What is the least number must be added to each of the numbers 511 19 and 37 so that they are in proportion?

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What number must be added to each of the numbers 5, 11, 19 and 37 so that they are in proportion?

Let x be added to 5, 11, 19 and 37 to make them in proportion.5 + x : 11 + x : : 19 + x : 37 + x⇒ (5 + x) (37 + x) = (11 + x) (19 + x)

⇒ 185 + 5x + 37x + x2 = 209 + 11x + 19x + x2

⇒ 185 + 42x + x2 = 209 + 30x + x2
⇒ 42x - 30x + x2 - x2 = 209 - 185⇒ 12x = 24⇒ x = 2

∴ Least number to be added = 2.

Concept: Concept of Proportion

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Page 2

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

Let x be subtracted from each term, then23 – x, 30 – x, 57 – x and 78 – x are proportional23 – 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.

Concept: Concept of Proportion

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