37. A piece of cloth costs Rs. 200. If the piece was 5m longer and each metre of cloth costs Rs. 2 less the cost of the piece would have remained unchanged. How long is the piece and what is the original rate per metre?
Answers
Let the piece of cloth is x metre and it costs Rs.y/metre. Then by the given conditions -
∴, xy=200 -------------(1) and
(x+5)(y-2)=200
or, xy+5y-2x-10=200
or, 200+5y-2x-10=200
or, 5y-2x-10=0
or, 5y=2x+10
or, y=(2x+10)/5
Putting in (1),
x(2x+10)/5=200
or, x(2x+10)=200×5
or, 2x²+10x=1000
or, x²+5x-500=0
or, x²+25x-20x-500=0
or, x(x+25)-20(x+25)=0
or, (x+25)(x-20)=0
either, x+25=0
or, x=-25
or, x-20=0
or, x=20
Since the length of the cloth can not be negative,
∴, x=20 metre
Putting in (1),
20y=200
or, y=10
∴, the cloth is 20 m long and the original rate is Rs. 10/metre. Ans.
Solutions :-
Let the length of a piece of cloth be x
And the original rate per metre be y
According to the question,
x × y = 200
=> xy = 200 _______(i)
(x + 5) × (y - 2) = 200
=> xy - 2x + 5y - 10 = 200 ________(ii)
Putting the value of xy in equation (ii) we get,
=> 200 - 2x + 5y - 10 = 200
=> - 2x + 5y = 10
=> y = (2x + 10)/5 ______(iii)
Putting the value of y in equation (i) we get,
=> x (2x + 10)/5 = 200
=> 2x² + 10x = 200 × 5
=> 2x² + 10x - 100 = 0
=> 2(x² + 5x - 500) = 0
=> x² + 5x - 500 = 0/2
=> x² + 25x - 20x - 500 = 0
=> x(x + 25) - 20(x + 25) = 0
=> (x + 25) (x - 20) = 0
=> x = - 25 or x = 20
Length of cloth be taken positively.
Putting the value of x in equation (iii) we get,
=> y = (2 × 20 + 10)/5
=> y = 50/5 = 10
Hence,
The length of a piece of cloth = 20 m
And The original rate per metre = Rs 10 per metre