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

Apr 26, 2017

$8 , 9$

#### Explanation:

Let the consecutive integers be $x \mathmr{and} x + 1$

The difference of their reciprocals is equal to $\frac{1}{72}$

$\rightarrow \frac{1}{x} - \frac{1}{x + 1} = \frac{1}{72}$

Simplify the left side of the equation

$\rightarrow \frac{\left(x + 1\right) - \left(x\right)}{\left(x\right) \left(x + 1\right)} = \frac{1}{72}$

$\rightarrow \frac{x + 1 - x}{{x}^{2} + x} = \frac{1}{72}$

$\rightarrow \frac{1}{{x}^{2} + x} = \frac{1}{72}$

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

$\rightarrow {x}^{2} + x = 72$

$\rightarrow {x}^{2} + x - 72 = 0$

Factor it

$\rightarrow \left(x + 9\right) \left(x - 8\right) = 0$

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 $9$