Question

How do you find two consecutive even integers, the sum of whose reciprocals is 3/4?

Answer 1

`1/2 + 1/4 =3/4`

Explanation:

Let the first number be `x`
The second is `x+2`

`1/x+ 1/(x+2) = 3/4`

`color(red)(4x(x+2)xx1)/x + color(red)((4x(x+2)xx1)/((x+2))`
= `color(red)((4x(x+2)xx3)/4 " multiply by" color(red)(4x(x+2)`

`color(red)(4cancelx(x+2)xx1)/cancelx + color(red)((4xcancel(x+2)xx1)/(cancel(x+2))`
= `color(red)((cancel4x(x+2)xx3)/cancel4 " multiply by" color(red)(4x(x+2)`

`4(x+2) + 4x = 3x(x+2)`

`4x + 8 +4x =3x^2 + 6x`
`3x^2-2x-8=0`

`(3x+4)(x-2)=0`

`x=2 or x = -4/3 "(reject)"`

The numbers are 2 and 4.
Check:
`1/2 + 1/4 =3/4`