From a rope `11 m` long, two pieces of lengths `2(3)/(5) m` and `3(3)/(10) m` are cut off. What is the length of the remaining rope?

Options

A. `5(1)/(10)m `
B. `5(2)/(10)m `
C. `4(1)/(10)m `
D. `3(1)/(10)m `

Answer

To find the length of the remaining rope after cutting off two pieces, we will follow these steps:

### Step 1: Identify the total length of the rope and the lengths of the pieces to be cut off.
- Total length of the rope = 11 m
- Length of the first piece = \(2 \frac{3}{5}\) m
- Length of the second piece = \(3 \frac{3}{10}\) m

### Step 2: Convert the mixed fractions into improper fractions.
1. For the first piece \(2 \frac{3}{5}\):
- Multiply the whole number (2) by the denominator (5) and add the numerator (3):
\[ 2 \times 5 + 3 = 10 + 3 = 13 \]
- So, \(2 \frac{3}{5} = \frac{13}{5}\) m.

2. For the second piece \(3 \frac{3}{10}\):
- Multiply the whole number (3) by the denominator (10) and add the numerator (3):
\[ 3 \times 10 + 3 = 30 + 3 = 33 \]
- So, \(3 \frac{3}{10} = \frac{33}{10}\) m.

### Step 3: Add the lengths of the two pieces together.
To add \(\frac{13}{5}\) and \(\frac{33}{10}\), we need a common denominator:
- The least common multiple (LCM) of 5 and 10 is 10.
- Convert \(\frac{13}{5}\) to a fraction with a denominator of 10:
\[ \frac{13}{5} = \frac{13 \times 2}{5 \times 2} = \frac{26}{10} \]
- Now, add the two fractions:
\[ \frac{26}{10} + \frac{33}{10} = \frac{26 + 33}{10} = \frac{59}{10} \]

### Step 4: Subtract the total length of the cut pieces from the total length of the rope.
- Total length of the rope = 11 m = \(\frac{110}{10}\) m (converting to a fraction with a denominator of 10).
- Now, subtract the total length of the cut pieces:
\[ \frac{110}{10} - \frac{59}{10} = \frac{110 - 59}{10} = \frac{51}{10} \]

### Step 5: Convert the improper fraction back to a mixed number.
- Divide 51 by 10:
- 51 divided by 10 is 5 with a remainder of 1.
- Therefore, \(\frac{51}{10} = 5 \frac{1}{10}\) m.

### Conclusion:
The length of the remaining rope is \(5 \frac{1}{10}\) m.

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