Question

A piece of cloth costs 200$.if the piece was 5m longer, and the cost of each metre of clth was 2$ less, the cost of the piece would have remained unchanged. How long is the piece and what is its original number of persons.

Answer 1

The piece is 20 m long and costs $10 per meter

Explanation:

The total cost of the cloth in $ can be expressed as

C = xu

Where x is the length of the cloth in m, and u is the unit cost in $/m

The problem states that if the length was 5 meters more and the unit cost was $2 less, the cost would not change, that is

C = (x+5)(u-2)

Operating:

C = xu-2x+5u-10

Since C = xu, then

-2x+5u=10 [1]

We also know that

xu=200 [2]

Isolating u in [1] we get

`u=(10+2x)/(5)`

Replacing into [2]

`x(10+2x)/(5)=200`

Simplifying

`10x+2x^2=1000 \\=> x^2+5x-500=0`

Factoring

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

Which gives x=20, x=-25. We can only use the positive value of x, so

x=20 m and therefore

u=200/20=10$

So the piece is 20 m long and costs $10 per meter

Answer 2

20 m long,

each piece of cloth for $10

Explanation:

Let x m be the length of the piece of cloth.

Original cloth:

Length = x m

Cost = $200

Cost per meter
`(\$200)/(x)`

Changed cloth:

Length = x + 5 m (5 m longer)

Cost = $200

Cost per meter
`(\$200)/(x)-\$2` (each metre of cloth was $2 less)

Hence,

`\left((200)/(x)-2\right)\cdot (x+5)=200`

Multiply the equation by x:

`(200-2x)(x+5)=200x\\ \\200x+1,000-2x^2-10x=200x\\ \\-2x^2-10x+1,000=0\\ \\x^2+5x-500=0\\ \\D=5^2-4\cdot(-500)=25+2,000=2,025\\ \\x_(1,2)=(-5\pm√(2,025))/(2)=(-5\pm 45)/(2)=-25,\ 20`

The length cannot be negative, so x = 20 m, each metre for $10.