Question

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

Answer 1

There are infinitely many, but the one midway between `-2/6` and `-1/6` is `-1/4`

Explanation:

Note that `-1/6` and `-2/6` are rational numbers, with `-2/6 < -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:

`1/2((-1/6)+(-2/6)) = (-1/12)+(-2/12) = -3/12 = -1/4`

Answer 2

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 `-(2)/(6)` and `-(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.

`-(2)/(6) = -(4)/(12)`

`-(1)/(6) = - (2)/(12)`

Now you can see that `-(3)/(12)` lies right between them

`larr` ...... `-(4)/(12)` ........ `-(3)/(12)` .......... `- (2)/(12)` ........ `rarr`.

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

Example:
Find three fractions between `(1)/(5)` and `(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.

`larr`...... `(2)/(10)` .......... `(3)/(10)` .......... `(4)/(10)` .... `rarr`

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

This time, try 25ths as the denominator.

`(1)/(5) = (5)/(25)`

`(2)/(5) = (10)/(25)`

Now you can easily see four fractions lying between `(1)/(5)` and `(2)/(5)`

`(1)/(5) = (5)/(25)`

`(6)/(25)`

`(7)/(25)`

`(8)/(25)`

`(9)/(25)`

`(2)/(5) = (10)/(25)`