An insect is crawling up on a vertical wall. It crawls up `12 (5)/(7)` inch. Then it slips back `3 (11)/(14)` inch and then again crawls up `5 (1)/(6)` inch. How far is it from the starting point?

Options

A. `11 (6)/(25)` inch
B. `12 (2)/(21)` inch
C. `14 (2)/(21)` inch
D. `13 (5)/(28)` inch

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. **Crawling Up**: The insect crawls up `12 (5)/(7)` inches.
- Convert to improper fraction:
\[ 12 \frac{5}{7} = \frac{12 \times 7 + 5}{7} = \frac{84 + 5}{7} = \frac{89}{7} \]

2. **Slipping Back**: The insect slips back `3 (11)/(14)` inches.
- Convert to improper fraction:
\[ 3 \frac{11}{14} = \frac{3 \times 14 + 11}{14} = \frac{42 + 11}{14} = \frac{53}{14} \]

3. **Crawling Up Again**: The insect crawls up `5 (1)/(6)` inches.
- Convert to improper fraction:
\[ 5 \frac{1}{6} = \frac{5 \times 6 + 1}{6} = \frac{30 + 1}{6} = \frac{31}{6} \]

### Step 2: Calculate the Total Distance from the Starting Point
Now we will calculate the total distance the insect has crawled up and slipped back:
\[ \text{Total Distance} = \left(\frac{89}{7} - \frac{53}{14} + \frac{31}{6}\right) \]

### Step 3: Find the Least Common Multiple (LCM)
To add and subtract these fractions, we need to find the LCM of the denominators (7, 14, and 6):
- Prime factors:
- 7: \(7\)
- 14: \(2 \times 7\)
- 6: \(2 \times 3\)
- LCM: \(2 \times 3 \times 7 = 42\)

### Step 4: Convert Each Fraction to Have a Common Denominator
1. Convert \(\frac{89}{7}\) to have a denominator of 42:
\[ \frac{89}{7} = \frac{89 \times 6}{42} = \frac{534}{42} \]

2. Convert \(\frac{53}{14}\) to have a denominator of 42:
\[ \frac{53}{14} = \frac{53 \times 3}{42} = \frac{159}{42} \]

3. Convert \(\frac{31}{6}\) to have a denominator of 42:
\[ \frac{31}{6} = \frac{31 \times 7}{42} = \frac{217}{42} \]

### Step 5: Combine the Fractions
Now we can combine the fractions:
\[ \frac{534}{42} - \frac{159}{42} + \frac{217}{42} = \frac{534 - 159 + 217}{42} \]
Calculating the numerator:
\[ 534 - 159 + 217 = 534 + 217 - 159 = 592 \]
Thus, we have:
\[ \frac{592}{42} \]

### Step 6: Simplify the Fraction
Now we simplify \(\frac{592}{42}\):
- Divide both the numerator and the denominator by 2:
\[ \frac{592 \div 2}{42 \div 2} = \frac{296}{21} \]

### Step 7: Convert to Mixed Number
Convert \(\frac{296}{21}\) to a mixed number:
- Divide 296 by 21:
\[ 296 \div 21 = 14 \quad \text{(whole number)} \]
- Remainder:
\[ 296 - (21 \times 14) = 296 - 294 = 2 \]
Thus, we have:
\[ 14 \frac{2}{21} \]

### Final Answer
The insect is \(14 \frac{2}{21}\) inches from the starting point.

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