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Question
If a, b, c, d are in proportion, prove that (2a + 3b) (3c + 5d) = (2c + 3d) (3a + 5b)
Theorem
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Solution
`a/b = c/d`
ad = bc
L.H.S.
= (2a + 3b) (3c + 5d)
= 2a ⋅ 3c + 2a ⋅ 5d + 3b ⋅ 3c + 3b ⋅ 5d
= 6ac + 10ad + 9bc + 15bd
R.H.S.
= (2c + 3d) (3a + 5b)
= 2c ⋅ 3a + 2c ⋅ 5b + 3d ⋅ 3a + 3d ⋅ 5b
= 6ac + 10bc + 9ad + 15bd
Use the given condition:
ad = bc
Thus,
10ad = 10bc
9ad = 9bc
L.H.S. = 6ac + 10ad + 9bc + 15bd
= 6ac + 19bc + 15bd
R.H.S. = 6ac + 10bc + 9ad + 15bd
= 6ac + 19bc + 15bd
L.H.S. = R.H.S.
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Chapter 7: Ratio and proportion - Exercise 7B [Page 126]
