Question

If a, b, c and d are in proportion, prove that: (5a + 7b) (2c – 3d) = (5c + 7d) (2a – 3b).

Answer 1

It is given that

a, b, c, d are in proportion

Consider `a/b = c/d = k`

a = bk, c = dk

LHS = (5a + 7b)(2c – 3d)

LHS = (5bk + 7b)(2dk – 3d) ...[Substituting the values]

LHS = b(5k + 7) d(2k – 3)  ...[Taking out the common terms]

LHS = bd (5k + 7)(2k - 3)

LHS = bd [5k (2k - 3) + 7(2k - 3)]

LHS = bd (10k² - 15k + 14k - 21)

LHS = bd (10k² - k - 21) ... (I)

 

RHS = (5c + 7d)(2a – 3b)

RHS = (5dk + 7d)(2bk – 3b)  ...[Substituting the values]

RHS = d(5k + 7) b(2k – 3) ...[Taking out the common terms]

RHS = bd (5k + 7)(2k – 3)

RHS = bd [5k(2k - 3) + 7(2k - 3)]

RHS = bd (10k² - 15k + 14k - 21)

LHS = bd (10k² - k - 21) ... (II)

From (I) and (II),

Therefore, LHS = RHS.