Question

A positive integer is 4 less than another. If the sum of the reciprocal of the smaller and twice the reciprocal of the larger is frac(5,6), then find the two integers.

Answer 1

Let the smaller number is x.

So the bigger number is x+4.

Reciprocal of smaller number is
`(1)/(x)` and reciprocal of larger number is
`(1)/(x+4)`

SO our equation is

`(1)/(x) + 2* (1)/(x+4) = (5)/(6)`

`(1)/(x) + (2)/(x+4) = (5)/(6)`

`(x+4 + 2x)/(x(x+4)) = (5)/(6)`

`(3x+4)/(x(x+4)) = (5)/(6)`

by cross multiplication rule

`6(3x+4) = 5(x(x+4))`

`18x+ 24= 5x^2 + 20x`

`5x^2 + 20x - 18x - 24 = 0`

`5x^2 +2x - 24 = 0`

Now we can solve it by factorization.

As we can see -24*5 = -120.

So we can write -120 as 12 * (-10) = -120 because 12 - 10 = 2

SO we can write expression as

`5x^2 + 12x - 10x - 24 = 0`

`x(5x+12) - 2(5x+12) = 0`

`(5x+12)(x-2) = 0`
So x = 2 or x =
`(-12)/(5)`

But we have to choose positive integer. SO we will choose x = 2.

SO the smaller number is 2 and bigger number is 6.