7/24.
If the question were what number is halfway between 3 and 4 we would simply add the numbers together and divide by 2. (3 + 4) / 2 = 7/2 = 3.5
We can do exactly the same thing with fractions.
So, we have (1/4 + 1/3) / 2. To start we convert both fractions so that they have the same denominator which allows us to add them; we will change both fractions into twelfths = (3/12 + 4/12) / 2 = 7/12/2 = 7/(2*12) = 7/24.
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9/16
7/32 is between 1/4 and 3/16
0.75 ---------------the halfway number
There is no "the" rational" number since there is an infinity of them. Start with the number halfway between the two: (1/10+1/11)/2 = 21/220. Then try halfway between 1/10 and 21/220, and then halfway between 1/10 and that on. And so on.
Halfway between 9 and 1/2 and 10.