Doubt Classes

Ratio

Basic Concept of Ratios

We will discuss here about the basic concept of ratios.

Definition: The ratio of two like quantities a and b is the fraction ab$\frac{a}{b}$, which indicates how many times b is the quantity a. In other words, their ratio indicates their relative sizes.

If x and y are two quantities of the same kind and with the same units such that y ≠ 0; then the quotient xy$\frac{x}{y}$ is called the ratio between x and y.

Let the weights of two persons be 40 kg and 80 kg. Clearly, the weight of the second person is double the weight of the first person because 80 kg = 2 × 40 kg.

Therefore, WeightofthefirstpersonWeightofthesecondperson$\frac{Weightofthefirstperson}{Weightofthesecondperson}$ = 40kg80kg$\frac{40kg}{80kg}$ = 12$\frac{1}{2}$.

We say, the ratio of the weight of the first person to the weight of the second person is 12$\frac{1}{2}$ or 1 : 2.

The ratio of two like quantities a and b is the quotient a ÷ b, and it is written as a : b (read a is to b).

In the ratio a : b, a and b are called terms of the ratio, a is called the antecedent or first term, and b is called the consequent or second term. Then, ratio of two quantities = antecedent : consequent.

Example: The ratio of heights of two persons A and B whose heights are 6 ft and 5 ft is 6ft5ft$\frac{6ft}{5ft}$, i.e., 65$\frac{6}{5}$ or 6 : 5. Here, 6 is the antecedent and 5 is the consequent.

Important Properties of Ratios

Some of the important properties of ratios are discussed here.

1. Ratio mn$\frac{m}{n}$ has no unit and can be written as m : n (read as m is to n).

2. The quantities m and n are called terms of the ratio. The first quantity m is called the first term or the antecedent and the second quantity n is called the second term or the consequent of the ratio m : n.

The second term of a ratio cannot be zero.

i.e., (i) In the ratio m : n, the second term n cannot be zero (n ≠ 0).

(ii) In the ratio n : m, the second term cannot be zero (m ≠ 0).

3. The ratio of two unlike quantities is not defined. For example, the ratio between 5 kg and 15 meters cannot be found.

4. Ratio is a pure number and does not have any unit.

5. If both the terms of a ratio are multiplied by the same non-zero number, the ratio remains unchanged.

If two terms of a ratio be multiplied by any number except zero, then there is no change in the value of the ratio because; m : n = mn$\frac{m}{n}$ = kmkn$\frac{km}{kn}$= km : kn

If both the terms of a ratio are divided by the same non-zero number, the ratio remains unchanged.

m : n = mn$\frac{m}{n}$ = mknk$\frac{\frac{m}{k}}{\frac{n}{k}}$ = mk$\frac{m}{k}$ : nk$\frac{n}{k}$, (k ≠ 0)

In other words, the ratio of m and n is the same as the ratio of the quantities km and kn, or mk$\frac{m}{k}$ and nk$\frac{n}{k}$, where k ≠ 0.

6. If two quantities are in the ratio m : n then the quantities will be of the form m ∙ k and n ∙ k, where k is nay number, k ≠ 0. Thus, if the ratio of two quantities x and y is 3 : 4, x and y can be 6 and 8 (k = 2), 9 and 12 (k = 3), and so on.

7. If m is k % of n then the ratio m : n = k : 100. Also, if m : n = p : q then m = pq$\frac{p}{q}$× 100% of n = pq$\frac{p}{q}$ × n.

8. A ratio must always be expressed in its lowest terms.

The ratio is in its lowest terms, if the H.C.F. of its both the terms is 1 (unity).

For example;

(i) The ratio 3 : 7 is in its lowest terms as the H.C.F. of its terms 3 and 7 is 1.

(ii) The ratio 4 : 20 is not in its lowest terms as the H.C.F. of its terms 4 and 20 is 4 and not 1.

9. Ratios m : n and n : m cannot be equal unless m = n

i.e. m : n ≠ n : m, unless m = n

In other words, the order of the terms in a ratio is important.

Learn More