A string of length, 221 metre is cut into two parts such that one part is `9/4th` as long as the rest of the string, then the difference between the larger piece the shorter piece is

Options

A. 58 m
B. 53 m
C. 85 m
D. none of these

Answer

To solve the problem, we need to find the lengths of the two parts of the string and then determine the difference between them. Let's break it down step by step.

### Step 1: Define the lengths of the two parts
Let the length of the shorter part of the string be \( x \) meters. According to the problem, the longer part is \( \frac{9}{4} \) times as long as the shorter part. Therefore, the length of the longer part can be expressed as:

\[ \text{Length of longer part} = \frac{9}{4}x \]

### Step 2: Set up the equation
The total length of the string is given as 221 meters. Therefore, we can set up the following equation:

\[ x + \frac{9}{4}x = 221 \]

### Step 3: Combine like terms
To combine the terms on the left side, we need a common denominator. The term \( x \) can be rewritten as \( \frac{4}{4}x \):

\[ \frac{4}{4}x + \frac{9}{4}x = 221 \]

This simplifies to:

\[ \frac{4x + 9x}{4} = 221 \]

\[ \frac{13x}{4} = 221 \]

### Step 4: Solve for \( x \)
To eliminate the fraction, multiply both sides of the equation by 4:

\[ 13x = 221 \times 4 \]

Calculating the right side:

\[ 13x = 884 \]

Now, divide both sides by 13 to find \( x \):

\[ x = \frac{884}{13} \]

Calculating this gives:

\[ x = 68 \]

### Step 5: Find the lengths of both parts
Now that we have \( x \), we can find the lengths of both parts:

- Shorter part: \( x = 68 \) meters
- Longer part: \( \frac{9}{4}x = \frac{9}{4} \times 68 = 153 \) meters

### Step 6: Calculate the difference between the two parts
The difference between the longer part and the shorter part is:

\[ \text{Difference} = 153 - 68 = 85 \text{ meters} \]

### Final Answer
The difference between the larger piece and the shorter piece is **85 meters**.

---