## Proportion & Variation

This lesson is to be read after you read the Ratio & Partnership lesson. This chapter begins with the various kinds of proportion, methods to solve these questions and concludes with different types of Variation. It is to be noted that Proportion can also be applied in concepts such as Geometry (relationship between two figures, similar triangles, etc.) and Variation can also be applied in concepts such as Time & Speed, Time & Work etc. Therefore, it is imperative for you to have a thorough understanding of this lesson.

### 1. Proportion

Proportion involves equality of ratios that have $2$ terms. If $\bm{a, b, c}$ and $\bm{d}$ are in proportion, then the following are different ways to write the same.

$a : b = c : d$             OR            $a : b :: c : d$             OR            $\dfrac{a}{b} = \dfrac{c}{d}$

If two ratios are equal, then they are in proportion. For example, as $1 : 2 = 3 : 6$, these ratios are in proportion.

$\bm{a}$ and $\bm{d}$ are called extremes, and $\bm{b}$ and $\bm{c}$ are called means. Upon simplifying, we note that the product of extremes equals the product of means.

$\dfrac{a}{b} = \dfrac{c}{d} \implies ad = bc$

### Example $1$

The first, third and fourth terms of a proportion are $15, 3$ and $18$ respectively. What is the second term?

### Solution

$\dfrac{15}{x} = \dfrac{3}{18} \implies x = 90$

Answer: $90$

### Example $2$

A picture, $15$ cm long and $21$ cm wide, is too big for a frame. However, to retain the quality of the photograph, its length and width should be kept proportional. What is the width of the reduced photograph if its length is $12$ cm?

### Solution

Let the new width be $w$.

$\dfrac{15}{21} = \dfrac{12}{w} \implies w = 16.8$

Answer: $16.8$ cm