The cost of a piece of cable wire is Rs.35/-. If the length of the piece of wire is 4 meters more each meter costs Rs. 1/- less, the cost would remain unchanged. What is the length of the wire ?

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Define the Variables
Let the length of the wire be \( x \) meters.

**Hint:** Start by defining the unknowns clearly to set up the problem.

### Step 2: Determine the Cost per Meter
The total cost of the wire is Rs. 35. Therefore, the cost per meter of the wire is given by:
\[ \text{Cost per meter} = \frac{35}{x} \]

**Hint:** Remember that the total cost divided by the length gives the cost per unit length.

### Step 3: Set Up the New Conditions
According to the problem, if the length of the wire is increased by 4 meters, the cost per meter decreases by Rs. 1. The new length will be \( x + 4 \) meters, and the new cost per meter will be:
\[ \text{New cost per meter} = \frac{35}{x + 4} \]
We also know that this new cost per meter is Rs. 1 less than the original cost per meter:
\[ \frac{35}{x + 4} = \frac{35}{x} - 1 \]

**Hint:** Set up an equation based on the conditions given in the problem.

### Step 4: Cross-Multiply to Eliminate Fractions
Cross-multiplying the equation gives:
\[ 35x = 35(x + 4) - x(x + 4) \]

**Hint:** Cross-multiplying helps to simplify the equation by removing fractions.

### Step 5: Expand and Rearrange the Equation
Expanding the equation:
\[ 35x = 35x + 140 - x^2 - 4x \]
Rearranging gives:
\[ 0 = -x^2 - 4x + 140 \]
or
\[ x^2 + 4x - 140 = 0 \]

**Hint:** Always rearrange the equation to set it to zero for easier factorization or application of the quadratic formula.

### Step 6: Factor the Quadratic Equation
Now we need to factor the quadratic equation:
\[ x^2 + 14x - 10x - 140 = 0 \]
This can be factored as:
\[ (x - 10)(x + 14) = 0 \]

**Hint:** Look for two numbers that multiply to the constant term and add to the coefficient of the linear term.

### Step 7: Solve for \( x \)
Setting each factor to zero gives:
\[ x - 10 = 0 \quad \Rightarrow \quad x = 10 \]
\[ x + 14 = 0 \quad \Rightarrow \quad x = -14 \]

Since the length cannot be negative, we discard \( x = -14 \).

**Hint:** Always consider the context of the problem when interpreting solutions.

### Step 8: Conclusion
The length of the wire is \( x = 10 \) meters.

**Final Answer:** The length of the wire is 10 meters.

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