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Ratio and Proportion:


Ratio gives us a relation between two quantities having similar unit. The ratio of A to B is written as A:B or A/B, where A is called the antecedent and B the consequent.


Proportion is an expression in which two ratios are equal. For example, A/B = C/D ==> A:B::C:D

Here, AD = BC

Properties of Ratios & Proportion:

==> 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 used in the ratio of three quantities.

If a:b = c:d, i.e. a/b = c/d, then

b/a = d/c, this is the property of Invertendo.

If a:b = c:d, i.e. a/b = c/d, then

a/c = b/d, this is the property of Alternendo.

==> If a/b = c/d, then (a+b)/c = (c+d)/d – this property is called Componendo

==> Also, (a-b)/c = (c-d)/d – this property is called Dividendo

It also follows that:

==> (a+b) / (a-b) = (c+d) / (c-d)

==> This property is called Componendo and Dividendo

==> If a/b = c/d = e/f …., then (a+c+e+…) / (b+d+f+…) = each individual ratio i.e. a/b or c/d

==> If A>B then (A+C) / (B+C) < A/B, where A, B and C are natural numbers. In a proportion it should be remembered that – Product of means = product of extremes, i.e. b*c = a*d.


Types of Proportion:

Continued Proportions:

We can say that a, b and c are in continued proportion, if a/b = b/c, b2 =ac à b = √ac

Here we can say that a is called first proportion, c is called third proportion and b is called mean proportion.

Also, if two nos. are given, and you are required to find mean proportion, then it should be written as – a:x :: x:b,

And is third proportion is to be computed, then it should be written as – a:b : b:x.

Direct proportion:

If X is directly proportional to Y, that means any increase or decrease in any of two quantities will have proportionate effect on the other quantity. If X increases then Y will also increase and vice-versa.

Inverse Proportion:

If X is inversely proportional to Y, that means any increase or decrease in any of two quantities will have inverse proportionate effect on the other quantity. This means if X increases, then Y decreases and if X decreases then Y increases and vice-versa for Y.

Applications of Ratio and Proportion


To find out profit-shaving ratio on the basis of capital contribution.

Example 1: Ram, Rohan and Ravi are partners in a firm. Ram contributed ₹10,000 for 6 months, where as Rohan and Ravi, both contributed ₹7500 for the full year. If at the end of the year, profit is ₹ 2500, then find Ram’s share of profit?


Proportionate capital of Ram, Rohan and Ravi

Proportionate = 10000*6 : 7500*12 : 7500*12 = 60000 : 90000 : 90000 or ratio = 2:3:3

Ram’s share = 2500*(2/8) = ₹ 625


“Mixtures and alligations” is about mixing different qualities of goods in order to get desired levels/percentage/concentration of different objects.

Quick method:

This rule helps us in solving questions where two varieties (of different prices) are mixed to get a new variety with a new Average price.

Quantity of cheaper variety / Quantity of dearer variety = (Price of Dearer variety – Average price) / (Average price – Price of cheaper variety) => c/d = (d-m) / (m-c)

Then, (Cheaper quantity) : (Dearer quantity) = (d-m) : (m-c) à c/d = (d-m) / (m-c)

Points to Remember:

==> If in a partnership the investments made by first, second and third partners are X1, X2, X3 respectively, the time period be t1, t2, t3 then the ratio of profits is given by X1t1 : X2t2 : X3t3.

==> If X1, X2, X3 is the ratio of investments and P1 : P2 : P3 be the ratio of Profit then time periods are given by = P1/ X1 : P2/ X2 : P3/ X3

==> If P1 : P2 : P3 is the ratio of profit on investments and t1 : t2 : t3  be the ratio of time periods, then the ratio of investments will be = P1/ t1 : P2/ t2 : P3/ t3

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