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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)/(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)
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