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

May 3, 2016

There are two possibilities:

• $5$ and $6$
• $- 6$ and $- 5$

#### Explanation:

$\frac{1}{5} \cdot \frac{1}{6} = \frac{1}{30}$

$\frac{1}{- 6} \cdot \frac{1}{- 5} = \frac{1}{30}$

May 3, 2016

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

#### Explanation:

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

The reciprocals of these two integers are $\frac{1}{a}$ and $\frac{1}{b}$.

The product of the reciprocals is $\frac{1}{a} \times \frac{1}{b} = \frac{1}{a b}$.

Thus, we know that $\frac{1}{a b} = \frac{1}{30}$.

Multiply both sides by $30 a b$ or cross-multiply to show that $a b = 30$.

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 $a b = 30$ we know that $n \left(n + 1\right) = 30$.

To solve $n \left(n + 1\right) = 30$, 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 $\left(n + 6\right) \left(n - 5\right) = 0$, 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 $\frac{1}{30}$:

$\frac{1}{- 6} \times \frac{1}{- 5} = \frac{1}{30}$

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

$\frac{1}{5} \times \frac{1}{6} = \frac{1}{30}$