How old am I?

Algebra Level pending

Benedict's age this year is twice his younger brother, Bob's age a certain number of years ago. Interestingly, Bob current age is the same as Benedict's age the same number of years ago. If their sum of their current ages is x years and their age difference is x k \frac{x}{k} years, find the value of k.


The answer is 7.

Noel Lo by Noel Lo , 24, Singapore

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1 solution

Noel Lo
Mar 12, 2015

Let the age difference be b. So Benedict's age this year would be x + b 2 \frac{x+b}{2} while that of Bob's current age is x − b 2 \frac{x-b}{2} . A certain number of years ago, Benedict's age was x − b 2 \frac{x-b}{2} while Bob's age was 1 2 \frac{1}{2} * x + b 2 \frac{x+b}{2} = x + b 4 \frac{x+b}{4} the same number of years ago.. Now we have

x + b 2 \frac{x+b}{2} - x − b 2 \frac{x-b}{2} = x − b 2 \frac{x-b}{2} - x + b 4 \frac{x+b}{4} since the age difference between the two brothers is constant.

Simplifying the equation, we have 3 ( x + b ) 4 \frac {3(x+b)}{4} = x-b and hence 3x+3b = 4x-4b so x = 7b. Now b = x 7 \frac{x}{7} and the result follows.

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