Question

Insert three rational numbers between the following rational numbers and write them in descending order:

`6/11` and `8/5`

Answer 1

Given: Insert three rational numbers between `6/11` and `8/5` and write them in descending order.

Stepwise Calculation:

1. Identify the two numbers:

`a = 6/11, b = 8/5`

2. Find the difference b – a:

`b - a = 8/5 - 6/11`

= `(8 xx 11)/(5 xx 11) - (6 xx 5)/(11 xx 5)`

= `88/55 - 30/55`

= `58/55`

3. We want to insert n = 3 rational numbers between a and b.

4. Divide the interval b – a into n + 1 = 4 equal parts:

`(b - a)/(n + 1) = (58//55)/4`

= `58/220`

= `29/110`

5. Find the three rational numbers:

`a + (b - a)/4` 

= `6/11 + 29/110`

= `60/110 + 29/110`

= `89/110`

`a + (2(b - a))/4`

= `6/11 + 2 xx 29/110`

= `60/110 + 58/110`

= `118/110`

= `59/55`

`a + (3(b - a))/4`

= `6/11 + 3 xx 29/110`

= `60/110 + 87/110`

= `147/110`

6. Write the four numbers in descending order:

`8/5 = 176/110 > 147/110 > 59/55 > 89/110 > 6/11 = 60/110`

Three rational numbers between `6/11` and `8/5` are `89/110, 59/55, 147/110`. 

In descending order, these numbers are `147/110 > 59/55 > 89/110`.