The ratio of the ages of Aunt Agatha and Uncle Roger is 5 : 6, the same as the…

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The ratio of the ages of Aunt Agatha and Uncle Roger is 5 : 6, the same as the ratio of the ages of their children, Suzanne and Bernard. If the sum of the ages of Suzanne and Aunt Agatha five years ago was 60, what will be the sum of the ages of Bernard and Uncle Roger five years hence?

  1. A.

    84

  2. B.

    94

  3. C.

    76

  4. D.

    96

Attempted by 15 students.

Show answer & explanation

Correct answer: B

Concept

When two pairs of quantities share the same ratio 5 : 6, each pair can be written with the SAME pair of multipliers scaled by its own constant: the 5-part as 5 times a constant and the 6-part as 6 times that constant. A change of the same number of years to every age shifts each person's age by that fixed amount, so a sum of n ages changes by (that amount) times n. These two ideas let us connect one pair's known sum to the other pair's unknown sum.

Application

  1. Write the ages with multipliers: Aunt Agatha = 5x, Uncle Roger = 6x (ratio 5 : 6); Suzanne = 5y, Bernard = 6y (same ratio 5 : 6).

  2. Five years ago each age was 5 less, so Suzanne was 5y - 5 and Aunt Agatha was 5x - 5. Their sum then was (5y - 5) + (5x - 5) = 60.

  3. Simplify: 5x + 5y - 10 = 60, hence 5x + 5y = 70, so 5(x + y) = 70 and x + y = 14.

  4. The present sum of Bernard and Uncle Roger is 6y + 6x = 6(x + y) = 6 * 14 = 84.

  5. Five years hence each of them is 5 older, so the required sum = (6y + 5) + (6x + 5) = 6(x + y) + 10 = 84 + 10 = 94.

Cross-check

Take a concrete set satisfying x + y = 14, say x = 8 and y = 6. Then today Uncle Roger = 48, Aunt Agatha = 40, Suzanne = 30, Bernard = 36. Five years ago Suzanne + Aunt Agatha = 25 + 35 = 60 (matches the given condition). Five years hence Bernard + Uncle Roger = 41 + 53 = 94, which is independent of the particular x and y chosen.

Required sum five years hence = 94.

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