Question: A piece of cloth costs Rs 800. If the piece was 5 m longer and the rate of cost of cloth per meter was Rs 8 less, the cost of the piece would have remained unchanged. How long is the piece and what is the rate of cost per meter?


Solution:
Given,
A piece of cloth costs Rs 800.

Let the length of the cloth be x m and the rate of cost per meter be Rs y /m.

So, the cost of cloth = xy
$or, 800 = xy$
$or, x = \frac{800}{y}$ - (i)

According to the question,
If the price was 5 m longer and the rate of cost of cloth per meter was Rs 8 less, the cost of the piece would have remained unchanged.

$or, (x +5) × (y -8) = 800$
Put value of x from equation (i)
$or, (\frac{800}{y} + 5) ( y -8) = 800$
$or, \frac{800 + 5y}{y} × (y -8) = 800$
$or, (800+5y)(y -8) = 800y$
$or, 800(y -8) +5y(y-8) = 800y$
$or, 800y -6400 + 5y² -40y = 800y$
$or, 5y² +800y -40y -6400 -800y = 0$
$or, 5y² + 760 -800y -6400 = 0$
$or, 5y² -40y -6400 = 0$
$or, 5(y² -8y - 1280) = 0$
$or, y² -8y -1280 = 0$
$or, y² - (40-32)y -1280 = 0$
$or, y² -40y +32y -1280 = 0$
$or, y(y -40) +32(y -40) = 0$
$or, (y +32) (y -40) = 0$

Either,
$y +32 = 0$
$\therefore y = -32$

Or,
$y -40 = 0$
$\therefore y = 40$

The rate of cost per meter can not be negative number. So, y = 40.

Put value of y in equation (i), we get,
$or, x = \frac{800}{40}$
$\therefore x = 20$

So, (x,y) = (20,40)

Therefore, the required length of the piece of cloth is 20m and the rate of cost per meter is Rs 40/m.

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