Question

A piece of cloth costs rs 200. if the piece was 5 m longer and each metre of cloth costs rs.2 less,the cost of the piece would have remain unchanged . How long is the piece and what is the original rate per metres

Answer 1

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

Answer 2

$\textbf{\underline{\underline{According\:to\:the\:Question}}}$

Assumption

Piece of cloth is p metre

Cost :-

= Rs.n/metre

Situation

p × n = 200 ....... (1)

Also

(p + 5)(n - 2) = 200

pn + 5n - 2p - 10 = 200

200 + 5n - 2p - 10 = 200

5n - 2p - 10 = 0

5n = 2p + 10

$\tt{\rightarrow n=\dfrac{2p+10}{5}}$

Substitute the value in (1)

$\tt{\rightarrow\dfrac{p(2p+10)}{5}=200}$

p(2p + 10) = 200 × 5

2p² + 10p = 1000

p² + 5p - 500 = 0

p² + 25p - 20p - 500 = 0

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

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

(p + 25)=0

p = -25

p - 20 = 0

p = 20

$\textbf{\underline{Negative\;value\;is\;rejected}}$

p = 20m

Substitute the value in (1),

20n = 200

n = 10

$\Large{\boxed{\sf\:{Cloth=20m\;long\;Original\;Price=10\;m}}}$