Question

A positive integer is twice another.the sum of the reciprocal of the two positive integer is 3/14. Find the integers

Answer 1

`\huge\boxed{14\ \text{and}\ 7}`

Explanation:

`n,\ m-\text{positive integer}\\\\n=2m-\text{a positive integer is twice another}\\\\(1)/(n)+(1)/(m)=(3)/(14)-\text{the sum of the reciprocal of the two positive integer is }\ (3)/(14)\\\\\text{We have the system of equations:}\\\\\left\{\begin{array}{ccc}n=2m&(1)\\(1)/(n)+(1)/(m)=(3)/(14)&(2)\end{array}\right`

`\text{Substitute (1) to (2):}\\\\(1)/(2m)+(1)/(m)=(3)/(14)\\\\(1)/(2m)+(1\cdot2)/(m\cdot2)=(3)/(14)\\\\(1)/(2m)+(2)/(2m)=(3)/(14)\\\\(1+2)/(2m)=(3)/(14)\\\\(3)/(2m)=(3)/(14)\Rightarrow2m=14\qquad\text{divide both sides by 2}\\\\(2m)/(2)=(14)/(2)\\\\\boxed{m=7}`

`\text{Substitute it to (1):}\\\\n=2\cdot7\\\\\boxed{n=14}`