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CHAPTER 7

Decimals, Ratio, Proportion, and Percent

7.1. Decimals

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 decimal points, add the numbers in columns asif they were whole numbers, and insert a decimal in the answer immediatelybeneath 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

1

2 7. DECIMALS, RATIO, PROPORTION, AND PERCENT

(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.

7.2. OPERATIONS WITH DECIMALS 3

(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 11

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.


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.

7.3. RATIO AND PROPORTION 13

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


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 15

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.


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

7.4. PERCENT 17

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


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.

7.4. PERCENT 19

(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


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 percent - christian …facstaff.cbu.edu/wschrein/media/m152...

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