## Answers

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

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