Certainly! Let's solve this step-by-step:
1. Identify the Variables:
- Let [tex]\( T \)[/tex] represent Tim's age.
- Pete is 6 years younger than Tim, so Pete's age [tex]\( P \)[/tex] can be expressed as [tex]\( P = T - 6 \)[/tex].
2. Set Up the Equation for Combined Age:
- We know the combined age of Tim and Pete is 66 years. Therefore, we can write the equation as:
[tex]\[
T + P = 66
\][/tex]
3. Substitute Pete's Age:
- Substitute [tex]\( P \)[/tex] with [tex]\( T - 6 \)[/tex] into the equation:
[tex]\[
T + (T - 6) = 66
\][/tex]
4. Solve for Tim's Age:
- Simplify the equation:
[tex]\[
T + T - 6 = 66
\][/tex]
[tex]\[
2T - 6 = 66
\][/tex]
- Add 6 to both sides:
[tex]\[
2T = 72
\][/tex]
- Divide both sides by 2:
[tex]\[
T = 36
\][/tex]
- So, Tim is 36 years old.
5. Determine Pete's Age:
- Since Pete is 6 years younger than Tim:
[tex]\[
P = T - 6 = 36 - 6 = 30
\][/tex]
- So, Pete is 30 years old.
6. Ratio of Their Ages:
- The ratio of Tim's age to Pete's age is:
[tex]\[
\frac{T}{P} = \frac{36}{30} = \frac{6}{5}
\][/tex]
7. Distribution of Money Based on Age Ratio:
- The total amount of money is £770. We need to divide this money in the ratio 6:5.
- First, find the total parts of the ratio:
[tex]\[
6 + 5 = 11
\][/tex]
- Tim's share is:
[tex]\[
\text{Tim's share} = \frac{6}{11} \times 770 = \frac{4620}{11} = 420
\][/tex]
- Pete's share is:
[tex]\[
\text{Pete's share} = \frac{5}{11} \times 770 = \frac{3850}{11} = 350
\][/tex]
So, the final amounts each son gets are:
- Tim receives £420.
- Pete receives £350.
Thus, Tim gets £420 and Pete gets £350.
