Question

The sum of the reciprocals of two consecutive even integers is 7/24. Write an equation that can be used to find the two integers. Find the two integers.

Answer 1

Assume the first even integer is x.
The next even integer will be 2 digits after this digit. So the next even integer will be x+2.

The reciprocals of these two even integers can be written as $\frac{1}{x}$ and $\frac{1}{x+2}$ respectively.

The sum of their reciprocals is 7/24. So we can set up the equation as:

$\frac{1}{x}+ \frac{1}{x+2}= \frac{7}{24}$

Taking LCM on left hand side:

$\frac{x+2+x}{x(x+2)}= \frac{7}{24} \\ \\ \frac{2x+2}{ x^{2} +2x}= \frac{7}{24}$

Cross Multiplying the denominators:

$24(2x+2)=7({ x^{2} +2x}) \\ \\ 48x+48=7 x^{2} +14x \\ \\ 7 x^{2} +14x-48x-48=0 \\ \\ 7 x^{2} -34x-48=0$

Using the quadratic formula to solve the above equation:
$7 x^{2} -34x-48=0 \\ \\ x= \frac{34+- \sqrt{1156+1344} }{14} \\ \\ x= \frac{34+- \sqrt{2500} }{14} \\ \\ x=\frac{34+-50 }{14} \\ \\ x=6 \\ x=- \frac{8}{7}$

Since, the value of x can only be an integer, we discard the fractional value and keep x=6

So the first even integer is 6 and the next even integer is 8. The sum of reciprocals of 6 and 8 is 7/24