To solve the equation [tex]\( X + \frac{X}{7} + \frac{1}{11}\left(X + \frac{X}{7}\right) = 60 \)[/tex], we will evaluate each of the given solution options to determine which one satisfies the equation.
### Given Options:
- [tex]\( x = 48.125 \)[/tex]
- [tex]\( x = 52.5 \)[/tex]
- [tex]\( x = 60.125 \)[/tex]
- [tex]\( x = 77 \)[/tex]
Let's check each option one by one.
### Option 1: [tex]\( x = 48.125 \)[/tex]
1. Calculate [tex]\( \frac{X}{7} \)[/tex]:
[tex]\[
\frac{48.125}{7} = 6.875
\][/tex]
2. Calculate [tex]\( X + \frac{X}{7} \)[/tex]:
[tex]\[
48.125 + 6.875 = 55
\][/tex]
3. Calculate [tex]\( \frac{1}{11} \left( X + \frac{X}{7} \right) \)[/tex]:
[tex]\[
\frac{1}{11} \times 55 = 5
\][/tex]
4. Sum the terms:
[tex]\[
48.125 + 6.875 + 5 = 60
\][/tex]
Since the left-hand side equals the right-hand side (60), [tex]\( x = 48.125 \)[/tex] satisfies the equation.
### Option 2: [tex]\( x = 52.5 \)[/tex]
1. Calculate [tex]\( \frac{X}{7} \)[/tex]:
[tex]\[
\frac{52.5}{7} \approx 7.5
\][/tex]
2. Calculate [tex]\( X + \frac{X}{7} \)[/tex]:
[tex]\[
52.5 + 7.5 = 60
\][/tex]
3. Calculate [tex]\( \frac{1}{11} \left( X + \frac{X}{7} \right) \)[/tex]:
[tex]\[
\frac{1}{11} \times 60 \approx 5.4545
\][/tex]
4. Sum the terms:
[tex]\[
52.5 + 7.5 + 5.4545 \approx 65.4545
\][/tex]
This does not equal 60, so [tex]\( x = 52.5 \)[/tex] does not satisfy the equation.
### Option 3: [tex]\( x = 60.125 \)[/tex]
1. Calculate [tex]\( \frac{X}{7} \)[/tex]:
[tex]\[
\frac{60.125}{7} \approx 8.5893
\][/tex]
2. Calculate [tex]\( X + \frac{X}{7} \)[/tex]:
[tex]\[
60.125 + 8.5893 \approx 68.7143
\][/tex]
3. Calculate [tex]\( \frac{1}{11} \left( X + \frac{X}{7} \right) \)[/tex]:
[tex]\[
\frac{1}{11} \times 68.7143 \approx 6.2467
\][/tex]
4. Sum the terms:
[tex]\[
60.125 + 8.5893 + 6.2467 \approx 74.961
\][/tex]
This does not equal 60, so [tex]\( x = 60.125 \)[/tex] does not satisfy the equation.
### Option 4: [tex]\( x = 77 \)[/tex]
1. Calculate [tex]\( \frac{X}{7} \)[/tex]:
[tex]\[
\frac{77}{7} = 11
\][/tex]
2. Calculate [tex]\( X + \frac{X}{7} \)[/tex]:
[tex]\[
77 + 11 = 88
\][/tex]
3. Calculate [tex]\( \frac{1}{11} \left( X + \frac{X}{7} \right) \)[/tex]:
[tex]\[
\frac{1}{11} \times 88 = 8
\][/tex]
4. Sum the terms:
[tex]\[
77 + 11 + 8 = 96
\][/tex]
This does not equal 60, so [tex]\( x = 77 \)[/tex] does not satisfy the equation.
### Conclusion
The correct solution to the equation [tex]\( X + \frac{X}{7} + \frac{1}{11}\left(X + \frac{X}{7}\right) = 60 \)[/tex] is:
[tex]\[
\boxed{48.125}
\][/tex]
