Ratio & Proportions:

Ratios are the way of expressing the relative sizes of two different quantities. Ratios are expressed as, x:y or \(\frac{x}{y}\). Like fractions, even ratios can be expressed in the form of their reduced equivalent forms. Meaning, 2:3 is same as 4:6.

A proportion is an equation relating two different ratios as, a:b = c:d. If a proportion is in the form of a:b = b:c. Then it is called to be in continuous proportion. We cross multiply the fractions to solve proportions.

Rules of Ratio:

Ratios have some basic rules to solve which are,

If \(\frac{a}{b} = \frac{c}{d} = \frac{e}{f}\), then \(\frac{{a+c+e}}{{b+d+f}}\). This is termed as property of equal ratios. Keep in mind, this holds true for any number of equal ratios.

If \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{{a+b}}{{a-b}} = \frac{{c+d}}{{c-d}}\). This is termed as componendo and dividendo.

If \(\frac{a}{b} = \frac{c}{d} = \frac{e}{f}\), then a:c:e = b:d:f.

Useful Tricks

When a question is asked to find A:B:C:D, given that A:B = m:n, B:C = o:p and C:D = q:r. We have a simple trick to solve this kind of question which can save time during exams. This is explained in the following picture.




In case of only three variables, it is solved as,


If a question is asked such that, a container contains a mixture of oil and water in ratio M:N, with the total amount of liquid, is x litres. What amount of water to be so that the ration becomes A:B?

In this case use the following formula,
\(\frac{x(MB-NA)}{A(M+N)}\)

For example,
A tank consists of a mixture of milk and water in ratio 2:3. The total amount of mixture is 10 litres. How much water should be added, so that the ratio becomes 1:3?

Using the above formula, the amount of water to be added = \(\frac{10(2*3 - 3*1)}{1(2+3)} = \frac{6}{1}\) litres