### 17.1. Ratio and Proportion

#### 17.1.1. Ratio

In our work so far, we have often dealt with expressions of the form

There are several different ways of looking at such an expression. We can say that a/b is

• A fraction, with a numerator a and a denominator b

• A quotient, where a is divided by b

• The ratio of a to b

Thus a ratio can be thought of as a fraction or as the quotient of two quantities. Another way to write a ratio is to use a colon (:) instead of a fraction line. Thus the ratio a/b can also be written

a:b

#### 17.1.2. Dimensionless Ratios

For a ratio of two physical quantities, it is usual to express the numerator and denominator in the same units, so that they cancel and leave the ratio dimensionless.

## Example 1:

A corridor is 8 ft wide and 12 yd long. Find the ratio of length to width.

Solution: We first express the length and width in the same units, say, feet.

12 yd = 36 ft

So the ratio of length to width is

Note that this ratio carries no units.

Dimensionless ratios are handy because you do not have to worry about units. For this reason they are often used in technology.

## Example 2:

The fuel-air ratio is the ratio of the mass of fuel to air in a combustion chamber. The turns ratio is the ratio of turns of wire in the secondary winding of a transformer to the number of turns in ...

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