Ratio, Proportion & Variation


The ratio of two qualities a and b in the same units, is the fraction a/b and we write it as a:b. In the ratio, a:b, we call a as the first term of antecedent and b, the second term consequent. For example, the ratio 5:9 represents 5/9 with antecedent 5 and consequent 9.

The multiplication or division of each term of 9 ratio by the same non-zero number does not affect the ratio.

Compounded Ratio

The compounded ratio of the ratios a:b, c:d, e:f is ace:bdf

Duplicate Ratio

If a:b, then duplicate ratio is a2: b2

Triplicate Ratio

a3 : b3 is called triplicate ratio of a : b

Sub-Duplicate Ratio

If a:b, then sub-duplicate ratio is √a:√b

Componendo and Divedendo

If a/b=c/d, then (a+b)/(a-b)=(c+d)/(c-d)


The equality of two ratios is called proportion. If a:b=c:d, we write a:b::c:d and we say that a, b, c and d are in proportion. Here, a and b are called extremes, while b and c are called means.

Product of means = Product of extremes

Thus, a:b::c:d implies that (b*c)=(a*d)

Continued Proportion

If three quantities a, b and c are such that a:b :: b:c, then b2 = ac and a, b and c are in continued proportions.

Fourth Proportional

If a:b::c:d, then d is called the fourth proportional to a, b and c

Third Proportional

If a:b::b:c, then c is called third proportional to a and b

Mean Proportional (Second Proportional)

Mean proportional between a and b is √(a*b)


If two quantities x and y are related in such a way that as the quantity x changes it also brings a change in the second quantity y, then the two quantities are in variation.

Direct Variation

The quantity x is in direct variation to y if an increase in x makes y to increase proportionally. Also a decrease in x makes y to decrease proportionally. You can say that x is directly proportional to y, if x = ky for some constant k.

For example, cost is directly proportional to the number of articles bought.

Indirect Variation

The quantity x is in inverse variation to y if an increase in x makes y to decrease proportionally. Also, a decrease in x makes y to increase proportionally. Ypu can say that x is inversely proportional to y, if xy = k (x=k/y) for some constant k.

For example, the time taken by a vehicle in covering a certain distance is inversely proportional to the speed of the vehicle.

Joint Variation

If there are more than two quantities x, y and z and x varies with both y and z, then x is in joint variation to y and z. It can be expressed as x = kyz. Where, k is the constant of proportionality. For example, men doing a work in some number of days working certain hours a day.