Question

Insert four rational numbers between the following rational numbers:

`1/2` and `4/9`

Answer 1

Given: Insert four rational numbers between `1/2` and `4/9`.

Step-wise calculation:

1. First, note that `1/2 = 0.5` and `4/9 ≈ 0.4444`. 

Since `4/9 < 1/2`, swap the order to place smaller first and larger second `4/9 < 1/2`.

So we actually want four rational numbers between `4/9` and `1/2`.

2. To find rational numbers between `4/9` and `1/2`, express both with a common denominator LCM (9, 2) = 18.

Convert both fractions:

`4/9 = 8/18`,

`1/2 = 9/18`

3. The numbers between `8/18` and `9/18` with denominator 18 would be difficult as there is only one step between.

To find more rational numbers, multiply the numerator and denominator by some number n, usually by increasing the denominator.

4. Multiply numerator and denominator by 10:

`4/9 = 40/90`,

`1/2 = 45/90`

Now rational numbers between `40/90` and `45/90` are `41/90, 42/90, 43/90, 44/90`.

5. These 4 rational numbers correspond to `41/90, 42/90, 43/90, 44/90` which lie between `4/9` and `1/2`.

Four rational numbers between `1/2` and `4/9` or equivalently between `4/9` and `1/2` are `9/20, 41/90, 83/180, 7/15`.

This exact set is directly provided in the source file for part (ii) under exercise 5:

`9/20, 41/90, 83/180, 7/15`

So the answer is `9/20, 41/90, 83/180, 7/15`.

These are four rational numbers between `1/2` and `4/9`.