Question

What fraction is between `1/3` and `1/2` ?

Answer 1

`2/5`

Explanation:

`1/2=0.5`

`1/3=0.333333`

So

`2/5=0.4`

Answer 2

There are infinitely many possibilities, but the midpoint between `1/3` and `1/2` is `5/12`

Explanation:

For any two numbers `a` and `b`, the average `1/2(a+b)` lies between them.

So to find a fraction between `1/3` and `1/2` we can form the average:

`1/2(1/3+1/2) = 1/2(2/6+3/6) = 1/2(5/6) = 5/12`

Answer 3

`5/12`

Explanation:

First, make the denominators equal:

`(1/2) * (3/3) = 3/6`

`(1/3) * (2/2) = 2/6`

Now figure out the middle number between the numerators:

`2/6 < x < 3/6`

`x = 2.5/6`

Since `2.5` is a decimal, multiply both the numerator and denominator to make the `2.5` into a whole number:

`(2.5/6) * (2/2) = 5/12`

Check the answer:

`(1/3) < (5/12) < (1/2) " "?`

`1/2` is the same thing as `6/12`

`1/3` is the same thing as `4/12`

so

`(4/12) < (5/12) < (6/12)` is true!

Answer 4

There are many ... but `5/12` is exactly half-way between them.

Explanation:

There are many fractions between `1/2 and 1/3`

If you use the LCD you end up with `3/6 and 2/6`.
The values between `2 and 3` are all fractions.

However, use a larger value in the denominator:

`6/12 and 4/12`

Now it is easy so see that a fraction exactly between them is `5/12`

Consider an even bigger value in the denominator:

`12/24 and 8/24`

Now we have the fractions: `9/24, 10/24, 11/24` lying between them.

You can continue in this using larger and larger values and each time you will find more and more fractions between `1/2 and 1/3`

There are infinitely many equivalent fractions.

Answer 5

`2/5` is one of them.

Explanation:

The fraction `(a+c)/(b+d)` is between `color(red)(a/b)` and `color(blue)(c/d)` for any integers `a,c` and positive integers `b,d`.

Proof:
Assuming that `a/b< c/d`

`color(red)(a/b)=color(red)(a/b)*(b+d)/(b+d)`

`=color(red)(a/b)*b/(b+d)+color(red)(a/b)*d/(b+d)<`

`< a/b*b/(b+d)+c/d*d/(b+d)<`

`< color(blue)(c/d)*b/(b+d)+color(blue)(c/d)*d/(b+d)`

`=color(blue)(c/d)*(b+d)/(b+d)=color(blue)(c/d)`

The middle term
`a/cancel(b)*cancel(b)/(b+d)+c/cancel(d)*cancel(d)/(b+d)=a/(b+d)+c/(b+d)=(a+c)/(b+d)`

That means
`a/b< (a+c)/(b+d)< c/d`

In our case the fraction between `1/3` and `1/2` is `(1+1)/(3+2)=2/5`

It's a nice trick if you need a fraction between fractions quick.

Ford's circles are closely related to that.