a piece of cloth costs rs 200.if the piece were 5m longer and each metre of cloth costed rs 2 less the cost of the piece would have remained unchanged.how long is the piece and what is he original rate per metre?
Answers
Let the length of the cloth be ‘x’ and cost of cloth per meters be Rs y
Then, xy = 200
and, y = 200/x → (1)
Given that, if the piece were 5 m longer, and each meter of cloth costs Rs 2 less
So, we get,
⇒ (x+5)(y – 2) = 200
⇒xy – 2x + 5y – 10 = 200
Putting ‘y’ value from (1),
We get, 200 – 2x + 5(200/x) – 10 = 200
⇒ (1000/x) – 2x = 200 – 190
⇒ 1000 – 2x² = 10x
⇒ 500 – x² = 5x
⇒ x²+5x – 500 = 0
⇒ x² + 25x – 20x – 500 =0
⇒ x(x+25) – 20(x+25) = 0
⇒ (x – 20) (x + 25) = 0
∴ x = 20 or x = -25
But length cannot be negative, therefore x = 20 m
Thus, y = 200/x
= 200/20
= 10
Therefore, the length of the cloth be 20 m and cost of cloth per meter is Rs 20.
There's an alternate method too:
Let the length of the original piece be x
Now, it's price = Rs.200
So, Price per m = 200/x .....(1)
Now, if the cloth was 5 m longer, its length = x+5
And, Price per m would be 200/x+5 .....(2)
(As total price is unchanged)
Or, Price per m would be Rs.2 less than the original one i.e.
200/x -2 .....(3)
Equating (2) & (3),
We get, 200/x+5 = 200/x - 2
⇒ 200/x+5 = 200-2x/x
⇒ 200x/x+5 = 200 -2x
⇒ 200x = (x+5)(200-2x)
⇒ 200x = 200x -2x² +1000 -10x
Cancelling 200x from both sides,
We get, -2x² -10x +1000 = 0
Solving the quadratic equation, we'll get the answer. I'm not presenting it again as I have done it in the first method.
Now, Substituting values in the formula,
So, Root 1 = 10 + √ [10² - 4(- 2) (1000) ] / -4
= 10 + 90 / -4
= 100 / -4
= -25
And, Root 2 = 10 -√ [10² - 4 (-2) (1000) ] / -4
= 10 - 90 / -4
= -80 / -4
= 20
Now, the length of the original piece cannot be negative.
So, the length of original piece is 20 m (Root 2)
And, original rate = 200/x
= 200/20
= 10 Rs./m
Hope This Helps :)
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