what fractions are between 1/4 and 1/3?

Question

10 fractions between 1/4 and 1/3

Answer 1

Infinitely many!  For example take the average of those fractions. 

Repeat by taking the average of 1/3 and the above answer.

Repeat the process an infinite number of times!

Answer 2

We can find infinitely many fractions between 1/3 and 1/4.

1) 1/2(1/3 +1/4) = 1/2(4/12 + 3/12) = 7/24.                            7/24 is between 1/4 &1/3

 

2) 1/2(1/3 + 7/24)=1/2(8/24+7/24)=15/48=5/16                    15/48 is between 7/24 &1/3, but 7/24>1/4

                                                                                                    

3) 1/2(15/48+1/3)=1/2(5/16+1/3)=1/2(15/48+16/48)=31/96

...

We can organize this process to find some more fractions.

Here are the fractions on the number line:

 

      ...   1/4                       7/24          15/48    31/96   1/3

 

All the new fractions will fall in the last right interval.

 

Here is another way of thinking about it.

1/4 = 0.25,

1/3 = 0.333...

3/10; 33/100; 333/1000; 3333/10000; 33,333/100,000; 333,333/1,000,000; ... all these fractions will be between 1/4 and 1/3.

As it is been noted before, there are infinitely many fractions that lie between any two different numbers.

Answer 3

Often, fractions are reduced to their lowest form.  In this problem, 1/4 might have previously been 2/8 or 3/12 or 4/16 or ...

Also, 1/3 might have been 2/6 or 3/9 or 4/12 or 5/15 or ...

 

If we select fractions with the same denominators, we get these:

     1/4                   1/3

    3/12                  4/12

    6/24                  8/24

    9/36                 12/36

   .. . .                   . . .

[note: the Least Common Multiple (LCM) of 4 and 3 is 12, so these fractions have denominators that are multiples of 12.

 

For example, if we choose 36 as a denominator, then 10/36 and 11/36  are between 1/4 and 1/3.  Choose a larger denominator, and you get more fractions between 1/4 and 1/3.

 

Now, also realize that a common denominator considers endpoints that are exactly equal to 1/4 and 1/3.  There are also fractions that have other denominators:  for example, if 1/4 = 9/36 and 1/3 = 12/36 are endpoints, and 10/36 and 11/36 are between them, other fractions, like 11/37 are also between those endpoints.

 

So, larger and larger denominators provide more and more fractions that are between 1/4 and 1/3.  That's why there are an infinite number of fractions between 1/4 and 1/3.

 

Answer 4

30/120 and 40/120, well that's nine!