Question

What number should be subtracted from each of the numbers 23, 30, 57 and 78 so that the remainders are in proportion ?

Answer 1

Let x be subtracted from each term, then
23 – x, 30 – x, 57 – x and 78 – x are proportional
23 – x : 30 – x : : 57 – x : 78 – x
⇒ `(23 – x)/(30 – x) = (57 – x)/(78 – x)`
⇒ (23 – x) (78 – x) = (30 – x) (57 – x)
⇒ 1794 – 23x – 78x + x² = 1710 – 30x – 57x + x²
⇒ x² – 101x + 1794 = x² – 87x + 1710
⇒ x² – 101x + 1794 – x² + 87x – 1710 = 0
⇒ –14x + 84 = 0
⇒ 14x = 84
∴ x = `(84)/(14)` = 6
Hence 6 is to be subtracted.