From a rope of length `38 3/5m,` a piece of length `5 3/38`m is cut off. The length of the remaining rope is :

Options

A. `190 99/33`
B. `33 90/190`
C. `33 99/190`
D. None of these

Answer

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

### Step 1: Convert Mixed Numbers to Improper Fractions
The length of the rope is given as \( 38 \frac{3}{5} \) meters. We need to convert this mixed number into an improper fraction.

\[ 38 \frac{3}{5} = \frac{(38 \times 5) + 3}{5} = \frac{190 + 3}{5} = \frac{193}{5} \text{ meters} \]

### Step 2: Convert the Cut Off Length to an Improper Fraction
The length of the piece cut off is \( 5 \frac{3}{38} \) meters. We will convert this mixed number into an improper fraction as well.

\[ 5 \frac{3}{38} = \frac{(5 \times 38) + 3}{38} = \frac{190 + 3}{38} = \frac{193}{38} \text{ meters} \]

### Step 3: Find a Common Denominator
To subtract the two fractions, we need a common denominator. The denominators are 5 and 38. The least common multiple (LCM) of 5 and 38 is 190.

### Step 4: Convert Both Fractions to the Common Denominator
Now we will convert both fractions to have the common denominator of 190.

For \( \frac{193}{5} \):
\[ \frac{193}{5} = \frac{193 \times 38}{5 \times 38} = \frac{7314}{190} \]

For \( \frac{193}{38} \):
\[ \frac{193}{38} = \frac{193 \times 5}{38 \times 5} = \frac{965}{190} \]

### Step 5: Subtract the Two Fractions
Now we can subtract the two fractions:

\[ \frac{7314}{190} - \frac{965}{190} = \frac{7314 - 965}{190} = \frac{6349}{190} \]

### Step 6: Simplify the Result
The fraction \( \frac{6349}{190} \) cannot be simplified further since 6349 and 190 have no common factors other than 1.

### Final Answer
The length of the remaining rope is \( \frac{6349}{190} \) meters.

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