Question

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

Answer 1

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

From (1) and (2)

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

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