The shortcut for dividing by 100 is to move the decimal point two places to the left.THINK
85.4% 85.4% = 85.4 ÷ 100 = 0.854 OR 85.4% = 0.854 Answer
258% 258% = 258 ÷ 100 = 2.58 OR 258% = 2.58 Answer
2.3% 2.3% = 02.3% 0.023 Answer
0.75% 0.75% = 00.75% 0.0075 Answer
►►A
►►B
►►C
Convert the following percents to their equivalent decimals.
Zero place digit needed.
►►DTwo zero place digits needed.
460 Chapter 5 — Percents
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
►►B
6214
%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.
14= ÷ =1 4 0 25. so 62.25% = 62
14
% = 0.6225 Answer
1413
%
13
1 3 0 3 0 33= ÷ = ≈. .
1413
% .33%≈ 14 14.33% = 0.1433 Answer
Before looking at the remaining two techniques for converting between fractions and percents, think about the fi rst four conversion techniques, as summarized in the table below.
Fraction Decimal Percent
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 fi nal two conversion techniques.
1340
)40 13 000120
10080
200200
0
0 325..
−
−
−
0.325
0.325
and
× 100%32.5%
32.5%32 5100
.0.325
0.325
and
3251000
1340 reduce
Model 2
461Section 5.1 — Converting between Percents, Decimals, and Fractions
Technique
Step 1 — Convert the fraction or mixed number to a decimal.
Step 2 — Convert the decimal to a percent.
Converting a Fraction or a Mixed Number 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.
Step 1
Step 2 0.875% = 87.5% Answer
Step 1
Step 2 2.50% = 250% Answer
78
212
►►A
►►B
78
0 875= .
212
2 5= .
)8 7 0000 875
64
60
56
40
40
0
.
.
−
−
−
Model 2 Special Case: The Denominator of the Fraction is 100
Convert to a percent: 37
100
37100
= 37% Answer
If the denominator of the fraction is 100, the numerator IS the percent number.
)2 1 00 5
10
0
.
.
−
462 Chapter 5 — Percents
Model 3
Convert to a percent:
Step 1
Step 2 0.45% = 45% Answer
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.
Special Case: The Denominator is a Factor of 100
9
20 920
0 45= .
)20 9 000 45
8 0
100
100
0
.
.
−
−
920
9 520 5
45100
= ××= = 45% by the definition of a percent.
Model 4 Special Case: The Result of Step 1 is a Repeating Decimal
Convert the fraction to a percent.
Round to the nearest hundredth percent, if necessary.
Step 1
Step 2 0.8333% = 83.33% Answer
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.
5
6
56
0 8333≈ . )6 5 00000
48
20
18
20
18
20
18
20
18
2
0 83333..
−
−
−
−
−
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,
Another is to present the repeating decimal part as a fraction.
That is,
56
83 3= . %
56
8313
= % )6 5 00
48
20
18
2
0 83..
−
−
2
60 83
1
3= .
463Section 5.1 — Converting between Percents, Decimals, and Fractions
Technique
Step 1 — Convert the percent to its equivalent decimal.
Step 2 — Convert the decimal to its equivalent fraction or mixed number and reduce.
Converting a Percent to a Fraction or Mixed Number
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
0.06% Step 1 00.06% = 0.0006
Step 2
3.2% Step 1 03.2% = 0.032
Step 2
►►A
►►B
0 00066
10 0006 2
10 000 2.
, ,= = ÷
÷= 3
5000 Answer
0 03232
1 00032 4
1 000 48 2
250 2.
, ,= = ÷
÷= ÷
÷= 4
125 Answer
Model 2 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%
215%
When the percent number is a whole number, there is no advantage to using its decimal equivalent. Simply write the percent number over 100 (defi nition of percent) and reduce.
54 0 5454 2100 2
% .= = ÷÷= 27
50 Answer
5454 2100 2
% = ÷÷= 27
50 Answer
►►B 215215100
215
100
3
20% = = = 23
20 Answer
Convert the following to fractions or mixed numbers. Reduce the fractions to lowest terms.
464 Chapter 5 — Percents
Model 3 Special case: The Percent Number Contains a Fraction
Convert the following to fractions. Reduce to lowest terms.
►►A
Note: This procedure assures that the resulting fractions will be the exact equivalent of the percent.
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.
►►B
2316
%
2316
2316
10023
16
1100
1396
1100
% = = × = × = 139600
Answer (reduced fully)
►►C
212
%
212
212
1100
52
1
100
1
20% = × = × = 140
Answer
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:
212
02 5 0 02525 25
1000 25140
% . % .= = = ÷÷
=
3313
%
3313
3313
1100
1003
1
100
1
1% = × = × = 13
Answer
465Section 5.1 — Converting between Percents, Decimals, and Fractions
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:
Example 2: Fraction to Decimal
A. Conversion:
Validation: (Convert 0.625 to a fraction)
B. Conversion:
Validation: 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.
Example 3: Decimal to Percent
Conversion: 1.723 = 172.3% Answer
Validation: (Convert 172.3% to a decimal.)
172.3% = 1.723
0.22 = 1150
Answer
)50 11 0010 0
100100
0
.−
−
0.22
(Convert to a decimal)1150
58= 0 625. Answer
0.625 =625 25
1000 2525 540 5
÷÷
= ÷÷= 5
8
512
≈ 0 417. Answer (rounded)
4171000
417 12 5 10005004 5000
=
× = ×≈
512
?
?
continues on the next page
466 Chapter 5 — Percents
Example 4: Percent to Decimal
A. Conversion: 2.3% = 0.023 Answer
Validation: (Convert 0.023 to a percent)
0.023% = 2.3%
B. Conversion:
Validation: (Convert 0.6225 to a percent)
Example 5: Fraction to Percent
A. Conversion:
Validation: (Convert 45% to a fraction)
B. Conversion:
Validation: Change the rounded 83.33% to its decimal equivalent. Then validate the equality of the decimal fraction and the original fraction.
Example 6: Percent to Fraction
Conversion:
Validation:
6214
% = 0 6225. Answer
0 6225 62 25 6225100
. . % %% = = = 6214
%
920= 45% Answer
45 0 4545 5
100 5% .= = ÷
÷= 9
20
56
≈ 83 33. % (rounded)
83 33 0 83338 333
10 0008 333 6 5 10 000
49 998 50 000
. % .,,
, ,, ,
=
=
× = ×≈
56
?
?
0.6% = 3500
Answer
(Convert to a percent.)3
500
)500 3 0003000
0
0 006..
−0 006. % = 0.6%
467Section 5.1 — Converting between Percents, Decimals, and Fractions
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)
Example:
< 1.0
Example: 0.4
< 100%
Example: 40%
improper fraction or mixed number (> 1)
Example:
> 1.0
Example: 1.75
> 100%
Example: 175%
Example: Convert 0.22 to a fraction.
Prediction: Since 0.22 < 1, the fraction will be proper (< 1).
Actual answer:
Example: Convert to a percent.
Prediction: Since < 1, the percent will be < 100%.
Actual answer: 87.5%
Example: Convert 1.6 to a percent.
Prediction: Since 1.6 > 1.0, the percent will be > 100%.
Actual answer: 160%
You can, in fact, refi ne 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
< 0.5 < 50%
< 0.1 < 10%
> 0.5 > 50%
> 0.75 > 75%
> 2 > 2.0 > 200%
25
134
THINK
1150
THINK
7
8
THINK
< 12
< 110
> 12
> 34
7
8
How Estimation/Prediction Can Help
468 Chapter 5 — Percents
Example: Convert 0.22 to a fraction.
Prediction: Since 0.22 < 0.5, the fraction will be < ½ .
Actual answer:
Example: Convert to a percent.
Prediction:
Actual answer: 87.5%
Example: Convert 2.8 to a percent.
Prediction: 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: Since 240% > 200%, the fraction will be a mixed number > 2 and
Since 40% < 50%, the fraction part of the mixed number will be
Actual answer:
Example: Convert 0.07 to a fraction.
Prediction: Since 0.07 < 0.1, the fraction will be
Actual answer:
Example: Convert 0.07 to a percent.
Prediction: Since 0.07 < 0.1, the percent will be < 10%.
Actual answer: 7%
THINK
1150
<⎛⎝⎜⎜⎜
⎞⎠⎟⎟⎟
1
2
25
50 or
7
8
THINK Since 7
8 , the percent will be >⎛⎝⎜⎜⎜
⎞⎠⎟⎟⎟ >3
4
6
875%
THINK
THINK
225
THINK < 1
10.
THINK
7100
<⎛⎝⎜⎜⎜
⎞⎠⎟⎟⎟
1
10
10
100 or
< 1
2.
469Section 5.1 — Converting between Percents, Decimals, and Fractions
Technique
Step 1 — Convert all numbers to the same equivalent form.
Step 2 — Order them as you would for your chosen form.
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 effi cient.
Model
List the following in order of their values, from lowest to highest: , 0.115, 11%
Step 1: Write as decimals. Step 2: Order the decimals using trailing zeros. Rank
0.111 2
0.115 is in decimal form 0.115 3
11% = 0.11 0.110 1
List the following in order of their values, from lowest to highest:
Step 1: Write as decimals. Step 2: Order the decimals using trailing zeros. Rank
0.75 = 0.75 0.7500 3
0.0075 1
7.5% = 07.5% 0.0750 2
►►A
►►B
19
0 1 0 111= ≈. .
Answer : 11%19
, 0.115 ,
1
9
)9 1 000
9
10
9
10
9
1
0 111..
−
−
−
0 753
47 5. % . %, ,
Answer : 34
% , 7.5% , 0.75
34
% .75%= =00 0 0075.
TTECHNIQUEECHNIQUE
Ordering a Mixed Set of Decimals, Fractions, and Percents
470 Chapter 5 — Percents
AADDRESSING DDRESSING CCOMMON OMMON EERRORSRRORS
Issue Incorrect Process Resolution Correct Process
Losing track of the percent sign (%) while converting a percent that contains a decimal or fraction
Convert to a decimal.
Convert 2.35% to a fraction.
Answer:
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.
Convert to a decimal.
Convert 2.35% to a fraction.
Making an error in the shift of the decimal point when converting between decimals and percents
Convert 1.5 to a percent.
1.5 = 0.015%
Convert 2.27% to a decimal.
2.27% = 227
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 Percentpoint shifts right
Decimal Percentpoint shifts left
Convert 1.5 to a percent.
1.5 > 1 so the percent will be greater than 100%
1.5 × 100% = 150%
Convert 2.27% to a decimal.
2.27% < 100%, so the decimal will be less than 1.
02.27% = 0.0227
135
1 6
01 6
% . %
. %
=
= 0.016
135
% 135
%
)5 3 030
0
0 6..
−
2 35 235
100
7
20. % =
2720
2 35 0 0235235 5
10 000 5
. % .
,
=
= ÷÷
= 472,000
THINK
THINK
5) 00 6
)5 3)30
52
. 0=
% 2=
471Section 5.1 — Converting between Percents, Decimals, and Fractions
Issue Incorrect Process Resolution Correct Process
Not presenting the required fraction form when converting from a percent to a fraction
Convert 0.8% to a fraction.
First convert the percent to a decimal. Then write the decimal as a decimal fraction and reduce to lowest terms.
The fraction equivalent of a decimal or percent should not contain a decimal point.
It must be in the form of a simple fraction—a whole number over a whole number.
Convert 0.8% to a fraction.
Misinterpreting the denominator for the decimal fraction when converting from a decimal to a fraction
Convert 0.125 to its fraction equivalent.
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.
Convert 0.125 to its fraction equivalent.
0 80 8 4100 4
. %.= ÷÷
Answer = 0 225.
0 8 00 8 0 008. % . % .= =
0 0088
1000
1
125. =
= 1125
0 125125 1251000 125
. = ÷÷
= 18
0 125125 125
10 000 125
5
400
180
1
80
.,
= ÷÷
= =
%80100
120,
25 1212500
472 Chapter 5 — Percents
PPREPARATION REPARATION IINVENTORYNVENTORY
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
473
ACTIVITY
Converting between Percents,
Decimals, and Fractions
PPERFORMANCE ERFORMANCE CCRITERIARITERIA
• 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
CCRITICAL RITICAL TTHINKING HINKING QQUESTIONSUESTIONS
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 fi nal answer for a decimal to fraction conversion?
Section 5.1
474 Chapter 5 — Percents
4. Why does
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?
•
•
•
•
100100
1001% = = ?
475Section 5.1 — Converting between Percents, Decimals, and Fractions
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?
476 Chapter 5 — Percents
10. How can you go about ordering a mixed set of fractions, decimals, and percents?
• 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.
TTIPS FOR IPS FOR SSUCCESSUCCESS
Fraction Decimal Percent
Begin with a Fraction0.75 75%
Begin with a Decimal0.5 50%
0.25Begin with a Percent
25%
34
14
25100
←
12
50100
←
• Use estimation/prediction to assure that your answer “makes sense.”
• Know some of the most common conversions:
1
1010
1
425
1
250 75 33= = = = =% % % %, , ,
3
4,
1
3
11
366
2
3% %,
2
3=
477Section 5.1 — Converting between Percents, Decimals, and Fractions
DDEMONSTRATE EMONSTRATE YYOUR OUR UUNDERSTANDINGNDERSTANDING
Convert to a Fraction or
Mixed NumberProcess Answer Validation (optional)
a) 0.03
b) 0.156
c) 8.42
1. Convert each of the following to a simple fraction or mixed number. Reduce to lowest terms.
Convert to a Decimal Process Answer Validation (optional)
a)
b)
2. Convert each of the following to a decimal. Round to the nearest thousandth for non-terminating decimals.
1740
1500
478 Chapter 5 — Percents
Convert to a Decimal Process Answer Validation (optional)
c)
d)
e)
f)
27
11320
445
716
479Section 5.1 — Converting between Percents, Decimals, and Fractions
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
3. Convert each of the following to a percent.
Convert to a Decimal Process Answer Validation
(optional)
a) 34%
b) 2.57%
c) 0.74%
d) 382%
e) 3%
f)
g)
4. Convert each of the following to a decimal. Round to the nearest ten-thousandth, if necessary.
2412
%
623
%
480 Chapter 5 — Percents
Convert to a Percent Process Answer Validation (optional)
a)
b)
c)
d)
5. Convert each of the following to a percent. Round to the nearest hundredth percent, if necessary.
316
750
338
112
481Section 5.1 — Converting between Percents, Decimals, and Fractions
Convert to a Percent Process Answer Validation (optional)
e)
f)
134
1512
100
Convert to a Fraction or
Mixed NumberProcess Answer Validation (optional)
a) 125%
b) 38%
6. Convert each of the following to a simple fraction or mixed number. Reduce to lowest terms.
482 Chapter 5 — Percents
Convert to a Fraction or
Mixed NumberProcess Answer Validation (optional)
c)
d)
e) 0.74%
f) 0.8%
g) 62.5%
6123
%
334
%
483Section 5.1 — Converting between Percents, Decimals, and Fractions
7. Order the following from lowest to highest value:
8. Order the following from highest to lowest value:
3
838 0 34, %, . , 37%
5 31
200 555. %, , . , 5
1
2%,
1
2
TEAM EXERCISESTEAM EXERCISES
1. In the grids below, fi ll 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 Chapter 5 — Percents
2. Fill in 17% of this grid.
3. What percent of this fi gure 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?
485Section 5.1 — Converting between Percents, Decimals, and Fractions
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 SolutionWhat is Wrong Here? Identify the Errors Correct Process
1) Write the equivalent fraction for 0.72%.
The fraction form should not contain a decimal point.
Convert the percent to a decimal, then to a fraction.
Reduce the fraction.
2) Change 5.5% to a fraction.
3) Write 1.12% as a fraction.
4) Write 0.005 as a percent.
IDENTIFY AND CORRECT THE ERRORSIDENTIFY AND CORRECT THE ERRORS
00 72 0072. % .=
= ÷÷
=72 410 000 4
182 500, ,
18 22 500 2
91 250
÷÷
=, ,
Answer:9
1 250,
486 Chapter 5 — Percents
235
Worked SolutionWhat is Wrong Here? Identify the Errors Correct Process
5) Convert 17.3 to a percent.
6) Convert to a percent.
487Section 5.1 — Converting between Percents, Decimals, and Fractions
Convert each of the following to a decimal. Round to the nearest ten-thousandth, if necessary.
1. 5.
2. 126% 6. 48%
3. 7.
4. 8.
Convert each of the following to a percent. Round to the nearest hundredth percent, if necessary.
9. 0.325 13. 0.006
10. 14.
11. 15.
12. 0.23 16.
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. 24.
Order the following from smallest to largest.
25.
26.
3
5
4
9
23
4%
45
8
5
11
121
6%
5
16
11
25
1
3
51
4
13
5
51
4% 66
2
3%
5
80 632 62 9, . , . %,
11
16
1 4 13
8. , , ,
29
20 142.5%
ADDITIONAL EXERCISESADDITIONAL EXERCISES