From a rope of length `20""1/2` m, a piece of length `3""5/8` m is cut off. Find the length of the remaining rope.

Options

A. `16""7/8` m
B. `9""5/8` m
C. `15""3/8` m
D. `19""5/9` m

Answer

To solve the problem of finding the length of the remaining rope after cutting off a piece, we can follow these steps:

### Step-by-Step Solution:

1. **Convert the lengths to improper fractions:**
- The total length of the rope is given as \(20 \frac{1}{2}\) m.
- Convert this to an improper fraction:
\[ 20 \frac{1}{2} = \frac{20 \times 2 + 1}{2} = \frac{40 + 1}{2} = \frac{41}{2} \text{ m} \]
- The length of the piece cut off is \(3 \frac{5}{8}\) m.
- Convert this to an improper fraction:
\[ 3 \frac{5}{8} = \frac{3 \times 8 + 5}{8} = \frac{24 + 5}{8} = \frac{29}{8} \text{ m} \]

2. **Set up the equation for the remaining length:**
- Let \(D\) be the length of the remaining rope.
- The equation will be:
\[ D = \text{Total Length} - \text{Cut Off Length} \]
- Substituting the values we have:
\[ D = \frac{41}{2} - \frac{29}{8} \]

3. **Find a common denominator:**
- The denominators are 2 and 8. The least common multiple (LCM) of 2 and 8 is 8.
- Convert \(\frac{41}{2}\) to have a denominator of 8:
\[ \frac{41}{2} = \frac{41 \times 4}{2 \times 4} = \frac{164}{8} \]

4. **Subtract the fractions:**
- Now we can subtract:
\[ D = \frac{164}{8} - \frac{29}{8} = \frac{164 - 29}{8} = \frac{135}{8} \]

5. **Convert the improper fraction back to a mixed number:**
- To convert \(\frac{135}{8}\) to a mixed number, divide 135 by 8:
\[ 135 \div 8 = 16 \quad \text{(whole number part)} \]
- The remainder is:
\[ 135 - (16 \times 8) = 135 - 128 = 7 \]
- Therefore,
\[ \frac{135}{8} = 16 \frac{7}{8} \text{ m} \]

### Final Answer:
The length of the remaining rope is \(16 \frac{7}{8}\) m.