#### Question

Given four quantities a, b, c and d are in proportion. Show that `(a - c)b^2 : (b - d)cd = (a^2 - b^2 - ab) : (c^2 - d^2 - cd)`

#### Solution

Let `a/b = c/d = k`

`=> a = bk and c = dk`

`L.H.S = ((a - c)b^2)/((b -d)cd)`

`= ((bk - dk)b^2)/((b - d)d^2k)`

`= b^2/d^2`

R.H.S = `(a^2 - b^2 - ab)/(c^2 - d^2 - cd)`

`= (b^2k^2 - b^2 - bkb)/(d^2k^2 - d^2 - dkd)`

`= (b^2(k^2 - 1 - k))/(d^2(k^2 - 1 - k))`

`= b^2/d^2`

=> L.H.S = R.H.S

Hence proved

Is there an error in this question or solution?

Solution Given Four Quantities A, B, C and D Are in Proportion. Show that (A - C)B^2 : (B - D)Cd = (A^2 - B^2 - Ab) : (C^2 - D^2 - Cd) Concept: Proportions.