Question

A piece of cloth costs rupees 75. If the piece is four meters longer and each meter costs rupees 5 less, the cost remains unchanged. What is the length of the piece?
A.12 meters
B.8 meters
C.10 meters
D.6 meters

Answer 1

Hint: First assume 2 variables for length and cost. Now using two conditions given in question find two equations in terms of two variables. By using a substitution method, solve the both equations and then find values of both the variables. By which you get the value of the length of the original piece. This is the required result.

Complete step-by-step answer:
Given condition in the question can be written as:
The cost of a cloth piece is Rs.75
The next condition can be written in the form of:
If length is increased by 4, cost is decreased by 5, total cost is Rs.75
Let us assume the length of the cloth piece is given by xm.
Let us assume the cost of cloth pieces is Rs.y per m.
By condition (1), we can say that the cost of piece:
\[Total{\rm{ cost = xy}}\]
By substituting value of 75 in the equation, we get it as:
\[xy = 75{\rm{ }}\] …………..(1)
By dividing with x on both sides we get value of y as:
\[y = \dfrac{{75}}{x}\] …………..(2)
Now using second condition, length of new piece is given by:
\[x + 4\]
Now using second condition, cost per m of new piece is given by:
\[y - 5\]
Now using second condition, total cost of new piece is given by:
\[\left( {x + 4} \right)\left( {y - 5} \right) = {\rm{ new cloth cost}}\]

Substitution method: The method of solving a system of equations. It works by solving one of the equations for one of the variables to get in terms of other variables, then plugging this back into another equation, solving for the other variable. By this you can find both the variables. This method is generally used when there are 2 variables. For more variables it will be tough to solve.

By substituting value of 75 in the equation, we get it as:
\[\left( {x + 4} \right)\left( {y - 5} \right) = 75\]
By using distribution law,\[a\left( {b + c} \right) = ab + ac\]we get equation:
\[xy - 5x + 4y - 20 = 75\]
By using substitution method, we get the equation to be:
\[75 - 5x + 4y - 20 = 75\]
By simplifying we get the equation in form \[4y - 5x = 20\]
By substituting equation (2) in the equation, we get:
\[\dfrac{{4\left( {75} \right)}}{x} - 5x = 20\]
By taking least common multiple and simplifying, we get:
\[5{x^2} + 20x - 300 = 0\]
By taking 5 common and dividing with 5, we get it as:
\[{x^2} + 4x - 60 = 0\]
By factorization method, we get 2 numbers whose sum is 4, product is -60 as 10, -6.
By using this we write:
\[{x^2} + 10x - 6x - 60 = 0\]
By taking x, -6 common from first, last two terms, we get:
\[x\left( {x + 10} \right) - 6\left( {x + 10} \right) = 0{\rm{ }} \Rightarrow \left( {x - 6} \right)\left( {x + 10} \right) = 0\]
Roots of above all x=6, -10, as x is length negative not possible.
The length of the original piece is 6m.
Therefore, the option (d) is correct.

Note: Be careful while writing equations from the conditions. While substituting don’t forget to write the constant 75.4m longer means \[{\rm{x}} + {\rm{4}}\]. Students confuse and write 4x which is wrong. If you want, you can calculate y and check whether it is satisfying or not (just as verification).