Let B represent the present age of the boy, F represent the present age of the father, and G represent the present age of the grandfather.
Age Difference Logic
The problem states that the age difference between the boy and his father is the same as the age difference between the father and grandfather. This can be written as:
$F - B = G - F$
Rearranging this equation gives:
$2F = G + B$ (Equation 1)
Past Age Calculation
Five years ago, the grandfather's age was $G - 5$ and the father's age was $F - 5$. The problem states the grandfather was twice as old as the father at that time:
$G - 5 = 2(F - 5)$
Simplify this equation:
$G - 5 = 2F - 10$
$G = 2F - 10 + 5$
$G = 2F - 5$ (Equation 2)
Boy's Age Determination
Now substitute the expression for G from Equation 2 into Equation 1:
$2F = (2F - 5) + B$
Simplify the equation by subtracting $2F$ from both sides:
$0 = -5 + B$
Solve for B:
$B = 5$
Therefore, the present age of the boy is 5 years.
Important Questions from Problem on Ages (Notes)
- At present, Sharad is three times as old as his son, and his daughter is 3 years younger than the son. If the sum of the ages of these three people 3 years ago was 63 years, then Sharad's present age (in years) is:
A father said to his son, "I was as old as you are when I became your father." If the current age of father is 52 years, the age of son after 10 years will be___________.
- The sum of the age of A and 5 times the age of B is 55 years. When 3 times the age of A is added to 7 times the age of B, the result is 99 years. The sum of the ages (in years) of A and B is:
- The sum of the age of A and 3 times the age of B is 43 years. When 5 times the age of A is added to 2 times the age of B, the result is 98 years. The sum of the ages (in years) of A and B is:
- Eight years ago, the age of the mother was four times the age of her son. Eight years hence, the mother's age will be two times the age of her son. Find the ratio of the present age of the mother to the age of the son.
