Ratios and Proportions

  • a/b is the ratio of a to b; b is not 0.

  • When two ratios are equal, they are said to be in proportion.

  • If a/b = c/d and can be written as a: b: : c: d.
    where "a" and "d" are called "extremes" and "c" and "b" the means.

  • For a, b, c, d to be in proportion the product of the extremes = the product of the means. i.e. ad = bc

  • Direct proportion: When a/b = k or a = kb then "a" is directly proportional to "b", where k is a constant.

  • Inverse proportion: When "a" and "b" are so related that ab = k, a constant, then "a" and "b" are said to be inversely proportional to each other.

  • If a sum of money S is divided in the ratio a : b : c then the three parts are

  • (i) S(a / a+b+c)

  • (ii) S(b / a+b+c)

  • (iii) S(c / a+b+c)

  • If a : b = m : n and b : c = p : q then a : b : c = mp : np : nq

  • If A and B are two partners investing in the ratio of m : n for the same period of time, then the ratio of profits is m : n

  • If the investment is in the ratio m : n and the period in the ratio p : q then the ratio of profits is mp : nq.

  • If m kg of one kind costing 'a" rupees/kg is mixed with "n" kg of another kind costing Rs.b/kg, then the price of the mixture is (ma + nb)/(m+n)

  • If "a" varies as "b", then a = kb, where "k" is called the constant of proportionality. (Direct variation).