Question

The product of the reciprocal of 2 consecutive integers is 1/30. What are the numbers?

Answer 1

There are two possibilities:

• `5` and `6`
• `-6` and `-5`

Explanation:

`1/5*1/6 = 1/30`

`1/(-6)*1/(-5) = 1/30`

Answer 2

There are two possibilities: `-6,-5` and `5,6`

Explanation:

Call the two integers `a` and `b`.

The reciprocals of these two integers are `1/a` and `1/b`.

The product of the reciprocals is `1/axx1/b=1/(ab)`.

Thus, we know that `1/(ab)=1/30`.

Multiply both sides by `30ab` or cross-multiply to show that `ab=30`.

However, this doesn't really solve the problem: we have to address that fact that the integers are consecutive. If we call the first integer `n`, then the next consecutive integer is `n+1`. Thus, we can say that instead of `ab=30` we know that `n(n+1)=30`.

To solve `n(n+1)=30`, distribute the left-hand side and move the `30` to the left hand side as well to obtain `n^2+n-30=0`. Factor this into `(n+6)(n-5)=0`, which implies that `n=-6` and `n=5`.

If `n=-6` then the next consecutive integer is `n+1=-5`. We see here that the product of their reciprocals is `1/30`:

`1/(-6)xx1/(-5)=1/30`

If `n=5` then the next consecutive integer is `n+1=6`.

`1/5xx1/6=1/30`