Certainly! Let's break down the problem step-by-step:
1. Understand the Problem:
- Ten years ago, the ratio of the father's age to the son's age was 11:3.
- Five years from now, the father's age will be 10 years older than twice the son's age at that time.
- We need to determine after how many years the son's age will be equal to the present age of his father.
2. Setup Variables:
- Let [tex]\( F \)[/tex] be the father's age 10 years ago.
- Let [tex]\( S \)[/tex] be the son's age 10 years ago.
3. Establish Equations Using Given Conditions:
Condition 1: Ratio of Ages 10 Years Ago
- According to the problem, 10 years ago the ratio of the father's age to the son's age was [tex]\( \frac{F}{S} = \frac{11}{3} \)[/tex].
- This gives us the equation:
[tex]\[
F = \frac{11}{3}S
\][/tex]
Condition 2: Age Relationship 5 Years From Now
- Five years after 10 years later (15 years from the ages we considered initially), the father's age will be 10 years older than twice the son's age.
- This gives us the equation:
[tex]\[
F + 15 = 2(S + 15) + 10
\][/tex]
4. Solve the Equations Simultaneously:
Solve the system of equations given:
[tex]\[
F = \frac{11}{3}S
\][/tex]
[tex]\[
F + 15 = 2S + 40
\][/tex]
Substituting [tex]\( F \)[/tex] from the first equation into the second equation:
[tex]\[
\frac{11}{3}S + 15 = 2S + 40
\][/tex]
Clear the fraction by multiplying all terms by 3:
[tex]\[
11S + 45 = 6S + 120
\][/tex]
Simplify and solve for [tex]\( S \)[/tex]:
[tex]\[
11S - 6S = 120 - 45
\][/tex]
[tex]\[
5S = 75
\][/tex]
[tex]\[
S = 15
\][/tex]
Now, find [tex]\( F \)[/tex]:
[tex]\[
F = \frac{11}{3} \times 15
\][/tex]
[tex]\[
F = 55
\][/tex]
5. Find the Current Ages:
- Since these ages are from 10 years ago, we add 10 years to both.
- Current age of father [tex]\( = F + 10 = 55 + 10 = 65 \)[/tex].
- Current age of son [tex]\( = S + 10 = 15 + 10 = 25 \)[/tex].
6. Determine After How Many Years the Son's Age Will Equal the Father's Current Age:
- We want to find [tex]\( x \)[/tex] such that the son's age + [tex]\( x \)[/tex] years equals the father's current age.
- Set up the equation:
[tex]\[
25 + x = 65
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 65 - 25
\][/tex]
[tex]\[
x = 40
\][/tex]
Therefore, the son will be equal to the present age of his father after 40 years.
