HFCC Math Lab
Arithmetic-7
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 ration 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
7 inches

=
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 10/09
1
Ex2: Express the ratio of 5cents to $1 as a fraction reduced to lowest terms.
5 cents
5
1
5cents


=
1dollar 100 cents 100 20
Ex3: Express the ratio 50 minutes : 2 hours as a fraction reduced to lowest terms.
50 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.
For example, in the ratio of 1
3
1
pounds to 3 pounds, both terms are mixed numbers. This
2
4
ratio written as a fraction reduced to lowest terms is:
3
1 pounds
 4

1
3 pounds
2
Ex4: Express the ratio of
7
pounds
4
7
pounds
2
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: Express the ratio 3

(Since
5
is a ratio, we do not change it to a mixed number.)
2
1
7
as a fraction reduced to lowest terms.
: 2
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 10/09
2
(Do not change
4
to a mixed number.)
3
PROPORTION
A proportion is a statement that one ratio is equal to another ratio. The common notation for a
a c
proportion is:

b d
This notation is read “a is to b as c is to d”.
The first term, a, of the proportion times the fourth term, d, is equal to the second term, b, times
the third term, c. These “cross products” of the terms in a proportion are always equal. The terms
a and d, are referred to as the extremes and the terms b and c are referred to as the means of the
proportion. Therefore, the product of the means is equal to the product of the extremes. In
a c
symbols, we write: If
=
then a  d  b  c
b d
In proportion problems, three of the terms are known and the fourth unknown term must be
x 6
solved for. For example, to solve for x in the proportion: = , set the product of the means
3 9
equal to the product of the extremes. Then solve the resulting equation.
The product of the extremes is: 9  x  9 x
The product of the means is: 3  6  18
The product of the extremes  The product of the means
9x  18
Therefore:
Solve the equation for x :
Ex6: Solve the proportion
9 x 18

9
9
x2
4 x
for x .
=
7 21
The product of the means: 7  x  7 x
The product of the extremes: 4  21  84
The product of the means  The product of the extremes
Therefore:
Solve the equation for x:
Revised 10/09
7 x  84
7 x 84

7
7
x  12
3
Ex7: Solve the proportion
16 feet
24 feet
for x .

x seconds 15 seconds
The product of the means: 24  x  24 x
The product of the extremes: 16 15  240
The product of the means  The product of the extremes
Therefore:
Solve the equation for x:
24x  240
24 x 240

24
24
x  10
So, x  10 seconds.
Note: Often the terms of a ratio are fractions or mixed numbers.
5
x
Ex8: Solve the proportion  6 for x .
3 5
The product of the means:
5 15 5
3   .
6 6
2
The product of the extremes: 5  x  5x
The product of the extremes  The product of the means
Therefore:
Solve the equation for x:
Revised 10/09
5x 
5
2
1
1 5
 5x  
5
5 2
1
x
2
4
Ex9: Solve the proportion
5 12
for x

1 x
2
2
6
5 12
1


2  12 =
1
21
2
The product of the means:
The product of the extremes:
5  x  5x
The product of the extremes  The product of the means
5x  30
Therefore:
5 x 30

5
5
x6
Solve the equation for x:
1
16 pounds
3pounds
5
Ex10: Solve the proportion
for x .

$x
$5.50
The product of the means:
The product of the extremes:
3  x  3x
 1
16    5.50   16.2    5.50   89.1
 5
The product of the means  The product of the extremes
3 x  89.1
Therefore:
Solve the equation for x :
3x 89.1

3
3
x  29.7
So, x  $29.70
Revised 10/09
5
EXERCISES
A.
B.
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
1
meters to 55 centimeters
2
Solve the following proportions for x .
1.
5 15

9 x
2.
x 4

12 1
3.
12 3

x 8
4.
96 x

16 2
6.
x
75

100 125
8.
$x
$1000

14days 70 days
5.
7.
Revised 10/09
4 16

1 x
2
150 miles 300 miles

x hours
5 hours
6
9.
1
cents
x cents
4

18 minutes 8 minutes
2
35 feet 100 feet

7
x sec
sec
5
10.
SOLUTIONS TO EXERCISES A & B
A: 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
9.
5 quarts
5 quarts
5
1



200 gallons 800 quarts 800 160
11.
1
1
foot
foot
1 6 1 
1
2
2

    
2 yards 6 feet
2 1 2  12
12.
1
1 meters
150 centimeters 150 30
2



55 centimeters
55
11
55 centimeters
13.
2
5
1
2
1
1 feet
5 29 5 
10
3
3
2


 



5
5 29 3
6
29
3 1 
4 feet
4
6
6
6
14.
7
7
23
1
pounds
2
23 13 23 
23
8
8
8







1
1 13
8
4 8 2 
26
3 pounds
3
4
4
4
15.
9
9
6
mile
9
7
9

54
10
10






7
7
10 12 10 5

35
mile
12
12
2
Revised 10/09
40 seconds 40 seconds
40 1



2 minutes 120 seconds 120 3
8.
7
10.
2 weeks 14 days 14 7



10 days 10 days 10 5
B:
1. 5  x  9 15
5 x  135
x  27
3. 3  x  12  8
2. 1  x  12  4
x  48
4. 16  x  96  2
3 x  96
x  32
1
16
2
4x  8
x2
5. 4  x 
7. 300  x  150  5
300 x  750
16 x  192
x  12
6. 125  x  100  75
125 x  7500
x  60
8. 70  x  14 1000
70 x  14000
1
2
1
x  2 hours
2
x2
7
9. 35  x  100
5
35 x  140
x  200
x  $200
1
10. 18  x  2  8
4
18 x  18
x4
x 1
x  4 sec
x  1 cent
Revised 10/09
8