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Sagot :
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]
### 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]
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