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

Options

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

Answer

To solve the problem, we need to find the lengths of two parts of an 85 m long rod, where one part is \( \frac{2}{3} \) of the other part. Let's break it down step by step.

### Step 1: Define the shorter part
Let the shorter part be \( x \) meters.

### Step 2: Define the longer part
Since the total length of the rod is 85 meters, the longer part can be expressed as:
\[ \text{Longer part} = 85 - x \]

### Step 3: Set up the equation
According to the problem, the shorter part \( x \) is \( \frac{2}{3} \) of the longer part \( 85 - x \). We can write this relationship as:
\[ x = \frac{2}{3}(85 - x) \]

### Step 4: Clear the fraction
To eliminate the fraction, multiply both sides of the equation by 3:
\[ 3x = 2(85 - x) \]

### Step 5: Distribute on the right side
Now, distribute the 2 on the right side:
\[ 3x = 170 - 2x \]

### Step 6: Combine like terms
Next, add \( 2x \) to both sides to combine the \( x \) terms:
\[ 3x + 2x = 170 \]
\[ 5x = 170 \]

### Step 7: Solve for \( x \)
Now, divide both sides by 5 to find \( x \):
\[ x = \frac{170}{5} = 34 \]

### Step 8: Conclusion
The shorter part of the rod is \( 34 \) meters.