When you divide one quantity by the other, the resulting comparison is called a ratio. This ratio can be expressed as a fraction or the use of the “:” symbol.

10 tapes to 20 tapes = 10/20 = 1/2 or 1:2
3 days to 9 days = 3/9 = 1/3 or 1:3
3 cars to 4 cars = 3/4 = 3:4

When comparing ratios, the quantities must be expressed in the same units of measure. So for example if you wanted to express a ratio of 4 feet to 36 inches, you must make one of the following conversions:

4 feet = 48 inches Therefore the ratio is 48:36 which equals 4:3

36 inches = 3 feet Therefore the ratio is 4:3

A proportion is a statement that two ratios are equivalent. For example the ratios 48:36 = the ratio 4:3. This is also true with fractions ( 48/36 = 4/3 ).

Extremes and Means

In a proportion the outside numbers (1 and 4) are called the extremes. The inside numbers on the inside are known as the means.

There are specific properties of proportions. If you multiply the two extremes together, they should equal the product of multiplying the two means together. For example:

4 : 7 = 24 : 42

4 x 42 = 7 x 24

168 = 168

This principle makes it easy to solve for unknowns with proportions. In the following example we are trying to figure out the unknown quantity “n”.

9 : n = 54 : 120

Therefore: 9 x 120 = n x 54

1080 = n x 54

1080 / 54 = n

20 = n

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Practice Exercise:

Solve for the following unknown numbers “n”:

1) 1:6 = n:54

2) 3:6 = 18:n

3) 7:n = 35:105

4) n:30 = 30:300

5) 3:24 = 9:n

6) 2:8 = n:64

7) 3:n = 21:21

8) 5:n = 20:60

9) 9:18 = n:126

10) 4:12 = n:84

Answers:

1) 1:6 = 9:54

2) 3:6 = 18:36

3) 7:21 = 35:105

4) 3:30 = 30:300

5) 3:24 = 9:72

6) 2:8 = 16:64

7) 3:3 = 21:21

8) 5:15 = 20:60

9) 9:18 = 63:126

10) 4:12 = 28:84