Question

Insert four rational numbers between the following rational numbers:

`-2/5` and `1/3`

Answer 1

Given: Insert four rational numbers between `2/5` and `1/3`.

Step-wise calculation:

1. Start with the two given rational numbers: 

`a = -2/5, b = 1/3`

2. Find a common denominator to compare the two rational numbers: 

LCM (5, 3) = 15

3. Express the two numbers with denominator 15:

`-2/5 = -6/15`,

`1/3 = 5/15`

4. The task is to find four rational numbers between `-6/15` and `5/15`.

5. Use the formula to insert n rational numbers between a and b: 

`a_i = a + (i(b - a))/(n + 1)` for i = 1, 2, ..., n

Here n = 4.

6. Calculate: b – a 

= `5/15 - (-6/15)`

= `5/15 + 6/15`

= `11/15`

7. Calculate the increment step:

`(b - a)/(n + 1)`

= `(11/15)/5`

= `11/75`

8. Calculate the four rational numbers:

`a_1 = a + (1 xx 11)/75`

= `-6/15 + 11/75`

= `-30/75 + 11/75`

= `-19/75`

`a_2 = a + (2 xx 11)/75`

= `-30/75 + 22/75`

= `-8/75`

`a_3 = a + (3 xx 11)/75`

= `-30/75 + 33/75`

= `3/75`

= `1/25`

`a_4 = a + (4 xx 11)/75`

= `-30/75 + 44/75`

= `14/75`

The four rational numbers between `-2/5` and `1/3` are:

`-19/75, -8/75, 1/25, 14/75`

Or equivalently as simplified fractions:

`-19/75, -8/75, 1/25, 14/75`

These values lie strictly between `-2/5` and `1/3`.

`1/15, 2/15, 1/5, 4/15`

Which are valid rational numbers strictly between `-2/5` and `1/3` considering the order `-2/5 < 1/15 < 2/15 < 1/5 < 4/15 < 1/3`.