Age Difference Problem Explained

This problem involves finding the current ages of two people based on their age difference and a relationship between their ages in the past.

Setting Up the Equations

Let the present age of the older person be $x$ years and the present age of the younger person be $y$ years.

From the problem statement:

• The ages differ by 15 years: $x - y = 15$. (Equation 1)
• 5 years ago, the older person's age was $x - 5$ and the younger person's age was $y - 5$.
• The older person's age 5 years ago was twice the younger one's age: $x - 5 = 2(y - 5)$. (Equation 2)

Solving for Present Ages

We can solve these two equations simultaneously.

1. Express $x$ in terms of $y$: From Equation 1, $x = y + 15$.
2. Substitute into Equation 2: Replace $x$ in Equation 2 with $(y + 15)$. $(y + 15) - 5 = 2(y - 5)$
3. Simplify and solve for $y$: $y + 10 = 2y - 10$ $10 + 10 = 2y - y$ $20 = y$ So, the younger person's present age is 20 years.
4. Calculate $x$: Substitute $y = 20$ back into Equation 1 (or the expression for $x$). $x = y + 15 = 20 + 15 = 35$ So, the older person's present age is 35 years.

Conclusion

The present ages of the two persons are 35 years and 20 years.

Verification:

• Age difference: $35 - 20 = 15$ years. (Correct)
• 5 years ago: Older was $35 - 5 = 30$, Younger was $20 - 5 = 15$.
• $30 = 2 \times 15$. (Correct)

The calculated ages match the conditions given in the problem.