decimals, ratio, proportion, and percentfacstaff.cbu.edu/~wschrein/media/m151...

CHAPTER 7

Decimals, Ratio, Proportion, and Percent

Problem (Page 264). A street vendor had a basket of apples. Feelinggenerous one day, he gave away one-half of his apples plus 1 to the first strangerhe met, one-half of his remaining apples plus 1 to the next stranger he met,and one-half of his remaining apples plus 1 to the third stranger he met. If thevendor had one left for himself, with how many apples did he start?

Strategy 12 – Work Backward.

This strategy may be appropriate when

• The final result is clear and the initial portion of a problem is obscure.

• A problem proceeds from being complex initially to being simple at theend.

• A direct approach involves a complicated equation.

• A problem involves a sequence of reversible actions.

Solution.

The vendor finished with 1 apple.

To the third stranger he gave one-half his apples +1. So he must have had(1 + 1)2 = 4 when he met the third stranger.

To the second stranger he gave one-half his apples +1. So he must have had(4 + 1)2 = 10 when he met the second stranger.

To the first stranger he gave one-half his apples +1. So he must have had(10 + 1)2 = 22 when he met the first stranger.

So he started with 22 apples ⇤

113

114 7. DECIMALS, RATIO, PROPORTION, AND PERCENT

7.1. Decimals

Decimals are used to represent fractions.

Example. 3457.968

Expanded form:

3(1000) + 4(100) + 5(10) + 7(1) + 9⇣ 1

10

⌘+ 6

⇣ 1

100

⌘+ 8

⇣ 1

1000

⌘Thus

3457.968 = 3457968

1000.

This is read as: three thousand four hundred fifty-seven and nine hundredsixty-eight thousandths.

The decimal point is placed between the ones column and the tenths columnto show where the whole number ends and the decimal (or fractional) portionbegins.

Note.

(1) In some countries (such as India)the role of the comma and period in writingnumbers is interchanged.

(2) In some countries our z (“zee”) is pronounced “zed.”

(3) In some countries the fraction4

7is read as “four by seven.”

7.1. DECIMALS 115

A hundreds square can be used to represent tenths and hundredths.

A number line can also be used to picture decimals.

Example.9

100= (as a decimal)

.09

Example.452

10, 0000= (as a decimal)

400

10, 000+

50

10, 000+

2

10, 000=

4

100+

5

1000+

2

10, 000= .0452


Terminating decimals have a finite number of nonzero digits to the right of thedecimal point. Thus the denominator of the fractional part is a power of 10.

Theorem (Fractions with Terminating Decimal Representations).

Leta

bbe a fraction in simplest form. Then

a

bhas a terminating decimal

representation if and only if b contains only 2’s and/or 5’s in its primefactorization (since b can be expanded to a power of 10).

Example.

(1)7

32=

7

25=

7 · 55

25 · 55=

7 · 3125

105=

21875

100, 000= .21875.

(2)37

1600=

37

26 · 52=

37 · 54

26 · 56=

37 · 625

106=

23, 125

1, 000, 000= .02135

(3)5

8=

5

23=

5 · 53

23 · 53=

54

103=

625

1000= .625

Ordering Decimals

Terminating decimals can be compared using four methods:

(1) Hundreds square – the larger of two decimals has more shaded area.

Example..7 > .23

7.1. DECIMALS 117

.135 < .14

Note. Smaller decimals may have more nonzero digits than larger decimals.

(2) Number line – greater decimals are located to the right of smaller decimals.

Example..135 < .14

(3) Fraction method – compare the decimals as fractions (with a common de-nominator)

Example.

.135 =135

1000, .14 =

14

100=

140

1000

Since 135 < 140,135

1000<

140

1000, and so .135 < .14.


(4) Place-value method – compare place-values one at a time from left to rightjust as with whole numbers.

Example.

.135 < .14 since both have the same tenths digit, but .14 has a larger hundredthsdigit. Further digits cannot contribute enough to make a di↵erence.

Calculate mentally, using compatible decimal numbers, properties, and/or com-pensation:

(1)

7⇥ 3.4 + 6.6⇥ 7

=|{z}commutative

7⇥ 3.4 + 7⇥ 6.6 =|{z}distributive

7(3.4 + 6.6) = 7(10) = 70.

(2)

26.53� 8.95

=|{z}equal additions

26.58� 9 = 17.58.

(3)

5.89 + 6.27

=|{z}additive compensation

6 + 6.16 = 12.16.

(4)

(5.7 + 4.8) + 3.2

=|{z}associative+compatible

5.7 + (4.8 + 3.2) = 5.7 + 8 = 13.7.

(5)

0.5⇥ (639⇥ 2)

=|{z}commutative

0.5⇥ (2⇥ 639) =|{z}associative+compatible

(0.5⇥ 2)⇥ 639 = 1⇥ 639 = 639.

7.1. DECIMALS 119

(6)

6.5⇥ 12

(6 + .5)12 =|{z}distributive

6(12) + .5(12) = 72 + 6 = 78.

Theorem (Multiplying/Dividing Decimals by Powers of 10).

Let n be any decimal number and m represent any nonzero whole number.Mulitplying a number n by 10m is equivalent to forming a new number bymoving the decimal point of n to the right m places. Dividing a number nby 10m is equivalent to forming a new number by moving the decimal pointof n to the left m places.

Example.

(1)67.32⇥ 103 = 67320

(2)0.491 ÷ 102 = 0.00491

491

1000· 1

100=

491

100, 000= .00491


Fraction equivalents can often be used to simplify decimal calculations.

Decimal 0.05 0.1 0.125 0.2 0.25 0.375 0.4 0.5 0.6 0.625 0.75 0.8 0.875Fraction 1

20110

18

15

14

38

25

12

35

58

34

45

78

Calculate using fractional equivalents:

(1)

230⇥ .1 =

230⇥ 1

10= 23.

(2)

36⇥ 0.25 =

36⇥ 1

4= 9.

(3)

82⇥ 0.5 =

82⇥ 1

2= 41.

(4)

125⇥ .8 =

125⇥ 4

5= 100.

(5)

175⇥ 0.2 =

175⇥ 1

5= 35.

7.1. DECIMALS 121

(6)

0.6⇥ 35 =

3

5⇥ 35 = 21.

Decimals can be rounded to any specified place:

(1) Round 321.0864 to the nearest hunderdth.

321.09

(We use the “round a 5 up” method)

(2) Round 12.16231 to the nearest thousandth.

12.162

(3) Round 4.009055 to the nearest thousandth.

4.009

(4) Round 1.9984 to the nearest tenth.

2.0 (not 2)

(5) Round 1.9984 to the nearest hundredth.

2.00 (not 2 or 2.0)

Estimate the decimals given the various properties:

(1) 34.7⇥ 3.9 ⇡ (range, rounding to nearest whole number)

low is 34⇥ 3 = 102; high is 35⇥ 4 = 140; rounding is 35⇥ 4 = 140.

(2) 15.71 + 3.23 + 21.95 ⇡ (2 column front end with adjustment)

15 + 3 + 21 = 39 (same as low, so adjust)

.71 + .23 + .95 ⇡ 2, so overall estimate is 39 + 2 = 41.


(3) 13.7⇥ 6.1 ⇡ (one column front end and range)

front end is 10⇥ 6 = 60; low is 13⇥ 6 = 78; high is 14⇥ 7 = 98.

(4) 3.61 + 4.91 + 1.3 ⇡ (front end with adjustment)

front end is 3 + 4 + 1 = 8 (= low).

adjustment is .61 + .91 + .3 ⇡ 2, so estimate is 8 + 2 = 1.

Estimate by rounding to compatible numbers and fraction equivalents.

(1)

123.9 ÷ 5.3 ⇡125 ÷ 5 = 25.

(2)

87.4⇥ 7.9 ⇡90⇥ 8 = 720.

(3)

402 ÷ 1.25 ⇡

400 ÷ 5

4= 400⇥ 4

5= 320.

(4)

34, 546⇥ 0.004 ⇡

350⇥ .4 = 350⇥ 2

5= 140.

(5)

0.0024⇥ 470, 000 ⇡

.24⇥ 4700 ⇡ 1

4⇥ 4800 = 1200.

7.2. OPERATIONS WITH DECIMALS 123

7.2. Operations with Decimals

Addition

Example. 3.71 + 13.809

(1) Using fractions:

3.71 + 13.809 =371

100+

13, 809

1000=

3710

1000+

13, 809

1000=

17, 519

1000= 17.519

(2) Decimal approach – align the decimalm points, add the numbers in columnsas if they were whole numbers, and insert a decimal in the answer immedi-ately beneath the decial points of the numbers being added.

3.71+13.809��17.519

or

3.710+13.809��17.519

Subtraction

Example. 13.809� 3.71


13.809 � 3.71 =13, 809

1000� 371

100=

13, 809

1000� 3710

1000=

10, 099

1000= 10.099

(2) Decimal approach – as with addition.

13.809�3.71��10.099

or

13.809�3.710��17.519


Example. 14.3� 7.961

14.3�7.961��

=)14.300�7.961��6.339

Multiplication

Example. 7.3⇥ 11.41

(1) Estimate: 7⇥ 11 = 77


7.3⇥ 11.41 =73

10⇥ 1141

100=

73 · 1141

10 · 100=

83, 293

1000= 83.293

Note that the location of the decimal matches the estimate.

(3) Decimal approach – multiply as though without decimal points, and theninsert a decimal point in the answer so that the number of digits to theright of the decimal in the answer equals the sum of the number of digitsto the right of the decimal points in the numbers being multiplied.

7.3⇥ 11.41 = 11.41⇥ 7.3

Again, the placement of the decimal point makes sense in view of the esti-mate.


Example. 421.2⇥ .0076

Estimate:

400⇥ .01 = 400⇥ 1

100= 4

The placement of the decimal point corresponds with the estimate.

Division:

Example. 6.5 ÷ 0.026

(1) Estimate:

6 ÷ .03 = 6 ÷ 3

100= 6⇥ 100

3=

600

3= 200


6.5 ÷ 0.026 =65

10÷ 26

1000=

6500

1000÷ 26

1000=

6500

26= 250

(3) Decimal approach – replace the original problem by an equivalent problemwhere the divisor is a whole number


Example. 6.5 ÷ 0.026

(1) Estimate:

6 ÷ .03 = 6 ÷ 3

100= 6⇥ 100

3=

600

3= 200


6.5 ÷ 0.026 =65

10÷ 26

1000=

6500

1000÷ 26

1000=

6500

26= 250

(3) Decimal approach – replace the original problem by an equivalent problemwhere the divisor is a whole number

Example. 1470.3838 ÷ 26.57


Repeating Decimals

(1) Fractions in simplified form with only 2’s and 5’s as prime factors in thedenominator convert to terminating decimals.

Example.

Example.


(2) Fractions in simplified form with factors other than 2 and 5 in the denomi-nator convert to repeating decimals.

Example.5

12

5

12= .4166 · · · = .416 with 6 indicating the 6 repeats indefinitely.


Example.3

11

3

11= 0.27. The “27” is called the repetend. Decimals with a repetend are

called repeating decimnals. The number of digits in the repetend is the periodof the decimal.

Terminating decimals are decimals with a repetend of 0, e.g., 0.3 = 0.30.


Every fraction can be written as a repeating decimal. Ts see why this is so,

consider5

7. In dividing by 7, there are 7 possible remainders, 0 through 6. Thus

a remainder must repeat by the 7th division:

Example.5

7

5

7= 0.714285

Theorem (Fractions with Repeating, Nonterminating Decimal Represen-

tations). Leta

bbe a fraction in simplest form. Then

a

bhas a repeating

decimal representation that does not terminate if and only if b has a primefactor other than 2 or 5.


Example. Changing a repeating decimal into a fraction.

18.634 has a period of 3, so we use 103 = 1000.

Let n = 18.634. Then 1000n = 18634.634.

1000n = 18634.634634 · · ·�n = 18.634634 · · ·

��999n = 18616

n =18616

999

Example. Change .439 to a fraction.

.439 has a period of 1, so we use 101 = 10.

Let n = .439. Then 10n = .439.

10n = 4.39999 · · ·�n = .43999 · · ·

��9n = 3.96

n =3.96

9=

396

900=

44

100|{z}Notice n = .44

=11

25

So .439 = .44 = .440.

We have two decimal numerals for the same number. When 9 repeats, you cvandrop the repetend and increase the preivious digit by 1 to get a terminatingdecimal.

Theorem. Every fraction has a repeating decimal representation, andevery repeating decimal has a fraction representation.


7.3. Ratio and Proportion

Example. On a given farm, the ratio of cattle to hogs is 7 : 4. (This is read7 to 4.).

What this means:

1) For every 7 cattle, there are 4 hogs.

2) For every 4 hogs, there are 7 cattle.

3) Assuming there are no other types of livestock on the farm:

a)7

11of the livestock are cattle.

a)4

11of the livestock are hogs.

4)There are7

4as many cattle as hogs.

5) There are4

7as many hogs as cattle.

6) Again assuming no other types of livestock:

a) 7 of 11 livestock are cattle.

a) 4 of 11 livestock are hogs.

Definition. A ratio is an ordered pair of numbers, written a : b, withb 6= 0.

Note.

1) Ratios allow us to compare the relative sizes of 2 quantities.

2) The ratio a : b can also be represented by the fractiona

b.

7.3. RATIO AND PROPORTION 133

3) Ratios can involve any real numbers:

Example.

3.5 : 1 or3.5

1,

7

2:3

4or

7/2

3/4,p

2 : ⇡ or

p2

⇡

4) Ratios can be used to express 3 typres of comparisons:

a) part-to-part

A cattle to hog ratio of 7 : 4.

b) part-to-whole

A hog to livestock ratio of 4 : 11.

c) whole-to-part

Livestock to cattle ratio of 11 : 7.

Example. Suppose our farm has 420 cattle. How many hogs are there?

Solution. The cattle can be broken up into 60 groups of 7 (420÷7). therewould then be 60 corresponding groups of 4 hogs each, or 60 ·4 = 240 hogs. ⇤

Definition (Equality of Ratios).

Leta

band

c

dbe any two ratios. Then

a

b=

c

dif and only if ad = bc.

Note.

1) a and d are called the extremes and b and c are called the means

“a : b = c| {z }means

: d

| {z }extremes

if and only if ad = bc.”

“Two ratios are equal if and only if the product of the extremes equals theproduct of the means.”

2) Just as with fractions, if n 6= 0,an

bn=

a

bor an : bn = a : b.


Definition.

A proportion is a statement that 2 ratios are equal.

Example.

Write a fraction in simplest form that is equivalent to the ratio 39 : 91.

39 : 91 =39

91=

13 · 313 · 7 =

3

7Example.

Are the ratios 7 : 12 and 36 : 60 equal?.

Extremes: 7 · 60 = 420 Means: 12 · 36 = 432

The ratios are not equal.

Example.

Solve for the unknown in the proportionB

8=

214

18.

18B = 8·21

4=) 18B = 8

⇣2+

1

4

⌘=) 18B = 16+2 =) 18B = 18 =) B = 1

Example.

Solve for the unknown in the proportion3x

4=

12� x

6.

18x = 4(12� x) =) 18x = 48� 4x =) 22x = 48 =) x =48

22=

24

11Example.

Solve the follwing proportions mentally:

1) 26 miles for 6 hours is equal to for 24 hours.

104

7.3. RATIO AND PROPORTION 135

2) 750 people for each 12 square miles is equal to people for each 16square miles.

1000Example.

If one inch on a map represents 35 miles and two cities are 1000 miles apart,how many inches apart would the be on the map?

Use a table:scale actual

inches 1 xmiles 35 1000

We have1

35=

x

1000(notice how the unit align).

35x = 1000

x =1000

35=

200

7= 28

4

7⇡ 28.57

Example.

A softball pitcher has given up 18 earned runs in 39 innings. How many earnedruns does she give up per seven-inning game (ERA)

season gameearned runs 18 x

innings 39 7

18

39=

x

739x = 126

x =126

39=

42

13⇡ 3.23


7.4. Percent

Percent means per hundred and % is used to represent percent.

60 percent = 60% =60

100= .60

530 percent = 530% =53

100= 5.30

In general,

n% =n

100(definition).

Conversions:

(1) Percents to fractions – use the definition

Example.

37% =37

100

(2) Percents to decimals – go percent to fraction to decimal

Example.

67% =67

100= .67

Shortcut – drop % sign and move the dcimal two places to the left.

Example.

54% = .54

5% = .05

372% = 3.72

(3) Decimals to percents – reverse the shortcut of step (2) (move the decimaltwo places to the right and add the % sign.

7.4. PERCENT 137

Example.

.73 = 73%

2.17 = 217%

.235 = 23.5%

(4) Fractions to percents – go fraction to decimal to percent.

Note. fractions with terminating decimals (denominator only has 2’s and5’s as factors) can be expressed as a fraction with a denominator of 100.

Example.5

8=

625

1000=

62.5

100= .625 = 62.5%

3

7⇡ (long division) .429 = 42.9%

Common Equivalents

Percent Fraction5% 1

20

10% 110

20% 15

25% 14

3313%

13

50% 12

6623%

23

75% 34


Example. Find mentally:

196 is 200% of .

2x = 196 =) x =1

2⇥ 196 = 98

25% of 244= .1

4⇥ 244 = 61

40 is % of 32.40

32=

5

4= 1 +

1

4= 100% + 25% = 125%

731 is 50% of .1

2x = 731 =) x = 2⇥ 731 = 1462

16623% of 300 is .

1662

3% = 100% + 66

2

3% = 1 +

2

3⇣1 +

2

3

⌘300 = 300 + 200 = 500

Find 15% of 40.

15% = 10% + 5% =1

10+

1

20⇣ 1

10+

1

20

⌘40 = 4 + 2 = 6

Find 300% of 120.2⇥ 120 = 240

Find 3313% of 210.

1

3⇥ 210 = 70

7.4. PERCENT 139

Example. Estimate mentally:

21% of 34.1

5of 35 = 7

11.2% of 431.

(10 + 1)% =⇣ 1

10+

1

100

⌘of 430 = 43 + 4 = 47

Solving Percent Problems

(1) Grid approach.

Example. A car was purchased for $14,000 with a 30% down payment.How much was the down payment?

Let the grid below represent the total cost of $14,000. Since the down pay-ment is 30%, 30 of 100 squares are marked.

Each square represents14, 000

100= 140 dollars (1% of $14,000).

Thus 30 squares represent 30% of $14,000 or

30⇥ $140 = $4200.


(2) Proportion approach – since percents can be written as a ratio.

Example. A volleyball team wins 105 games, which is 70% of the gamesplayed. How many games were played?

percent actualwins 70 105

games 100 x70

100=

105

x=) 70x = 10, 500 =) x = 150 games played

Example. If Frank saves $28 of his $240 weekly salary, what percent doeshe save?

actual percentsaved 28 xsalary 240 100

28

240=

x

100=) 240x = 2800 =) x =

2800

240=

35

3

Frank saves 1123%.

(3) Equation approach (x is unknown; p, n, and a are fixed numbers).Translation of Problem Equation

(a) p% of n is x⇣ p

100

⌘n = x

(b) p% of x is a⇣ p

100

⌘x = a

(c) x% of n is a⇣ x

100

⌘n = a

7.4. PERCENT 141

Example. Sue is paid $315.00 a week plus a 6% comission on sales. Findher weekly earnings if the sales for the week are $575.00.

Translation (a): x =6

100· 575 = 34.5.

Salary = $315.00 + $34.50 = $349.50.

Example. A department store marked down all summer clothing 25%. Thefollowing week, remaining items were marked down 15% o↵ the sale price. WhenJohn bought 2 tank tops, he presented a coupon that gave him an additional20% o↵. What percent of the original price did John save?

solution.

x = percent saved, P = original price

Translation (c):x

100P = P � price John paid

= P � 80

100· (2nd markdown)

= P � 80

100·h 85

100· (1st markdown)

i

= P � 80

100·h 85

100·⇣ 75

100P

⌘ix

100P = P � .51P = .49P

x

100= .49

x = 49%

⇤

decimals, ratio, proportion, and percentfacstaff.cbu.edu/~wschrein/media/m151...

Documents