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

Jun 26, 2016

$\frac{1}{2} + \frac{1}{4} = \frac{3}{4}$

#### Explanation:

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

$\frac{1}{x} + \frac{1}{x + 2} = \frac{3}{4}$

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)

$4 \left(x + 2\right) + 4 x = 3 x \left(x + 2\right)$

$4 x + 8 + 4 x = 3 {x}^{2} + 6 x$
$3 {x}^{2} - 2 x - 8 = 0$

$\left(3 x + 4\right) \left(x - 2\right) = 0$

$x = 2 \mathmr{and} x = - \frac{4}{3} \text{(reject)}$

The numbers are 2 and 4.
Check:
$\frac{1}{2} + \frac{1}{4} = \frac{3}{4}$