UNIT Proportion F.Y.B.Com Prof.P.A.Navale Dept. of Commerce

Proportion Meaning: An equality of two ratios is called a Proportion. Four quantities are said to be in proportion if a: b = c: d (also written as a:b :: c:d). a, b, c, d are called the terms of the proportion a) First & fourth terms are called extremes b) Second & third terms are called means (or middle terms). c) Product of extremes = Product of means. (Cross Product Rule) Definitions: a) Charles McKeague: “A statement that two ratios are equal is called a proportion. If are two equal ratios, then the statement is called a proportion.” b) Ricardo Fierro: “Let a: b and c: d represent equivalent ratios. The equation a: b = c: d is called a proportion and is read as "a is to b as c is to d"

Proportion Properties of Proportion: 1) If a: b = c: d, then ad = bc (By cross multiplication). 2) If a : b = c : d, then b : a = d : c (Invertendo) 3) If a : b = c : d, then a : c = b : d (Alternendo) 4) If a : b = c : d, then a + b : b = c + d : d (Componendo) 5) If a : b = c : d, then a – b : b = c – d : d (Dividendo) 6) If a : b = c : d, then a + b : a – b = c + d : c – d (Componendo and Dividendo)

Proportion Types of Proportion Continued Proportion Direct Proportion Inverse Proportion Compound Proportion

Proportion 1) Continued Proportion: When three or more numbers are so related that the first to the second, the ratio of the second to the third, third to fourth, etc. are all equal, the numbers are said to be in continued proportion. Written as: a/b = b/c = c/d = d/e = ………………when a, b, c, d, e are in continued proportion. a) If a, b, c are in continued proportion, then middle term b is called the mean proportional between the first proportional a and third proportional c.   b) If a ratio is equal to the reciprocal of the other, then either of them is in inverse (reciprocal) proportion of the other. E.g. 3/4 is in inverse proportion of 4/3 and vice versa.

Proportion 2) Direct Proportion: If one quality is directly proportional to another it changes in the same way. As it increases, so does the others it decreases, the other decreases also. Example: If a person wants to buy one dozen pieces of soap, then he has to pay 240 Rs. If he wants to buy two dozen pieces of soap, he has to pay 480 Rs and so on. Solution: If x and y are in direct proportion, then division of x and y will be constant. In the above example, it sees that each ratio is the same. Hence, if we are dealing with quantities, which are related directly, (which are in direct proportion).

Proportion 3) Inverse Proportion: If one quantity is inversely proportional to another, it changes in the opposite way – as it increases, the other decreases. Example:  If 8 men take 4 days to build a wall, how long would it take 2 men (assuming they work at the same rate)? Solution: First, decide whether the problem is direct or inverse proportion. In this case, if less man is used, they will take longer, so it is inverse proportion. 8 men take 4 days 1 man takes 8 x 4 = 32 days 2 men take = 16 days Again we find the value of 1 by multiplying. Then divide to find the final answer.

Proportion 4) Compound Proportion: “The proportion involving two or more quantities is called Compound Proportion” Example: 195 men working 10 hour a day can finish a job in 20 days. How many men employed to finish the job in 15 days if they work 13 hours a day: Solution: Let x be the no. of men required Days Hours Men’s 20 10 195 15 13 x 20 x 10 x 195 = 15 x 13 x x

Proportion Direct Variation Inverse variation Joint variation Types of Variation: Direct Variation Inverse variation Joint variation

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