Ratio and proportion

Ratio And Proportion

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, b² = 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? 

(a) Rs. 210

(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