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Question
If a : b : : c : d, then prove that
`(4"a" + 9"b")/(4"c" + 9"d") = (4"a" - 9"b")/(4"c" - 9"d")`
Solution
`(4"a" + 9"b")/(4"c" + 9"d") = (4"a" - 9"b")/(4"c" - 9"d")`
`"a"/"b" = "c"/"d"`
Multiplying both sides by `4/9`
`=> "a"/"b" xx 4/9 = "c"/"d" xx 4/9`
`=> (4"a")/(9"b") = (4"c")/(9"d")`
Applying componendo and dividendo,
`(4"a" + 9"b")/(4"a" - 9"b") = (4"c" + 9"d")/(4"c" - 9"d")`
`=> (4"a" + 9"b")/(4"c" + 9"d") = (4"a" - 9"b")/(4"c" - 9"d")`
Hence, 4a + 9b : 4c + 9d = 4a - 9b : 4c - 9d
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