# Chapter 12 - Ratio and Proportion

## Introduction to Ratio

In our daily life we come to a conclusion through comparison on various occasions. Comparison by difference is a common task. But in certain situations, comparison by division makes better sense than comparison by taking the difference. The comparison by division is known as the Ratio.

### Comparison by Ratio

Ratio is a way of comparing quantities using division. We use this method to see how many times one quantity is to another quantity.

Example: If the price of a car is Rs. 5,00,000 and the price of a motor cycle is Rs. 50,000, then we can say

$${Price\:of\:the\:car}/{Price\:of\:the\:motor\:cycle} = {500000}/{50000} = {10}/{1}$$

Thus, the price of the car is 10 times more than that of a motor cycle. In the form of ratio, it can be written as Price of car : Price of motor cycle = 10 : 1

Example: Mother wants to divide Rs. 36 between her daughters Alisa and Purbasa in the ratio of their ages. Alisa is 15 and Purbasa is 12 years.

Ratio of age: ${15}/{12}$ = ${5}/{4}$

Sum of these parts = 5 + 4 = 9

This means if there is Rs. 9, Alisa will get Rs. 5 and Purbasa will get Rs. 4.

Therefore, Alisa's share = ${5}/{9}$ and Purbasa's share = ${4}/{9}$

Out of given Rs.36:

Alisa's Share = 36 × ${5}/{9}$ = 5 × 4 = 20

Purbasa's Share = 36 × ${4}/{9}$ = 4 × 4 = 16

Thus, their shares are Rs. 20 and Rs. 16.

### Facts Related to Ratio

• We denote ratio using the symbol ':'
• Two quantities can be compared only if they have same unit. Ratio of height and weight can be found. But we cannot compare height with weight.
• Different situations can give rise to same ratio (Ratio of 50 kg potato and 6 kg onion is 50:6 = 25:3. Ratio of 75 m road to 9 m road is 75:9 = 25:3)
• We can get equivalent ratios by multiplying or dividing the numerator and denominator by the same number. (5:7 = ${5}/{7}$ = ${5 × 3}/{ 7 × 3}$ = ${15}/{21}$ ≡ 15:21)

## Understanding Proportion

### Equivalent Fractions

Fractions that look different but represent the same quantity are called equivalent fractions. For example,

$${1}/{2} = {1}/{2} × {2}/{2} = {2}/{4}$$

Thus, ${1}/{2}$ and ${2}/{4}$ are equivalent fractions.

Similarly,

$${9}/{18} × {3}/{3} = {27}/{54}$$

Thus, ${9}/{18}$ and ${27}/{54}$ are equivalent fractions.

### Proportion

Ratios that look different but represent the same quantity are said to be in proportion. Like ratio, proportion is also a useful idea to compare things. If two ratios are equal, then they are in proportion.

For example, 2:4, 4:8, and 8:16 are in proportion because these ratios represent the same quantity. That is, 2 out of 4 parts is same as 4 out of 8 parts or 8 out of 16 parts.

Example

Suppose John is on a trip to his hometown in a car. He travels 70 km in the first one hour.
If he maintains the same speed for the next hour too, he can travel 70 kilometres more.
So, John travels 140 km. in 2 hours.
The ratio of time taken is 1:2 and ratio of distance travelled is 70:140 = 1:2.
These two ratios are in proportion.


## Unitary Method

In this method, comparison is done based on the value of each thing. First, we find the value of one unit. For example,

Value of 4 pencils = Rs. 40

Value of 1 pencil = ${40}/{4}$ = Rs.10

Now, we can calculate the price of any number of units. If it is required to calculate the value of 100 pencils, then:

Value of 1 pencil = Rs.10

Value of 100 pencils = 10 × 10 = Rs.100

Example: John wants to purchase four soaps. In a shop, he finds each soap costs Rs.30. In another shop, he finds a pack of four soaps at Rs.100.

To decide the best offer, he needs to resort to the unitary method.

In the second case, he has to find the price of each soap using unitary method.

• Price of 4 soaps = 100
• Price of each soap = ${100}/{4}$ = 25

In the first shop, he is getting soap at Rs.30 each, whereas he gets the same soap at Rs.25 in the second shop. So, John is definitely getting a better offer in the second shop.