Question

A piece of cloth costs Rs 35. If the piece was 4 m longer and each metre cost Rs 1 less, the cost would remain unchanged. How long is the piece?

Answer 1

Step 1:

Form a quadratic equation:

Let the length of the piece be $x\mathrm{metres}$.

Then, the cost of $1\mathrm{metre}$ will be $\mathrm{Rs}\frac{35}{x}$.

If the piece was $4\mathrm{metre}$ longer, then,

$\mathrm{new}\mathrm{length}=\left(x+4\right)\mathrm{metres}$

According to the question,

$\begin{array}{rcl}\left(x+4\right)\left(\frac{35}{x}-1\right)& =& 35\\ & \Rightarrow & 35-x+\frac{140}{x}-4=35\\ & \Rightarrow & \frac{35x-x^2+140-4x}{x}=35\\ & \Rightarrow & 35x-x^2+140-4x=35x\\ & \Rightarrow & x^2+4x-140=0\end{array}$

Step 2:

Determine the length of the piece:

Use factorization method to find the roots of equation.

$$\begin{gathered} x^2+4x-140=0 \\ \begin{array}{rcl}& \Rightarrow & x^2+14x-10x-140=0\\ & \Rightarrow & x\left(x+14\right)-10\left(x+14\right)=0\\ & \Rightarrow & \left(x+14\right)\left(x-10\right)=0\\ & \Rightarrow & x+14=0\mathrm{or}x-10=0\\ & \Rightarrow & x=-14\mathrm{or}x=10\end{array} \end{gathered}$$

But the length cannot be negative, rejecting $x=-14$

$\therefore x=10$

Hence, the length of the piece is $10\mathrm{metres}$.