A centipede was crawling in a straight line. First it crawled for `8 (3)/(7)` inch, then it crawled in the reverse direction for `5 (5)/(8)` inch. Once again it reversed direction and crawled for `11 (1)/(2)` inch in the original direction. How far away was it finally from the starting point?

Answer

To find out how far the centipede is from the starting point after crawling in different directions, we can break down the problem step by step.

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers into improper fractions for easier calculations.

1. **For `8 (3/7)`**:
- Convert: \( 8 \times 7 + 3 = 56 + 3 = 59 \)
- So, \( 8 \frac{3}{7} = \frac{59}{7} \)

2. **For `5 (5/8)`**:
- Convert: \( 5 \times 8 + 5 = 40 + 5 = 45 \)
- So, \( 5 \frac{5}{8} = \frac{45}{8} \)

3. **For `11 (1/2)`**:
- Convert: \( 11 \times 2 + 1 = 22 + 1 = 23 \)
- So, \( 11 \frac{1}{2} = \frac{23}{2} \)

### Step 2: Set Up the Equation
Now we can set up the equation based on the movements of the centipede:

- First, it crawls forward: \( \frac{59}{7} \)
- Then it crawls backward: \( -\frac{45}{8} \)
- Finally, it crawls forward again: \( +\frac{23}{2} \)

The total distance from the starting point is:
\[ \text{Total distance} = \frac{59}{7} - \frac{45}{8} + \frac{23}{2} \]

### Step 3: Find a Common Denominator
To perform the addition and subtraction, we need a common denominator. The least common multiple (LCM) of 7, 8, and 2 is 56.

- Convert each fraction:
1. \( \frac{59}{7} = \frac{59 \times 8}{7 \times 8} = \frac{472}{56} \)
2. \( \frac{45}{8} = \frac{45 \times 7}{8 \times 7} = \frac{315}{56} \)
3. \( \frac{23}{2} = \frac{23 \times 28}{2 \times 28} = \frac{644}{56} \)

### Step 4: Combine the Fractions
Now we can combine the fractions:
\[ \text{Total distance} = \frac{472}{56} - \frac{315}{56} + \frac{644}{56} \]
Combine the numerators:
\[ = \frac{472 - 315 + 644}{56} = \frac{801}{56} \]

### Step 5: Convert to Mixed Number
Now convert \( \frac{801}{56} \) into a mixed number:
- Divide 801 by 56:
- 801 ÷ 56 = 14 remainder 17
So, \( \frac{801}{56} = 14 \frac{17}{56} \).

### Final Answer
The centipede is \( 14 \frac{17}{56} \) inches away from the starting point.

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