To solve this problem, let's break it down step-by-step:
1. Define Variables:
- Let \(\(T\)\) be the amount of money Tony has saved.
- Let \(\(S\)\) be the amount of money Sarah has saved.
2. Analyze Given Information:
- The ratio of their savings is 7:11. This means for every £7 Tony has, Sarah has £11.
- Sarah has saved £24 more than Tony.
3. Express the Relationship Using the Ratio:
Since the ratio of Tony's savings to Sarah's savings is 7:11, we can write this as:
\(\[
\frac{T}{S} = \frac{7}{11}
\]\)
This can be rearranged to:
\(\[
11T = 7S
\]\)
4. Utilize the Difference in Savings:
Given that Sarah has saved £24 more than Tony:
\(\[
S = T + 24
\]\)
5. Substitute and Solve:
Substitute \(\(S\)\) from the relationship \(\(S = T + 24\)\) into the equation \(\(11T = 7S\)\):
\(\[
11T = 7(T + 24)
\]\)
Now distribute and simplify:
\(\[
11T = 7T + 168
\]\)
Combine like terms:
\(\[
11T - 7T = 168
\]\)
\(\[
4T = 168
\]\)
Solve for \(\(T\)\):
\(\[
T = \frac{168}{4}
\]\)
\(\[
T = 42
\]\)
6. Find Sarah's Savings:
Substitute \(\(T\)\) back into the equation \(\(S = T + 24\)\):
\(\[
S = 42 + 24
\]\)
\(\[
S = 66
\]\)
7. Calculate Total Savings:
Add Tony's savings and Sarah's savings to find the total savings:
\(\[
\text{Total savings} = T + S
\]\)
\(\[
\text{Total savings} = 42 + 66
\]\)
\(\[
\text{Total savings} = 108
\]\)
So, the total amount Tony and Sarah have saved together is £108.
