#### Question

If a : b = c : d, prove that: (6a + 7b) (3c – 4d) = (6c + 7d) (3a – 4b).

#### Solution

Given `a/b = c/d`

`=> (6a)/(7b) = (6c)/(7d)` (Multiplying each side by 6/7)

`=> (6a + 7b)/(7b) = (6c + 7d)/(7d)` (By componendo)

`=> (6a + 7b)/(6c + 7d) = (7b)/(7d) = b/d` .....(1)

Also `a/b = c/d`

`=> (3a)/(4b) = (3c)/(4d)` (Mutipling each side by 3/4)

`=> (3a - 4b)/(4b) = (3c - 4d)/(4d)` (By dividendo)

`=>(3a - 4b)/(4b) = (3c - 4d)/(4d)` (By dividendo)

`=> (3a - 4b)/(3c - 4d) = (4b)/(4d) = b/d` ....(2)

From (1) and (2)

`(6a + 7b)/(6c + 7d) = (3a - 4b)/(3c - 4c)`

(6a+ 7b)(3c - 4d) = (6c + 7d)(3a - 4b)

Is there an error in this question or solution?

Solution If a : B = C : D, Prove That: (6a + 7b) (3c – 4d) = (6c + 7d) (3a – 4b). Concept: Componendo and Dividendo Properties.