Question

A piece of cloth costs$Rs.35$. If the piece were \[4m\] longer and each meter costs $\operatorname{Re} .1$less, the cost would remain unchanged the length of the piece is ………..$m$

Answer 1

Hint: Here, we go through a unitary method for solving the question. First find the cost of 1m cloth and after applying the condition of question, find the cost of longer cloth. The unitary method is used to find the value of a single unit from a given multiple.

Let the length of the piece be $x$ $m$.
Cost price of cloth $ = Rs.35$
Cost price of $1m$ cloth $ = Rs.\dfrac{{35}}{x}$ by applying a unitary method.
According to the question,
Now the cost price of unit cloth become $\operatorname{Re} .1$less for the longer cloth i.e. $\left( {\dfrac{{35}}{x} - 1} \right)$
Cost of ${\text{(}}x + 4)m$cloth is
We find that the cost price of new unit cloth is $Rs.\left( {\dfrac{{35}}{x} - 1} \right)$
Therefore cost price of ${\text{(}}x + 4)m$is $Rs.{\text{(}}x + 4)\left( {\dfrac{{35}}{x} - 1} \right)$
According to the question, the cost price is the same.
$
   \Rightarrow {\text{(}}x + 4)\left( {\dfrac{{35}}{x} - 1} \right) = 35 \\
   \Rightarrow \;\;35 - x + \dfrac{{140}}{x} - 4 = 35 \\
   \Rightarrow \;\;35x - {x^2} + 140 - 4x = 35x \\
   \Rightarrow \;\;{x^2} + 4x - 140 = 0 \\
   \Rightarrow \;\;{x^2} + 14x - 10x - 140 = 0 \\
   \Rightarrow \;\;x(x + 14) - 10(x + 14) = 0 \\
   \Rightarrow \;\;(x + 14)(x - 10) = 0 \\
 $
$ \Rightarrow x + 14 = 0$And $x - 10 = 0$
$\therefore \;\;x = - 14\;$And $x = 10$
Length cannot be negative.
Hence the length of the piece is$10m$.

Note: Whenever we face such a type of question first calculate the cost of the unit. For this question we have to first assume the original length. And then solve according to the question statement to get an answer.