P varies directly as Q and inversely as R , also P = 2 3 when Q = 3 / 7 and R = 9 14 . Find Q , when P = 48 and R = 75 .

It is given that P varies directly as Q and inversely as R then we have P ∝ Q P ∝ 1 R Therefore, we have P = K Q R . . . i where K is any constant. Now, we have P = 2 3 when Q = 3 7 and R = 9 14 . Then from i , we have 2 3 = K 3 7 9 14 ⇒ K = 2 3 × 9 14 × 7 3 = 1 Putting this value in i , we get P = Q R ⇒ 48 = 1 · Q 75 ⇒ Q = 60

Given that x + y varies as z + 1 z and that x - y varies z - 1 z . Find the relation between x and z , provided that z = 2 when x = 3 and y = 1 .

Given: x + y ∝ z + 1 z x + y = k z + 1 z . . . i where, k is the constant of proportionality. And, x - y ∝ z - 1 z x - y = k ' z - 1 z . . . ii Now when x = 3 , y = 1 then z = 2 . Putting these values in i and ii , we get 4 = k 5 2 ⇒ k = 8 5 2 = k ' 3 2 ⇒ k ' = 4 3 Therefore, by i and ii , we have x + y = 8 5 z + 1 z . . . iii x - y = 4 3 z - 1 z . . . iv Eliminating y between these, we get 2 x = 44 z 15 + 4 15 z ⇒ 2 x = 22 15 z + 2 15 z