Download Decimals, Ratio, Proportion, and Percent

CHAPTER 7
Decimals, Ratio, Proportion, and Percent
Problem (Page 264). A street vendor had a basket of apples. Feeling
generous one day, he gave away one-half of his apples plus 1 to the first stranger
he 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 the
vendor 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 the
end.
• 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:
Thus
⇣1⌘
⇣ 1 ⌘
⇣ 1 ⌘
3(1000) + 4(100) + 5(10) + 7(1) + 9
+6
+8
10
100
1000
968
.
1000
This is read as: three thousand four hundred fifty-seven and nine hundred
sixty-eight thousandths.
3457.968 = 3457
The decimal point is placed between the ones column and the tenths column
to show where the whole number ends and the decimal (or fractional) portion
begins.
Note.
(1) In some countries (such as India)the role of the comma and period in writing
numbers is interchanged.
(2) In some countries our z (“zee”) is pronounced “zed.”
4
(3) In some countries the fraction is read as “four by seven.”
7
7.1. DECIMALS
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
50
2
+
+
=
10, 000 10, 000 10, 000
4
5
2
+
+
= .0452
100 1000 10, 000
115
116
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
Terminating decimals have a finite number of nonzero digits to the right of the
decimal point. Thus the denominator of the fractional part is a power of 10.
Theorem (Fractions with Terminating Decimal Representations).
a
a
Let be a fraction in simplest form. Then has a terminating decimal
b
b
representation if and only if b contains only 2’s and/or 5’s in its prime
factorization (since b can be expanded to a power of 10).
Example.
(1)
(2)
(3)
7
7
7 · 55
7 · 3125
21875
= 5= 5 5=
=
= .21875.
32 2
2 ·5
105
100, 000
37
37
37 · 54 37 · 625
23, 125
= 6 2= 6 6 =
=
= .02135
1600 2 · 5
2 ·5
106
1, 000, 000
Ordering Decimals
5
5
5 · 53
54
625
= 3= 3 3= 3=
= .625
8 2
2 ·5
10
1000
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 denominator)
Example.
135
14
140
, .14 =
=
1000
100 1000
135
140
Since 135 < 140,
<
, and so .135 < .14.
1000 1000
.135 =
118
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
(4) Place-value method – compare place-values one at a time from left to right
just as with whole numbers.
Example.
.135 < .14 since both have the same tenths digit, but .14 has a larger hundredths
digit. Further digits cannot contribute enough to make a di↵erence.
Calculate mentally, using compatible decimal numbers, properties, and/or compensation:
(1)
=
|{z}
commutative
(2)
7 ⇥ 3.4 + 6.6 ⇥ 7
7 ⇥ 3.4 + 7 ⇥ 6.6 |{z}
=
7(3.4 + 6.6) = 7(10) = 70.
distributive
(3)
26.53 8.95
=
26.58 9 = 17.58.
|{z}
(4)
5.89 + 6.27
=
6 + 6.16 = 12.16.
|{z}
equal additions
additive compensation
=
|{z}
(5.7 + 4.8) + 3.2
5.7 + (4.8 + 3.2) = 5.7 + 8 = 13.7.
associative+compatible
(5)
=
|{z}
commutative
0.5 ⇥ (2 ⇥ 639)
0.5 ⇥ (639 ⇥ 2)
=
(0.5 ⇥ 2) ⇥ 639 = 1 ⇥ 639 = 639.
|{z}
associative+compatible
7.1. DECIMALS
119
(6)
6.5 ⇥ 12
(6 + .5)12 |{z}
=
6(12) + .5(12) = 72 + 6 = 78.
distributive
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 by
moving the decimal point of n to the right m places. Dividing a number n
by 10m is equivalent to forming a new number by moving the decimal point
of n to the left m places.
Example.
(1)
(2)
67.32 ⇥ 103 = 67320
0.491 ÷ 102 = 0.00491
491
1
491
·
=
= .00491
1000 100 100, 000
120
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
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 0.875
1
1
1
1
1
3
2
1
3
5
3
4
7
Fraction 20
10
8
5
4
8
5
2
5
8
4
5
8
Calculate using fractional equivalents:
(1)
230 ⇥ .1 =
1
230 ⇥
= 23.
10
(2)
36 ⇥ 0.25 =
1
36 ⇥ = 9.
4
(3)
82 ⇥ 0.5 =
1
82 ⇥ = 41.
2
(4)
125 ⇥ .8 =
4
125 ⇥ = 100.
5
(5)
175 ⇥ 0.2 =
1
175 ⇥ = 35.
5
7.1. DECIMALS
(6)
0.6 ⇥ 35 =
3
⇥ 35 = 21.
5
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.
121
122
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
(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)
(2)
(3)
123.9 ÷ 5.3 ⇡
125 ÷ 5 = 25.
87.4 ⇥ 7.9 ⇡
90 ⇥ 8 = 720.
402 ÷ 1.25 ⇡
5
4
400 ÷ = 400 ⇥ = 320.
4
5
(4)
34, 546 ⇥ 0.004 ⇡
2
350 ⇥ .4 = 350 ⇥ = 140.
5
(5)
0.0024 ⇥ 470, 000 ⇡
1
.24 ⇥ 4700 ⇡ ⇥ 4800 = 1200.
4
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
13, 809
3710 13, 809
17, 519
+
=
+
=
= 17.519
100
1000
1000
1000
1000
(2) Decimal approach – align the decimalm points, add the numbers in columns
as if they were whole numbers, and insert a decimal in the answer immediately beneath the decial points of the numbers being added.
3.71
3.710
+13.809
+13.809
or
17.519
17.519
Subtraction
Example. 13.809
3.71
(1) Using fractions:
13.809
3.71 =
13, 809
1000
371
13, 809
=
100
1000
3710
10, 099
=
= 10.099
1000
1000
(2) Decimal approach – as with addition.
13.809
3.71
10.099
or
13.809
3.710
17.519
124
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
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
(2) Using fractions:
73 1141 73 · 1141 83, 293
⇥
=
=
= 83.293
10
100
10 · 100
1000
Note that the location of the decimal matches the estimate.
7.3 ⇥ 11.41 =
(3) Decimal approach – multiply as though without decimal points, and then
insert a decimal point in the answer so that the number of digits to the
right of the decimal in the answer equals the sum of the number of digits
to 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 estimate.
7.2. OPERATIONS WITH DECIMALS
125
Example. 421.2 ⇥ .0076
Estimate:
400 ⇥ .01 = 400 ⇥
1
=4
100
The placement of the decimal point corresponds with the estimate.
Division:
Example. 6.5 ÷ 0.026
(1) Estimate:
6 ÷ .03 = 6 ÷
3
100 600
=6⇥
=
= 200
100
3
3
(2) Using fractions:
6.5 ÷ 0.026 =
65
26
6500
26
6500
÷
=
÷
=
= 250
10 1000 1000 1000
26
(3) Decimal approach – replace the original problem by an equivalent problem
where the divisor is a whole number
126
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
Example. 6.5 ÷ 0.026
(1) Estimate:
6 ÷ .03 = 6 ÷
3
100 600
=6⇥
=
= 200
100
3
3
(2) Using fractions:
6.5 ÷ 0.026 =
65
26
6500
26
6500
÷
=
÷
=
= 250
10 1000 1000 1000
26
(3) Decimal approach – replace the original problem by an equivalent problem
where the divisor is a whole number
Example. 1470.3838 ÷ 26.57
7.2. OPERATIONS WITH DECIMALS
127
Repeating Decimals
(1) Fractions in simplified form with only 2’s and 5’s as prime factors in the
denominator convert to terminating decimals.
Example.
Example.
128
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
(2) Fractions in simplified form with factors other than 2 and 5 in the denominator convert to repeating decimals.
5
Example.
12
5
= .4166 · · · = .416 with 6 indicating the 6 repeats indefinitely.
12
7.2. OPERATIONS WITH DECIMALS
Example.
129
3
11
3
= 0.27. The “27” is called the repetend. Decimals with a repetend are
11
called repeating decimnals. The number of digits in the repetend is the period
of the decimal.
Terminating decimals are decimals with a repetend of 0, e.g., 0.3 = 0.30.
130
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
Every fraction can be written as a repeating decimal. Ts see why this is so,
5
consider . In dividing by 7, there are 7 possible remainders, 0 through 6. Thus
7
a remainder must repeat by the 7th division:
5
Example.
7
5
= 0.714285
7
Theorem (Fractions with Repeating, Nonterminating Decimal Represena
a
tations). Let
be a fraction in simplest form. Then
has a repeating
b
b
decimal representation that does not terminate if and only if b has a prime
factor other than 2 or 5.
7.2. OPERATIONS WITH DECIMALS
131
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
18616
n=
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
3.96 396
n=
=
=
9
900
So .439 = .44 = .440.
44
11
=
100
25
|{z}
Notice n = .44
We have two decimal numerals for the same number. When 9 repeats, you cvan
drop the repetend and increase the preivious digit by 1 to get a terminating
decimal.
Theorem. Every fraction has a repeating decimal representation, and
every repeating decimal has a fraction representation.
132
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
7.3. Ratio and Proportion
Example. On a given farm, the ratio of cattle to hogs is 7 : 4. (This is read
7 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:
7
a)
of the livestock are cattle.
11
4
a)
of the livestock are hogs.
11
7
4)There are as many cattle as hogs.
4
4
5) There are as many hogs as cattle.
7
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, with
b 6= 0.
Note.
1) Ratios allow us to compare the relative sizes of 2 quantities.
a
2) The ratio a : b can also be represented by the fraction .
b
7.3. RATIO AND PROPORTION
133
3) Ratios can involve any real numbers:
Example.
3.5 : 1 or
3.5
,
1
7 3
7/2
: or
,
2 4
3/4
p
2 : ⇡ or
p
2
⇡
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). there
would then be 60 corresponding groups of 4 hogs each, or 60 · 4 = 240 hogs. ⇤
Definition (Equality of Ratios).
a
c
a c
Let and be any two ratios. Then = if and only if ad = bc.
b
d
b d
Note.
1) a and d are called the extremes and b and c are called the means
“a : b| {z
= }c : d if and only if ad = bc.”
| means
{z }
extremes
“Two ratios are equal if and only if the product of the extremes equals the
product of the means.”
an a
2) Just as with fractions, if n 6= 0,
= or an : bn = a : b.
bn
b
134
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
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 13 · 3 3
39 : 91 =
=
=
91 13 · 7 7
Example.
Are the ratios 7 : 12 and 36 : 60 equal?.
Extremes: 7 · 60 = 420
The ratios are not equal.
Means: 12 · 36 = 432
Example.
B 2 14
Solve for the unknown in the proportion = .
8
18
⇣ 1⌘
1
18B = 8·2 =) 18B = 8 2+
=) 18B = 16+2 =) 18B = 18 =) B = 1
4
4
Example.
3x 12 x
Solve for the unknown in the proportion
=
.
4
6
18x = 4(12
x) =) 18x = 48
4x =) 22x = 48 =) x =
Example.
Solve the follwing proportions mentally:
1) 26 miles for 6 hours is equal to
for 24 hours.
104
48 24
=
22 11
7.3. RATIO AND PROPORTION
135
2) 750 people for each 12 square miles is equal to
square miles.
people for each 16
1000
Example.
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
x
miles 35 1000
1
x
=
(notice how the unit align).
35 1000
35x = 1000
1000 200
4
x=
=
= 28 ⇡ 28.57
35
7
7
Example.
We have
A softball pitcher has given up 18 earned runs in 39 innings. How many earned
runs does she give up per seven-inning game (ERA)
season game
earned runs 18
x
innings
39
7
18 x
=
39 7
39x = 126
126 42
x=
=
⇡ 3.23
39
13
136
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
7.4. Percent
Percent means per hundred and % is used to represent percent.
60
60 percent = 60% =
= .60
100
53
530 percent = 530% =
= 5.30
100
In general,
n
n% =
(definition).
100
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
= .67
100
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 decimal
two 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 and
5’s as factors) can be expressed as a fraction with a denominator of 100.
Example.
5
625
62.5
=
=
= .625 = 62.5%
8 1000
100
3
⇡ (long division) .429 = 42.9%
7
Common Equivalents
Percent Fraction
1
5%
20
10%
1
10
20%
1
5
25%
1
4
33 13 %
1
3
50%
1
2
66 23 %
2
3
75%
3
4
138
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
Example. Find mentally:
196 is 200% of
.
2x = 196 =) x =
25% of 244=
40 is
.
1
⇥ 196 = 98
2
1
⇥ 244 = 61
4
% of 32.
40 5
1
= = 1 + = 100% + 25% = 125%
32 4
4
731 is 50% of
.
1
x = 731 =) x = 2 ⇥ 731 = 1462
2
166 23 % of 300 is
.
2
2
2
166 % = 100% + 66 % = 1 +
3
3
3
⇣
⌘
2
1 + 300 = 300 + 200 = 500
3
Find 15% of 40.
Find 300% of 120.
Find 33 13 % of 210.
1
1
15% = 10% + 5% =
+
10 20
⇣1
⌘
1
+
40 = 4 + 2 = 6
10 20
2 ⇥ 120 = 240
1
⇥ 210 = 70
3
7.4. PERCENT
139
Example. Estimate mentally:
21% of 34.
11.2% of 431.
1
of 35 = 7
5
1 ⌘
(10 + 1)% =
+
of 430 = 43 + 4 = 47
10 100
Solving Percent Problems
⇣1
(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 payment is 30%, 30 of 100 squares are marked.
Each square represents
14, 000
= 140 dollars (1% of $14,000).
100
Thus 30 squares represent 30% of $14,000 or
30 ⇥ $140 = $4200.
140
7. DECIMALS, RATIO, PROPORTION, AND PERCENT
(2) Proportion approach – since percents can be written as a ratio.
Example. A volleyball team wins 105 games, which is 70% of the games
played. How many games were played?
percent actual
wins
70
105
games 100
x
70
105
=
=) 70x = 10, 500 =) x = 150 games played
100
x
Example. If Frank saves $28 of his $240 weekly salary, what percent does
he save?
actual percent
saved 28
x
salary 240
100
28
x
2800 35
=
=) 240x = 2800 =) x =
=
240 100
240
3
Frank saves 11 23 %.
(3) Equation approach (x is unknown; p, n, and a are fixed numbers).
Translation of Problem ⇣ Equation
p ⌘
(a) p% of n is x
n=x
100
⇣ p ⌘
(b) p% of x is a
x=a
100
⇣ x ⌘
(c) x% of n is a
n=a
100
7.4. PERCENT
141
Example. Sue is paid $315.00 a week plus a 6% comission on sales. Find
her weekly earnings if the sales for the week are $575.00.
6
Translation (a): x =
· 575 = 34.5.
100
Salary = $315.00 + $34.50 = $349.50.
Example. A department store marked down all summer clothing 25%. The
following week, remaining items were marked down 15% o↵ the sale price. When
John bought 2 tank tops, he presented a coupon that gave him an additional
20% o↵. What percent of the original price did John save?
solution.
x = percent saved,
P = original price
Translation (c):
x
P =P
100
=P
=P
=P
x
P =P
100
x
= .49
100
x = 49%
price John paid
80
· (2nd markdown)
100
i
80 h 85
·
· (1st markdown)
100 100
80 h 85 ⇣ 75 ⌘i
·
·
P
100 100 100
.51P = .49P
⇤