The sum of ages of a father and his son is 45 years. Five years ago, the product of their ages (in yrs.) was 124. The present age of father is :
36 years
Solving the Father and Son Age Problem
This question asks us to find the present age of a father given the sum of his and his son's ages now and the product of their ages five years ago. We can solve this using algebra by setting up equations based on the information provided.
Setting Up the Equations
Let's denote the present age of the father as \(F\) years and the present age of the son as \(S\) years.
- The sum of their present ages is 45 years. This gives us the equation:
\(F + S = 45\)
- Five years ago, the father's age was \(F - 5\) and the son's age was \(S - 5\).
- The product of their ages five years ago was 124. This gives us the equation:
\((F - 5)(S - 5) = 124\)
Solving the Equations
We have a system of two equations with two variables:
- \(F + S = 45\)
- \((F - 5)(S - 5) = 124\)
From the first equation, we can express \(S\) in terms of \(F\):
\(S = 45 - F\)
Now, substitute this expression for \(S\) into the second equation:
\((F - 5)((45 - F) - 5) = 124\)
\((F - 5)(40 - F) = 124\)
Expand the left side of the equation:
\(F(40) + F(-F) - 5(40) - 5(-F) = 124\)
\(40F - F^2 - 200 + 5F = 124\)
Combine like terms:
\(-F^2 + 45F - 200 = 124\)
Move all terms to one side to form a quadratic equation:
\(0 = F^2 - 45F + 200 + 124\)
\(F^2 - 45F + 324 = 0\)
Factoring the Quadratic Equation
We need to find two numbers that multiply to 324 and add up to -45. Let's list factors of 324:
Factors of 324:
| Factor 1 | Factor 2 | Sum |
|---|---|---|
| 1 | 324 | 325 |
| 2 | 162 | 164 |
| 3 | 108 | 111 |
| 4 | 81 | 85 |
| 6 | 54 | 60 |
| 9 | 36 | 45 |
We need a sum of -45, which means both factors must be negative. The pair (9, 36) sums to 45, so the pair (-9, -36) sums to -45 and their product is \((-9) \times (-36) = 324\).
So, we can factor the quadratic equation as:
\((F - 9)(F - 36) = 0\)
This gives us two possible values for \(F\):
- \(F - 9 = 0 \implies F = 9\)
- \(F - 36 = 0 \implies F = 36\)
Determining the Correct Age
We have two possible present ages for the father: 9 years and 36 years.
- If the father's age is 9 years, then the son's age is \(S = 45 - 9 = 36\) years. This is not logical as a father must be older than his son.
- If the father's age is 36 years, then the son's age is \(S = 45 - 36 = 9\) years. This is a logical age difference.
Let's check this solution with the second condition (product of ages five years ago):
- Father's age 5 years ago = \(36 - 5 = 31\) years.
- Son's age 5 years ago = \(9 - 5 = 4\) years.
- Product of their ages 5 years ago = \(31 \times 4 = 124\).
This matches the given information.
Conclusion
The present age of the father is 36 years.
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