A piece of cloth costs RS.75. If the piece were 4 metre longer and each metre costs RS.5 less, the total cost would remain unchanged. How long is the piece?

Answer

To solve the problem step by step, let's denote the length of the piece of cloth as \( x \) meters.

### Step 1: Establish the cost per meter
The total cost of the piece of cloth is Rs. 75. Therefore, the cost per meter is given by:
\[ \text{Cost per meter} = \frac{75}{x} \]

### Step 2: Determine the new length and new cost per meter
If the piece were 4 meters longer, the new length would be:
\[ \text{New length} = x + 4 \]
The new cost per meter, if each meter costs Rs. 5 less, would be:
\[ \text{New cost per meter} = \frac{75}{x + 4} \]
Since each meter costs Rs. 5 less, we can express this as:
\[ \frac{75}{x} - \frac{75}{x + 4} = 5 \]

### Step 3: Set up the equation
Now we will set up the equation based on the cost per meter:
\[ \frac{75}{x} - \frac{75}{x + 4} = 5 \]

### Step 4: Solve the equation
To eliminate the fractions, we can multiply through by \( x(x + 4) \):
\[ 75(x + 4) - 75x = 5x(x + 4) \]
This simplifies to:
\[ 75x + 300 - 75x = 5x^2 + 20x \]
Thus, we have:
\[ 300 = 5x^2 + 20x \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ 5x^2 + 20x - 300 = 0 \]
Dividing the entire equation by 5 simplifies it to:
\[ x^2 + 4x - 60 = 0 \]

### Step 6: Factor the quadratic equation
Next, we factor the quadratic:
\[ (x + 10)(x - 6) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives:
\[ x + 10 = 0 \quad \Rightarrow \quad x = -10 \quad (\text{not valid, as length cannot be negative}) \]
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]

### Conclusion
The length of the piece of cloth is:
\[ \text{Length} = 6 \text{ meters} \]