Question

The difference of the reciprocals of two consecutive integers is 1/72. What are the two integers?

Answer 1

`8,9`

Explanation:

Let the consecutive integers be `x and x+1`

The difference of their reciprocals is equal to `1/72`

`rarr1/x-1/(x+1)=1/72`

Simplify the left side of the equation

`rarr((x+1)-(x))/((x)(x+1))=1/72`

`rarr(x+1-x)/(x^2+x)=1/72`

`rarr1/(x^2+x)=1/72`

The numerators of the fractions are equal, so as the denominators

`rarrx^2+x=72`

`rarrx^2+x-72=0`

Factor it

`rarr(x+9)(x-8)=0`

Solve for the values of `x`

`color(green)(rArrx=-9,8`

Consider the positive value to get the correct answer

So, the integers are `8` and `9`