Question

1) Insert two rational numbers between 1/3 and 1/4 (with method no direct answer )

Answer 1

Answer:

7/24, 15/48

Step-by-step explanation:

Hi,

Given 2 rational numbers 1/3 and 1/4.

Let a = 1/4 and b = 1/3.

Now, for any 2 given rationals a and b, we know that (a + b)/2 is also rational

Hence , (1/3 + 1/4)/2 = 7/24 is also rational.

let c = 7/24.

Also, we know that ∀ a, b ∈ R, we have that

                        a ≤ (a + b)/2 ≤ b

Equality occurs if a =b.

Now, hence 1/4 < 7/24 < 1/3.

Hence, we have inserted one rational between 1/3 and 1/4.

Similarly consider pairs either 1/3 and 7/24 or 7/24 and 1/4,

say consider 1/3 and 7/24.

Now, (1/3 + 7/24)/2 = 15/48 lies between these 2 numbers

let this be d = 15/48

7/24 < 15/48 < 1/3

a = 1/4 <  7/24 < 15/48 < 1/3 = b

Hence , we have inserted 2 rationals 7/24 and 15/48 between 2 given

rational numbers.

Similarly, we can insert countable number of rationals.

These types of rational cuts were been proposed by 'Dedekind', hence

these cuts are known as 'Dedekind cuts'

Hope, it helped !

Answer 2

Answer:

5/18 and 11/36

Step-by-step explanation:

Make the denominators the same:

1/3 = (1x4)/(3x4) = 4/12

1/4 = (1x3)/(4x3) = 3/12

Write equivalent fractions until there is a difference of 2 in the numerators:

1/3 = 4/12 = 8/24 = 12/36

1/4 = 3/12 = 6/24 = 9/36

Write the fractions in between the two fractions:

9/36, 10/36, 11/36, 12/36

Write in simplest term:

10/36 = 5/18 (simplest term)

11/36 = 11/36 (simplest term)

Answer: The two rational number between 1/3 and 1/4 are 5/18 and 11/36