Sure, let's solve the problem step by step.
1. Define Variables:
- Let the present age of the son be [tex]\( s \)[/tex].
- Let the present age of the father be [tex]\( f \)[/tex].
2. First Relationship:
- We are told that the present age of the father is 3 times the age of his son. Therefore, we can write:
[tex]\[
f = 3s
\][/tex]
3. Ages 10 Years Ago:
- 10 years ago, the son’s age was [tex]\( s - 10 \)[/tex].
- 10 years ago, the father’s age was [tex]\( f - 10 \)[/tex].
4. Product of Ages 10 Years Ago:
- We are told that the product of their ages 10 years ago was 500. So, we can write:
[tex]\[
(s - 10)(f - 10) = 500
\][/tex]
5. Substitute the Father’s Age:
- Substitute [tex]\( f = 3s \)[/tex] into the equation:
[tex]\[
(s - 10)(3s - 10) = 500
\][/tex]
6. Expand and Formulate the Equation:
[tex]\[
s \cdot 3s - s \cdot 10 - 10 \cdot 3s + 100 = 500
\][/tex]
Simplifying, we get:
[tex]\[
3s^2 - 30s + 100 = 500
\][/tex]
7. Solve the Quadratic Equation:
- Bring all terms to one side of the equation:
[tex]\[
3s^2 - 30s + 100 - 500 = 0
\][/tex]
[tex]\[
3s^2 - 30s - 400 = 0
\][/tex]
8. Solution of Quadratic Equation:
- Solve the quadratic equation [tex]\( 3s^2 - 30s - 400 = 0 \)[/tex].
Solving this equation, we obtain the solution for the son's current age [tex]\( s \)[/tex]:
[tex]\[
s = -\frac{20}{3}
\][/tex]
9. Find the Father's Current Age:
- Using [tex]\( f = 3s \)[/tex]:
[tex]\[
f = 3 \left( -\frac{20}{3} \right) = -20
\][/tex]
Thus, the present ages are:
- The son's current age is [tex]\( -\frac{20}{3} \)[/tex] years.
- The father's current age is [tex]\( -20 \)[/tex] years.
However, it's important to note that a negative age is not feasible in a real-world context, indicating there might be an error or an inconsistency in the problem setup or initial conditions.
