Question

insert six rational number between 1/6 and 5/3​

Answer 1

Let, x = ⅙ and y = 5/3

n = 6

Formula: d = (y - x)/(n + 1)

⇒ d = (5/3 - ⅙)/(7) = (3/2) × ⅐ = 3/14

Then six rational numbers lying between ⅙ and 5/3 are ⅙ + 3/14, ⅙ + 2(3/14),..., ⅙ + 6(3/14).

∴ The numbers are 8/21,..., 61/42.

ALITER:

R = ½(a + b)

So, the rational numbers are,

R1 = ½(⅙ + 5/3) = 11/12

R2 = ½(⅙ + 11/6) = 1

R3 = ½(5/3 + 11/6) = 7/6

R4 = ½(7/2 + 5/3) = 31/12

R5 = ½(1 + ⅙) = 13/12

R6 = ½(31/12 + 1/6) = 33/24

More:

Try multiplying denominators of the given numbers of 100, 1000,... and then to the numerator with the same number. It becomes easier. So you learnt method 3.