To solve the problem, let's carefully break it down step-by-step.
1. Define Variables:
Let [tex]\( b \)[/tex] represent the brother's age.
Let [tex]\( s \)[/tex] represent the sister's age.
2. Given Relationship:
From the problem, we know that the brother is 5 years older than the sister. Therefore, we can write this as:
[tex]\[
b = s + 5
\][/tex]
3. Set Up the Equation:
According to the problem, if you take the fifth part of the brother's age, add 6 to it; and take the third part of the sister's age, add 5 to it; the results are equal. This can be expressed as:
[tex]\[
\frac{b}{5} + 6 = \frac{s}{3} + 5
\][/tex]
4. Substitute the Relationship into the Equation:
Substitute [tex]\( b = s + 5 \)[/tex] into the equation:
[tex]\[
\frac{s + 5}{5} + 6 = \frac{s}{3} + 5
\][/tex]
5. Solve for [tex]\( s \)[/tex]:
Simplify the equation to find the sister's age:
[tex]\[
\frac{s + 5}{5} + 6 = \frac{s}{3} + 5
\][/tex]
Subtract 5 from both sides:
[tex]\[
\frac{s + 5}{5} + 1 = \frac{s}{3}
\][/tex]
Multiply both sides by 15 to eliminate the denominators:
[tex]\[
3(s + 5) + 15 = 5s
\][/tex]
Simplify and solve for [tex]\( s \)[/tex]:
[tex]\[
3s + 15 + 15 = 5s
\][/tex]
[tex]\[
3s + 30 = 5s
\][/tex]
Subtract 3s from both sides:
[tex]\[
30 = 2s
\][/tex]
Divide by 2:
[tex]\[
s = 15
\][/tex]
6. Find the Brother's Age:
Now that we have the sister's age, we can find the brother's age using the given relationship:
[tex]\[
b = s + 5 = 15 + 5 = 20
\][/tex]
7. Conclusion:
Therefore, the sister's age is 15 years and the brother's age is 20 years.
