Question

If `(7"a" + 12"b")/(7"c" + 12"d")` then prove that `"a"/"b"="c"/"d"`

Answer 1

`(7"a" + 12"b")/(7"c" + 12"d") = (7"a" - 12"b")/(7"c" - 12"d")`

Applying alternendo, 

`(7"a" + 12"b")/(7"a" - 12"b") = (7"c" + 12"d")/(7"c" - 12"d")`

Applying componendo and dividendo, 

`(7"a" + 12"b" + 7"a" - 12 "b")/(7"a" + 12"b" - 7"a" + 12"b") = (7"c" + 12"d" + 7"c" - 12"d")/(7"c" + 12"d" - 7"c" + 12"d")`

`=> (14"a")/(24"b") = (14"c")/(24"d")`

Dividing both sides by  `14/24`

`"a"/"b" = "c"/"d"`

Hence, proved.