Download NAT 01 - Ratios - HFC Learning Lab

HFCC Math Lab
NAT – 01
Ratio and Proportion
Ratio
A ratio is the comparison of two quantities which have the same units. This comparison can be written in
three different ways:
1.
as a fraction
2.
as two numbers separated by the symbol “ : ”
3.
as two numbers separated by the word “ to ”
For example, the ratio of the lengths of two boards, one 8 feet long and the other 10 feet long, can be written:
1
8 feet
8 4
=
=
10 feet 10 5
2.
8 feet : 10 feet = 8 :10 = 4 : 5
3.
8 feet to 10 feet = 8 to 10 = 4 to 5
Note that when like units of measurement appear in both numerator and denominator of a fraction, we can
“cancel” the units common to both just as we can cancel numerical factors common to both.
If the units of the numerator and denominator are not alike, then the units must be converted to like units, if
possible.
For example, to find the ratio of 7 inches to one foot we must express both lengths in terms of feet or both in
terms of inches. It is easier to use inches and the ratio is expressed as:
7 inches
7 inches
7
=
=
1 foot
12 inches 12
Ex1:
Write the comparison of 18 quarts to 12 quarts as a ratio in simplest form using:
a) a fraction
a)
c)
(Remember: 1 foot = 12 inches)
18 quarts
12 quarts
=
18 3
=
12 2
b) “ : “
b)
c) “to”
18 quarts : 12 quarts = 18 :12 = 3 : 2
18 quarts to 12 quarts = 18 to 12 = 3 to 2
Revised 05/11
1
Ex2:
Express the ratio of 5 cents to $1 as a fraction reduced to lowest terms.
5 cents
5
1
5 cents
=
=
=
1dollar 100 cents 100 20
Ex3:
Express the ratio 50 minutes : 2 hours as a fraction reduced to lowest terms.
5 0 minutes 50 minutes
50 5
=
=
=
2 hours
120 minutes 120 12
In all ratios studied so far, the terms of the ratio (numerators and denominators) have been whole numbers.
This is not always the case. The terms of the ratio can be any kind of number; the only restriction is that the
denominator cannot be zero.
3
1
pounds to 3 pounds, both terms are mixed numbers. This ratio written as
4
2
a fraction reduced to lowest terms is:
For example, in the ratio of 1
3
1 pounds
= 4
=
1
3 pounds
2
Ex4:
Express the ratio of
7
pounds
4
7
pounds
2
Express the ratio 3
7
7
7 2
14
1
÷
= ⋅ =
=
4
2
4 7
28
2
2
4
to
as a fraction reduced to lowest terms.
3
15
2
3 = 2 ÷ 4 = 2 ⋅ 15 = 30 = 5
4
3
15
3
4
12
2
15
Ex5:
=
(Since
5
is a ratio, we do not change it to a mixed number.)
2
1
7
: 2
as a fraction reduced to lowest terms.
4
16
1
13
1
4
4 = 4 = 13 ÷ 39 = 13 ⋅ 16 = 4
7
39
4
16
4 1 39 3 3
2
16
16
3
Revised 05/11
2
(Do not change
4
to a mixed number.)
3
Ex6:
Express the ratio 3.1 to 9.3 as a fraction reduced to lowest terms.
3.1 3.1 × 10 31 1
=
=
=
9.3 9.3 × 10 93 3
Ex7:
Express the ratio 0.2 : 0.12 as a fraction reduced to lowest terms.
0.2
0.2 × 100
20 5
=
=
=
0.12 0.12 × 100 12 3
(Do not change
5
to a mixed number.)
3
Since decimals and percents represent fractions it is possible to present them as ratios. Recall that the
denominator of a fractional representation of a decimal is determined by the position of the last digit in the
decimal and that the denominator of a fractional representation of a percent is 100.
Ex8:
Express the decimal, 0.025, as a ratio in simplest form using:
a) a fraction
25
1
=
1000 40
b) “ : “
a)
0.025 =
c)
25 to 1000 = 1 to 40
Ex9:
Express 65% as a ratio in simplest form using:
a) a fraction
b) “ : “
65 13
=
100 20
b)
a)
65% =
c)
65 to 100 = 13 to 20
Revised 05/11
b)
c) “to”
25 : 1000 = 1 : 40
c) “to”
65 : 100 = 13 : 20
3
EXERCISES
Express the following ratios as fractions reduced to lowest terms:
1.
12
21
2.
24 : 64
3.
6 to 33
4.
25
200
5.
8 : 36
6.
10 to 105
7.
15 inches
27 inches
8.
40 seconds : 2 minutes
9.
5 quarts to 200 gallons
10.
2 weeks
10 days
11.
1
foot : 2 yards
2
12.
1
13.
2
1 feet
3
5
4 feet
6
14.
7
1
2 pounds : 3 pounds
8
4
15.
9
7
mile to
mile
10
12
16.
4.5 to 8.1
17.
0.7 to 0.35
1
meters to 55 centimeters
2
Express the decimals or percents as ratios in simplest form using:
a) a fraction
b) “ : “
c) “to”
18.
0.45
19.
28%
20.
0.008
21.
150%
Revised 05/11
4
SOLUTIONS TO EXERCISES
1.
12 4
=
21 7
2.
24 3
=
64 8
3.
6 2
=
33 11
4.
25 1
=
200 8
5.
8 2
=
36 9
6.
10 2
=
105 21
7.
15 inches 15 5
=
=
27 inches 27 9
8.
40 seconds 40 seconds
40 1
=
=
=
2 minutes 120 seconds 120 3
9.
5 quarts
5
1
5 quarts
=
=
=
200 gallons 800 quarts 800 160
10.
2 weeks 14 days 14 7
=
=
=
10 days 10 days 10 5
11.
1
1
foot
foot
1 6 1 1
1
2
= 2
=
÷ = ⋅ =
2 yards
2 1 2 6 12
6 feet
12.
1
1 meters
150 centimeters 150 30
2
=
=
=
55 centimeters
55
11
55 centimeters
13.
2
5
1
2
1 feet
1
5 29
5 6
10
3
3
2
=
=
= ÷
=
⋅
=
5
5
29 3
29
6
3 1 29
4 feet
4
6
6
6
14.
7
7
23
1
pounds
2
23 13 23 4
23
8
8
8
=
=
=
÷
=
⋅
=
1
1 13
8
4 8 2 13
26
3 pounds
3
4
4
4
15.
9
9
6
mile
9
7
9
12
54
10
10
=
=
÷
=
⋅
=
7
7
10 12 10 5
7
35
mile
12
12
16.
4.5 4.5 × 10 45 5
=
=
=
8.1 8.1 × 10 81 9
17.
0.7
0.7 × 100 70 2
=
=
=
0.35 0.35 × 100 35 1
2
Revised 05/11
5
18.
a) 0.45 =
45
9
=
100 20
b) 45 : 100 = 9 : 20
c) 45 to 100 = 9 to 20
19.
a) 28% =
28
7
=
100 25
b) 28 : 100 = 7 : 25
c) 28 to 100 = 7 to 25
20.
a) 0.008 =
8
1
=
1000 125
b) 8 : 1000 = 1 : 125
c) 8 to 1000 = 1 to 125
21.
a) 150% =
150 3
=
100 2
b) 150 : 100 = 3 : 2
c) 150 to 100 = 3 to 2
Revised 05/11
6