Ratio and Proportion
Hence similar is with the aggregate values. •If a number is to be proportionately changed in a given ratio then the antecedent refers to the given number. Hence find the proportionality constant (number / antecedent) and multiply this constant with the consequent to get the answer. If 25 is to be changed in ratio 5:7 then 25 is represented by 5, so constant is 25/5 = 5, hence answer is 5×7 = 35 •In a given ratio a : b •If a > b then ratio is of greater inequality •If a < b then ratio is of lesser / less inequality.
The inverse ratio is b : a •Duplicate ratio is a2 : b2 •Triplicate ratio is a3 : b3 •Subduplicate ratio is va : vb •Subtriplicate ratio is 3va : 3vb •Commensurable if a and b are integers •Incommensurable if a and b are not integers •The compounded ratio of (a1: b1), (a2 : b2) and (a3 : b3) is (a1. a2. a3) : (b1. b2. b3) [ that is product of the antecedents by product of the consequents ] •A ratio, multiplied with its inverse produces 1. •A ratio multiplied with itself produces duplicate ratio Continued ratio or proportion is the proportional relationship between 3 or more items.
Proportion brings about a continuous relationship between 3 or more items = relationship between 2 or more ratios •Let A : B = 2 : 3 and B : C = 5 : 7 . then A : B : C ? 2 : 3 : 7 or ? 2 : 5 : 7 •We need to bring parity at b and LCM of 3 and 5 is 15. So •A 2 > 2 x 5 = 10 •B 3 > 3 x 5 = 15 5 > 5 x 3 = 15 •C 7 > 7 x 3 = 21 •so A : B : C = 10 : 15 : 21 The mean proportion of A and B is X such that •A, X and B are in proportion •A, X and B are in geometric progression •X is the geometric mean •(A/X) = (X/B) •X2 = AB •In A : B : C : D •A and D are called extremes / extreme terms •B and C are called means / middle terms •A, B, C and D are in a geometric progression •(A/B) = (B/C) = (C/D) •A. D = B. C that is (product of extremes) = (product of means) •B2 = AC •C2 = BD •The third proportion of A and B is X such that •A, B and X are in proportion •A, B and X are in geometric progression B is the mean proportion of A and X •(A/B) = (B/X) •B2 = AX.