# What rational number is between -2/6 and -1/6 ?

Nov 21, 2017

There are infinitely many, but the one midway between $- \frac{2}{6}$ and $- \frac{1}{6}$ is $- \frac{1}{4}$

#### Explanation:

Note that $- \frac{1}{6}$ and $- \frac{2}{6}$ are rational numbers, with $- \frac{2}{6} < - \frac{1}{6}$.

Like any two distinct real numbers, there are infinitely many rational numbers between them.

Since they are both rational, their average is also a rational number.

We can arrive at the average by adding the two numbers then halving the result.

So:

$\frac{1}{2} \left(\left(- \frac{1}{6}\right) + \left(- \frac{2}{6}\right)\right) = \left(- \frac{1}{12}\right) + \left(- \frac{2}{12}\right) = - \frac{3}{12} = - \frac{1}{4}$

Nov 25, 2017

When two fractions are too close to have another one in between, just re-write them with larger denominators to spread them apart enough to fit other fractions in.

#### Explanation:

It looks like there is no room between $- \frac{2}{6}$ and $- \frac{1}{6}$, but you can stretch them apart to get more room by writing the same fractions, only this time using the common denominator of 12.

$- \frac{2}{6} = - \frac{4}{12}$

$- \frac{1}{6} = - \frac{2}{12}$

Now you can see that $- \frac{3}{12}$ lies right between them

$\leftarrow$ ...... $- \frac{4}{12}$ ........ $- \frac{3}{12}$ .......... $- \frac{2}{12}$ ........ $\rightarrow$.

If you need even more room, just use an even larger denominator.

Example:
Find three fractions between $\frac{1}{5}$ and $\frac{2}{5}$

Using tenths as the denominator doesn't open up enough room for 3 fractions to fit between them. Only one fraction (3/10) fits.

$\leftarrow$...... $\frac{2}{10}$ .......... $\frac{3}{10}$ .......... $\frac{4}{10}$ .... $\rightarrow$

But don't give up if that happens.
Just try again with a bigger denominator.

This time, try 25ths as the denominator.

$\frac{1}{5} = \frac{5}{25}$

$\frac{2}{5} = \frac{10}{25}$

Now you can easily see four fractions lying between $\frac{1}{5}$ and $\frac{2}{5}$

$\frac{1}{5} = \frac{5}{25}$

$\frac{6}{25}$

$\frac{7}{25}$

$\frac{8}{25}$

$\frac{9}{25}$

$\frac{2}{5} = \frac{10}{25}$