An 85m long rod is divided into two parts. If one part is `(2)/(3)` of the other part, then the longer part (in metres) is :

Options

A. 34
B. `56(2)/(3)`
C. 85
D. 51

Answer

To solve the problem step by step, we will denote the lengths of the two parts of the rod.

### Step-by-Step Solution:

1. **Define the Parts**:
Let the length of the first part be \( x \) meters. According to the problem, the second part is \( \frac{2}{3} \) of the first part. Therefore, the length of the second part can be expressed as:
\[ \text{Second part} = \frac{2}{3}x \]

2. **Set Up the Equation**:
The total length of the rod is given as 85 meters. Therefore, we can write the equation:
\[ x + \frac{2}{3}x = 85 \]

3. **Combine Like Terms**:
To combine the terms on the left side, we need a common denominator. The common denominator for \( x \) and \( \frac{2}{3}x \) is 3. Thus, we rewrite \( x \) as \( \frac{3}{3}x \):
\[ \frac{3}{3}x + \frac{2}{3}x = 85 \]
This simplifies to:
\[ \frac{5}{3}x = 85 \]

4. **Solve for \( x \)**:
To isolate \( x \), we multiply both sides of the equation by 3:
\[ 5x = 85 \times 3 \]
Calculating the right side gives:
\[ 5x = 255 \]
Now, divide both sides by 5:
\[ x = \frac{255}{5} = 51 \]

5. **Find the Second Part**:
Now that we have \( x = 51 \), we can find the length of the second part:
\[ \text{Second part} = \frac{2}{3} \times 51 = \frac{102}{3} = 34 \]

6. **Identify the Longer Part**:
We have the lengths of both parts: the first part is 51 meters and the second part is 34 meters. The longer part is:
\[ \text{Longer part} = 51 \text{ meters} \]

### Final Answer:
The longer part of the rod is **51 meters**.