Question

Insert 9 rational numbers between the following rational numbers:

`3/10` and `7/5`

Answer 1

Given: Insert 9 rational numbers between the rational numbers`3/10` and `7/5`.

Stepwise calculation:

1. Let the two rational numbers be `a = 3/10, b = 7/5`.

2. Convert to equivalent fractions with a common denominator or directly calculate the difference:

`b - a = 7/5 - 3/10`

= `14/10 - 3/10`

= `11/10`

3. We need to insert n = 9 rational numbers between a and b.

Using the formula for numbers between two rational numbers: 

Step size = `(b - a)/(n + 1)`

= `(11//10)/10`

= `11/100`

4. The (k)-th rational number between a and b is:

a + k × Step size = `3/10 + k xx 11/100`

5. Calculating each number for k = 1, 2,...,9:

 `k = 1 : 3/10 + 11/100`

= `30/100 + 11/100`

= `41/100`

`k = 2 : 3/10 + 2 xx 11/100`

= `30/100 + 22/100`

= `52/100`

= `13/25`

`k = 3 : 3/10 + 3 xx 11/100`

= `30/100 + 33/100`

= `63/100`

`k = 4 : 3/10 + 4 xx 11/100`

= `30/100 + 44/100`

= `74/100`

= `37/50`

`k = 5 : 3/10 + 5 xx 11/100`

= `30/100 + 55/100`

= `85/100`

= `17/20`

`k = 6 : 3/10 + 6 xx 11/100`

= `30/100 + 66/100`

= `96/100`

= `24/25`

`k = 7 : 3/10 + 7 xx 11/100`

= `30/100 + 77/100`

= `107/100`

`k = 8 : 3/10 + 8 xx 11/100`

= `30/100 + 88/100`

= `118/100`

= `59/50`

`k = 9 : 3/10 + 9 xx 11/100`

= `30/100 + 99/100`

= `129/100`

The 9 rational numbers between `3/10` and `7/5` are:

`41/100, 13/25, 63/100, 37/50, 17/20, 24/25, 107/100, 59/50, 129/100`

Alternatively, a similar set of 9 rational numbers provided is:

`46/75, 77/75, 36/25, 139/75, 34/15, 67/25, 232/75, 263/75, 98/25`

Or, simplified within the interval, as from the source: 

`2/5, 1/2, 3/5, 7/10, 4/5, 9/10, 1, 11/10, 6/5`

The calculation method is by equally dividing the interval between the two rational numbers into 10 parts 9 rational numbers inserted.

The methods are from the provided source, detailing the exact rational numbers between `3/10` and `7/5`.