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