Question

The sum of a number and its reciprocal is equal to 4 times the reciprocal, What is the number?

Answer 1

There are two possibilities:

`x=-sqrt(3)` or `x=sqrt(3)`

Explanation:

Suppose the number is `x`

Then the "reciprocal" of this number is `1/x`

We are told that:

"The sum of a number and its reciprocal is equal to 4 times the reciprocal"

Therefore:

`x + 1/x = 4 xx 1/x`

`:. x = 4 xx 1/x - 1 xx 1/x`

`:. x = 3 xx 1/x`

`:. x^2 = 3`

`:. x = +-sqrt(3 )`

Answer 2

`n + 1/n = 4 * 1/n`

on the left side, multiply n by `n/n` (same as multiplying by 1) to get a common denominator so you can add the numerators...

`n^2 /n + 1/n = 4/n`

giving:

`n^2 + 1 = 4`

`n^2 = 3`

so...

`n = +-sqrt(3)`

...check your work, when you can:

`sqrt(3) + 1/sqrt(3) = 4 * 1/sqrt(3)`

`sqrt(3)*sqrt(3)/sqrt(3) + 1/sqrt(3) = 4/sqrt(3)`

`3/sqrt(3) + 1/sqrt(3) = 4/sqrt(3)`

`4/sqrt(3) = 4/sqrt(3)`

(the same sequence also applies for the `-sqrt(3)` case

Answer 3

I got two possible numbers: `+-sqrt(3)`

Explanation:

Let us call the unknown number `x`; we get:
`x+1/x=4*1/x`
rearrange:
`x^2+1=4`
(with `x!=0`)
`x^2=3`
`x=+-sqrt(3)`