__What is Ratio?__

Ratio is a mathematical term used to compare two similar quantities expressed in the same units. The ratio of two terms ‘x’ and ‘y’ is denoted by x : y. In ratio x : y , we can say that x as the first term or antecedent and y, the second term or consequent.

In general, the ratio of a number x to a number y is defined as the quotient of the numbers x and y i.e. x/y.

**Example:** The ratio of 25 km to 100 km is 25:100 or 25/100, which is 1:4 or 1/4, where 1 is called the antecedent and 4 the consequent.

**Note that fractions and ratios are same; the only difference is that ratio is a unit less quantity while fraction is not. **

**Compound Ratio**

Ratios are compounded by multiplying together the fractions, which denote them; or by multiplying together the antecedents for a new antecedent, and the consequents for a new consequent. The compound of a : b and c : d is i.e. ac : bd.

__Properties of Ratio:__

**☑** a : b = ma : mb, where m is a constant

**☑** a : b : c = A : B : C is equivalent to a / A = b /B = c /C, this is an important property and has to be used in ratio of three things.

**☑**

i.e. the inverse ratios of two equal ratios are equal. This property is called **Invertendo. **

**☑**

i.e. the ratio of antecedents and consequents of two equal ratios are equal. This property is called **Alternendo.**

**☑**

This property is called **Componendo.**

**☑**

This property is called **Dividendo. **

**☑**

This property is called** Componendo - Dividendo. **

**☑** **☑**The incomes of two persons are in the ratio of a: b and their expenditures are in the ratio of c: d. If the saving of each person be Rs. S, then their incomes are given by-
**Example: **Annual income of A and B are in the ratio of 5: 4 and their annual expenses bear a ratio of 4: 3. If each of them saves Rs. 500 at the end of the year, then find the annual income.

__Dividing a Quantity Into a Ratio__

Suppose any given quantity ‘a’ is to be divided in the ratio of m : n.

Then,

__Proportion__

When two ratios are equal, the four quantities composing them are said to be in proportion.

If a/b=c/d, then a, b, c, d are in proportions.

This is expressed by saying that ‘a’ is to ‘b’ is to ‘c’ is to ‘d’ and the proportion is written as

**a : b :: c : d or a : b = c : d **

(product of means = product of extremes)

If there is given three quantities like a, b, c of same kind then we can say it proportion of continued.

a : b = b : c the middle number b is called mean proportion. a and c are called extreme numbers.

So, b^{2} = ac. (middle number)^{2} = ( First number x Last number ).

**Application:** These properties have to be used with quick mental calculations; one has to see a ratio and quickly get to results with mental calculations.

Example:
should quickly tell us that
**Q. A certain amount was to be distributed among A, B and C in the ratio 2 : 3 : 4, but was erroneously distributed in the ratio 7 : 2 : 5. As a result of this, B received Rs. 40 less. What is the actual amount? **

(b) Rs. 270

(c) Rs. 230

(d) Rs. 280

(e) None of these

**Q. Mixture of milk and water has been kept in two separate containers. Ratio of milk to water in one of the containers is 5 : 1 and that in the other container 7 : 2. In what ratio the mixtures of these two containers should be added together so that the quantity of milk in the new mixture may become 80%? **

(a) 2 : 3

(b) 3 : 2

(c) 4 : 5

(d) 1 : 3

(e) None of these