Difference of Cubes Ratio

CAT 2019 Slot 1 · QA · Medium · Number theory

All CAT 2019 Slot 1 questionsMore Number theory questionsAll CAT QA

The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157:3157:3, then the sum of the two numbers is

  1. A.

    50

  2. B.

    85

  3. C.

    95

  4. D.

    58

Answer

A

Explanation

Let the numbers be aa and bb, with ab=616ab = 616. a3−b3(a−b)3=1573  ⟹  (a−b)(a2+ab+b2)(a−b)3=(a−b)2+3ab(a−b)2=1+3ab(a−b)2=1573\frac{a^3 - b^3}{(a - b)^3} = \frac{157}{3} \implies \frac{(a-b)(a^2+ab+b^2)}{(a-b)^3} = \frac{(a-b)^2 + 3ab}{(a-b)^2} = 1 + \frac{3ab}{(a-b)^2} = \frac{157}{3}. 3ab(a−b)2=1543  ⟹  3(616)(a−b)2=1543  ⟹  (a−b)2=9×616154=9×4=36\frac{3ab}{(a-b)^2} = \frac{154}{3} \implies \frac{3(616)}{(a-b)^2} = \frac{154}{3} \implies (a-b)^2 = \frac{9 \times 616}{154} = 9 \times 4 = 36. a−b=6a - b = 6. (a+b)2=(a−b)2+4ab=36+4(616)=36+2464=2500  ⟹  a+b=50(a + b)^2 = (a - b)^2 + 4ab = 36 + 4(616) = 36 + 2464 = 2500 \implies a + b = 50.

Practise this under exam conditions

Sign in to solve it with a live timer, the on-screen CAT calculator, and streak and accuracy tracking across every question you attempt.

Solve in the workspace