# Ratio and Proportion

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## RATIO AND PROPORTION

Introduction:

Ratio is a comparison of two quantities by division. Ratio represents the relation that one quantity bears to the other. It is represented as a:b. In any ratio a:b, a is called Antecedent and B is called Consequent. It is an abstract (without units) quantity.

A ratio remains unaltered if its numerator and denominator are multiplied or divided by the same number, e.g. 4:3 is the same as the (4 x 10) : (3 x 10) ie 40:30.

Different Types of Ratios:

Duplicate Ratio: a2 : b2 is called duplicate ratio of a : b.

Triplicate Ratio: a3 : b3 is called triplicate ratio of a : b.

Sub – Duplicate Ratio: √a :√b is called sub-duplicate ratio of a : b.

Sub – Triplicate Ratio: 3√a :3√b is called sub-triplicate ratio of a : b

Compound Ratio: ab : cd is the compound ratio of a : c and b : d. It is the ratio of the product of the antecedents to that of the consequents of two or more given ratios.

Inverse Ratio: 1/a : 1/b is the inverse ratio of a : b.

Componendo and Dividendo: If a/b = c/d, then (a + b)/(a – b) = (c + d)/(c – d)

Proportion:

Proportion is a statement that two ratios are similar. When two ratios are equal, they make a proportion, i.e. if a/b = c/d, then a, b, c and d are in proportion. This is represented as a:b :: c:d. When a, b, c and d are in proportion, then a and d are called the Extremes and b and c are called the Means, also Product of the Means = Product of the Extremes i.e. bc = ad.

Continued Proportion: If these quantities a, b and c are such that a:b :: b:c, then b2 = ac and a, b and c are in continued proportion. Also the quantity c is called the third proportion of a and b.

Fourth Proportion: If four quantities a, b, c and x are such that a:b :: c:x, then ax = bc and x is called the fourth proportion of a, b and c.

Mean or second Proportion: If three quantities a, b and x are such that a:x ::x:b, then x2 = ab and x is called the mean of a and b. Also, if a:b = c:d, then the following properties holds good.

i) b:a = d:c (Invertendo)

ii) a:c = b:d (Alternendo)

iii) (a + b) : b = (c + d) : d (Componendo)

iv) (a – b) : b = (c – d) : d (Divendendo)

v) (a + b)/(a – b) = (c + d)/(c – d) (Componendo – Divendendo)

Variation:

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 decrease in x makes y to decrease proportionally it can be expressed as x = ky, where k is called the constant of proportionality.

Eg: Cost is directly proportional to the number of articles bought.

Inverse 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. It can be expressed as x = k/y, where k is a constant of proportionality.

Eg: 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 kyz, where k is constant of proportionality.

Eg: Men doing a work in some number of days working certain hours a day.

Distribution of Amount: If an amount A is distributed in the ratio a:b, then

First part =a/(a+b) x A; Second Part =b/(a + b) x A

Formulae:

1) If a:b :: b:c, then a/b = b/c => c = b2/a

2) If a:b :: c:d, then a/b = c/d => d = bc/a

3) If a:x :: x:b, then x = √ab (x is mean proportional)

4) If x/y = 1, then (x + a)/(y + a) = 1 and (x – a)/(y – a) = 1

5) If x/y > 1, then (x + a)/(y + a) < x/y and (x –a)/(y – a) > x/y

6) If x/y < 1, then (x + a)/(y + a) > x/y and (x –a)/(y – a)< x/y

7) If a/b = c/d = e/f = …………… = k (constant), then (a + c + e + ……..)/(b + d + f + ……..) = k