In this chapter we will learn method to solve direct variation problems utilizing the concept of ratio and proportion.

Let us first review the basic concept of direct variation.

## What is direct variation?

Direct variation is represented by following expression;

**y = k.** **x**

Where y and x are variables and k is constant.

The expression is called direct variable as;

⟹ If **value of x increases** the **value of y will also increase** proportionally.

⟹ If **value of x decrease**, the **value of y will also decrease**.

Hence, the value of x & y are in direct relationship with each other.

### Significance of k in direct variation

In direct variation the value of constant k will always remains the same.

Rewriting the above expression;

**y = k. x **

\mathtt{\frac{y}{x} \ =\ k}

It means that in direct variation,** the value of y & x will always maintain the same ratio.**

**Even if the value of y & x change, the ratio value k will always be the same**.

Let there are two set of values x1, y1 and x2, y2 for direct variation.

As the values maintain the same ratio, the expression can be written as;

\mathtt{\frac{y1}{x1} \ =\ \frac{y2}{x2} \ =\ k}

Remember this expression as it would help us to solve different problems related to direct variation.

I hope the concept is clear, let us now move to solve some problems.

## Direct Variation Problem using Proportion method

**Example 01**

Cost of 6 kg mangoes is 36$. Find the cost of 15kg mangoes.

**Solution**

It’s a questions of direct variation as the number of mangoes increases the cost will also increase.

Let;

x1 = 6 kg

y1 = 36$

x2 = 15 kg

y2 = ?

Using the direct variation proportion formula;

\mathtt{\frac{y1}{x1} \ =\ \frac{y2}{x2} \ }

Putting the values;

\mathtt{\frac{36}{6} \ =\ \frac{y2}{15}}\\\ \\ \mathtt{y2\ =\ \frac{36\ \times 15}{6}}\\\ \\ \mathtt{y2\ =\ 90\ \$}

Hence, **the cost of 15 kg mangoes is 90$**.

**Example 02**

A blue collar worker earns 240$ in 4 days. In how many days will he earn 540$.

**Solution**

This is a question of direct variation because as number of working days increases, the earning will also increase.

Let;

x1 = 4

y1 = 240

x2 = ?

y2 = 540

Putting the values in direct variation expression.

\mathtt{\ \frac{y1}{x1} \ =\ \frac{y2}{x2} \ }\\\ \\ \mathtt{\frac{240}{4} \ =\ \frac{540}{x2}}\\\ \\ \mathtt{x2\ =\ \frac{540\ \times 4}{240}}\\\ \\ \mathtt{y2\ =\ 9\ days}

Hence, **the worker will earn 540$ in 9 days**.

**Example 03**

A car travels 90 kilometers in 15 liter diesel. Find the distance travelled by car on 22 liters diesel.

**Solution**

Distance travelled and diesel usage are in direct variation. As the diesel consumption increase, the distance travelled by car will also increase.

Let;

x1 = 15

y1 = 90

x2 = 22

y2 = ?

Putting the value in direct variation expression;

\mathtt{\frac{y1}{x1} \ =\ \frac{y2}{x2} \ }\\\ \\ \mathtt{\frac{90}{15} \ =\ \frac{y2}{22}}\\\ \\ \mathtt{y2\ =\ \frac{90\ \times 22}{15}}\\\ \\ \mathtt{y2\ =\ 132}

Hence **in 22 liters oil, the car will travel 132 kilometers**.

**Example 04**

A writer can write 4500 words in 3 hours. How many words can he type in 11 hours.

**Solution**

Words typed and time taken are in direct variation. As the types word increases, the time of work will also increase.

Let;

x1 = 3

y1 = 4500

x2 = 11

y2 = ?

Putting the numbers in direct variation equation.

\mathtt{\frac{y1}{x1} \ =\ \frac{y2}{x2} \ }\\\ \\ \mathtt{\frac{4500}{3} \ =\ \frac{y2}{11}}\\\ \\ \mathtt{y2\ =\ \frac{4500\ \times 11}{3}}\\\ \\ \mathtt{y2\ =\ 16500}

Hence** in 11 hours, the writer can type 16500 words**.

**Example 05**

A fan consumes electricity of 900 watt hour in 12 hours. Find the electricity consumed by fan in 36 hours.

**Solution**

Electricity consumed and time are in direct variation. As the time of usage increase, the electricity consumed will also increase.

Let;

x1 = 12

y1 = 900

x2 = 36

y2 = ?

Putting the values in direct variation ratios.

\mathtt{\frac{y1}{x1} \ =\ \frac{y2}{x2} \ }\\\ \\ \mathtt{\frac{900}{12} \ =\ \frac{y2}{36}}\\\ \\ \mathtt{y2\ =\ \frac{900\ \times 36}{12}}\\\ \\ \mathtt{y2=\ 2700}

Hence **in 36 hours, 2700 watt hour of electricity is consumed**.