Question

Kate cut a 100-foot rope into two pieces. One piece was 5 feet longer than 4 times the length of the others. What is the length of each piece?

Answer 1

`color(blue)("Longer piece"=81" ft")`
`color(blue)("Shorter piece"=19" ft")`

Explanation:

Let the shorter piece of rope be

`x`.

Then the longer piece is 4 times this plus 5.

`4x+5`

The total length is 100 ft

`:.`

`4x+5+x=100`

Solve for `x`:

`5x=100-5=95`

`x=95/5=19`

Longer piece is:

`4x+5`

`4(19)+5=81`

Shorter piece is:

`x=19`

Answer 2

`81` feet & `19` feet

Explanation:

Let the `x` & `y` feet be the lengths of longer & shorter parts of rope of original length `100` feet.

The sum of lengths of cut-parts must be equal to the length of original rope hence

`x+y=100\ ............(1)`

Given that longer part of length `x` is `5` feet longer than `4` times the length of smaller part of length `y` hence

`x=4y+5`

`x-4y=5\ ..........(2)`

subtracting (2) from (1), we get

`x+y-(x-4y)=100-5`

`5y=95`

`y=95/5`

`y=19`

Setting above value of `y` in (1) as follows

`x+19=100`

`x=100-19`

`x=81`

Hence, the two parts of rope are `81` feet & `19` feet