## Concept of Ratio

There should be two quantities required to compare them with each other. For example, a : b, they are called terms(a and b). The first term(a) is called antecedent and the second term(b) is called the consequent. Generally, a ratio is expressed in the simplest form.

Two Quantities of Ratio.

## Remarks:

The order of the terms in a ratio is important, we can not write a : b to b : a. They are different.

Ratio exists only between quantities of the same kind as well as in the same units.

We can convert them into fractions whenever needed.

Fraction and Ratio.

### Types of Ratios

Some of the important ratios are:

Inverse Ratio- The new ratio obtained by reversing the terms of a ratio is called the inverse ratio. For example, the inverse ratio of 2 : 3 will be 3 : 2.

Compound Ratio- The new ratio formed by the product of the previous terms of two or more ratios and the product of the last terms is called a mixed ratio. For example, the mixed ratio of two ratios (a : b) and (c : d) will be (ad : bc). Similarly the mixed ratio of 2 : 3 , 4 : 5 and 6 : 7 will be 2 × 4 × 6 : 3 × 5 × 7 i.e. 48 : 105 or 16 : 35.

Duplicate Ratio- If a new ratio is made by mixing a ratio with the same, then it is called a square ratio. For example, the square ratio of 2 : 3 is 2² : 3³. That is 2 × 2 : 3 × 3 or 4 : 9.

## How To Solve Ratio?

When two numbers are divided by each other, the ratio of those numbers is obtained. As a and b are two numbers then a/b will be their ratio. If we want to find the ratio of two numbers 3 and 7, then 3/7 will be their ratio. It will be written as 3:7.

## Simple Ratio Problems(Illustrations)

What is the ratio of 2 to 3?

Solution: 2 + 3 = 5. We have 5 parts in the ratio of 2 : 3.

25 cm and 1 m. What is the ratio?

Solution: 1 m = 100 cm.

25 cm and 100 cm ratio of 25/100 = 1 : 4.

The sum of the two numbers is 60 and the difference is 6. What will be the ratio?

Solution: Required ratio of numbers:

(60 + 6) / (60 - 6) = 66/54 = 11/9 or 11 : 9.

## Solved Questions

What will be the inverse ratio of 5 : 6?

Solution: The inverse ratio of 5 : 6 is 6 : 5.

What will be the duplicate ratio of 3 : 4?

Solution: The duplicate ratio of 3 : 4 = 3² : 4² or 9 : 16

If a quantity is divided in the ratio of 5 : 7, the larger part is 100. Find the quantity.

Solution: Let the quantity be x.

Then the two parts will be 5x / 7 + 5 and 7x / 7 + 5.

Hence, if the larger part is 100, we get 7x / 5 + 7 = 100.

7x / 12 = 100

7x = 100 × 12

7x = 1200

x = 1200 / 7

x = 171.42

Therefore, the quantity is 171.42.

## Learning by Doing

Problems On Ratio:

1. Shilpa earns ₹150 in 4 hours and isha ₹300 in 7 hours. What will be the ratio of their earnings?

Ans: Shilpa earning: 150 X 4 = 600.

Isha earning: 300 X 7 = 2100.

Ratio= 600 : 2100

= 2 : 7

2. The ratio between the speeds of two trains is 4:5. If the second train runs 400 kms. in 6 hours. What will be the speed of the first train?

Ans: The speed of train in 1 hours : 400/6 = 66(approx)

Ratio = 4x/5x=y/66

= 4x X 66 = 5x X y

= 264 x = 5xy

= 264x/5x=y

= 52.8=y

The speed of the first train is 52.8 km/hr.

## Summary

The Ratio refers to the comparison of at least two quantities of each other. If a and b are two quantities of the same kind (in the same units), then the fraction a/b is called the ratio of a to b. It is written as a : b. The quantities a and b are called the terms of the ratio, a is called the first term or antecedent and b is called the second term or consequent. The ratio compounded of the two ratios a : b and c : d is ad : bc. A ratio compounded of itself is called its duplicate ratio. a² : b² is the duplicate ratio of a : b. For any ratio a : b, the inverse ratio is b : a.

## FAQs on Ratio Problems

1. Isha weighs 55 kg. If she reduces her weight in the ratio of 5: 4, find his reduced weight.

Let the previous weight be 5x.

5x = 55

x = 11

Therefore, 4 × 11 = 44 kg. The reduced weight is 44 kg.

2. Shubham leaves $ 1000000 behind. According to his wish, the money is to be divided between his wife and sister in the ratio of 3 : 2. Find the sum received by his wife.

We know if a quantity x is divided in the ratio a: b then the two parts are ax / a + b and bx / a + b.

= 3/5 × 1000000

= 3 × 200000

= 600000

Therefore, the sum received by his wife is $600000.

3. Ishita weighs 56.7 kg. If she reduced her weight in the ratio of 7 : 6, find her new weight.

Original weight of Ishita = 56.7 kg.

She reduces her weight by a ratio of 7 : 6.

Her new weight = (6 × 56.7)/7 = 6 × 8.1 = 48.6.