Question

The total cost of a certain length of a piece of cloth is 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 its original rate per metre. CBSE Board Paper 2015 abroad

Answer 1

Let length of the cloth = x 

Cost of cloth per m = y 

x * y = 200

y = 200/x ....(1)

 Conditions-

1.If the piece were 5m longer

2.Each metre of cloth costed Rs 2 less

(x+5)(y -2) = 200.

xy -2x + 5y -10 = 200

Substituting  value of (1)

1000/x - 2x = 200 - 190

1000 -2x²= 10x

500 -x²= 5x

x²+5x - 500 = 0

(x-20) (x+25) = 0

x= 20 (neglecting x= -25)

y = 200/x = 200/20 = 10

 Length of the piece of cloth is 20 m and rate per metre is Rs 10

Answer 2

Let the length of the original piece be x
Now, its price = Rs.200

Price per m = 200/x --Eq.1

Now, if the cloth was 5 m longer, its length = x+5
and
Price per m would be 200/x+5 --Eq.2
(As total price is unchanged)
Or
Price per m would be Rs.2 less than the original one i.e.
200/x -2 -- Eq.3

Equating Eq.2 and Eq.3,

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
Cut 200x from both sides
-2x² -10x +1000 = 0

Solve the quadratic equation, you'll get the answer .
Substituting values in the formula.

Root 1 = 10 +√[10² - 4(-2)(1000)]/ -4
= 10+90/-4
= 100/-4
= -25

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 = Rs.10