If two quantities are so related to each other that an increase (or decrease) in the first causes an increase (or decrease) in the second, then the two quantities are said to vary directly.

### Examples of Direct Proportion:

- The cost of an article varies directly with the number of articles. [More cost, More articles]
- The work done directly varies with the number of men at work.
- The distance covered by a car directly varies with its speed.
- More the money deposited in the bank, the more the interest earned.

### Direct Proportion Rule:

If two quantities ‘x' and ‘y' vary directly with each other then always

$\frac{x}{y}$ = k (Constant)### Examples on Direct Proportion

Given below are some examples on Direct Proportion.

**Example 1:**

The cost of 5 meters of a particular quality of cloth is $210. Tabulate the cost of 2, 4, 10 and 13 meters of cloth of the same type.

**Solution:**

Suppose the length of a cloth is x meters and its cost, in is $y.

x |
2 |
4 |
5 |
10 |
13 |

y |
y2 |
y3 |
210 |
y4 |
y5 |

As the length of the cloth increases, the cost of the cloth also increases in the same ratio. It is a case of direct proportion.

We make use of the relation of type $\frac{x1}{y1}$ = $\frac{x2}{y2}$

(i) Here x1 = 5 y1 = 210 and x2 = 2

Therefore, $\frac{x1}{y1}$ = $\frac{x2}{y2}$

$\frac{5}{210}$ = $\frac{2}{y2}$

5y2 = 2 x 210

y2 = $\frac{420}{5}$

y2 = 64

(ii) x3 = 4, then $\frac{5}{210}$ = $\frac{4}{y3}$

5 y3 = 4 x 210

y3 = $\frac{840}{5}$ = 168

(iii) x4 = 10, then $\frac{5}{210}$ = $\frac{10}{y4}$

5 y4 = 10 x 210

y4 = $\frac{2100}{5}$ = 420

(iv) x5 = 13, then $\frac{5}{210}$ = $\frac{13}{y5}$

5 y5 = 12 x 210

y5 = $\frac{(13\times210)}{5}$ = 546

**Example 2:**

An electric pole, 14 meters high, casts a shadow of 10 meters. Find the height of a tree that casts a shadow of 15 meters under similar conditions.

**Solution:**

Let the height of the tree be x meters. We form a table as shown below:

Height of the Object (in metres) |
14 |
x |

Length of the Shadow (in metres) |
10 |
15 |

Note: The more the height of the object, the more the length of the shadow.

Hence, this is the case of direct proportion. That is,

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

We have

$\frac{14}{10}$ = $\frac{x}{15}$

$\frac{(14\times15)}{10}$ = x

21 = x

Thus, height of the tree is 21 m

**Example 3:**

If 32 horses consume 112 kg of grass in a certain period, how much grass will be consumed by 11 horses during the same period?

**Solution:**

Fewer Horses, lesser the consumption of grass [Direct Variation]

32 horses consume = 112 kg

1 horse will consume = $\frac{112}{32}$ kg

11 horses will consume = $\frac{(112\times11)}{32}$

= 38. 5 kg.

**Example 4:**

If the weight of 65 coffee packets of the same size is 26 kg. What is the weight of 25 such packets?

**Solution:**

The fewer the packets, the lesser the weight [Direct Variation]

Weight of 65 packets = 26 kg

Weight of 1 packet = $\frac{26}{65}$ kg

Therefore, the weight of 25 packets = $\frac{(26\times25)}{65}$ kg

= 10 kg