Question

A positive integer is twice another. The difference of the reciprocals of the two positive integers is 1/8. Find the two integers?

Answer 1

The integers are 4 and 8.

Explanation:

Let's look at the first statement, "A positive integer is twice another." If we wanted to represent the first integer as the letter `a` and the second as `b`, we could write this as a mathematical expression:

`a=2b`

The next statement is "The difference of the reciprocals of the two positive integers is 1/8" Understanding that the reciprocal of `arArr1/a` and `brArr1/b`, we can write this as follows:

`abs(1/a-1/b)=1/8`

I put the absolute sign in there because we just know the difference between the two, and not necessarily the order. Because we wrote the earlier expression `a=2b` we can write the expression in terms of `b`:

`abs(1/(2b)-1/b)=1/8`

we need to make the denominators the same for both fractions on the left hand side, so we will multiply `1/b` by `2/2`:

`abs(1/(2b)-2/(2b))=1/8`

`abs(-1/(2b))=1/8`

`1/(2b)=1/8`

`2b=8`

`color(green)(b=4)`

now that we know `b`, we can go back and solve for `a`:

`a=2b`

`a=2(4)`

`color(green)(a=8)`