You know that 1 11 = 0 . 09 ¯ . Can you predict, what the decimal expansion of 3 11 , 4 11 , 5 11 , 6 11 are without actually doing the long division method? If so, how?

Given, 1 11 = 0 . 09 ∴ 3 11 = 3 × 0 . 09 = 0 . 27 ∴ 4 11 = 4 × 0 . 09 = 0 . 36 ∴ 5 11 = 5 × 0 . 09 = 0 . 45 ∴ 6 11 = 6 × 0 . 09 = 0 . 54

Express the following numbers in the form of p q , q ≠ 0 . 0 . 4

Let x = 0 . 4 ⇒ x = 0 . 44444 . . . ( i ) Multiplying both the sides of equation ( i ) by 10 , we get: 10 x = 4 . 4444444 . . . ( i i ) Subtracting equation i from equation i i , we get: 9 x = 4 x = 4 9 . Hence, the required answer is 4 9 .

Express the following numbers in the form of p q , q ≠ 0 . 0 . 8

Let x = 0 . 8 ⇒ x = 0 . 88888 . . . ( i ) Multiplying both the sides of equation ( i ) by 10 , we get: 10 x = 8 . 88888 . . . ( i i ) Subtracting equation ( i ) from equation ( i i ) , we get: 9 x = 8 x = 8 9 . Hence, the required answer is 8 9 .

Express the following numbers in the form of p q , q ≠ 0 . 0 . 68

Let x = 0 . 68 ⇒ x = 0 . 686868 . . . . ( i ) Multiplying both the sides of equation ( i ) by 100 , we get: 100 x = 68 . 686868 . . . . . . ( i i ) Subtracting equation i from equation i i , we get: 99 x = 68 x = 68 99 . Hence, the required answer is 68 99 .

Express the following numbers in the form of p q , q ≠ 0 . 1 . 32

Let x = 1 . 32 ⇒ x = 1 . 323232 . . . . . ( i ) Multiplying both the sides of equation ( i ) by 100 , we get: 100 x = 132 . 323232 . . . . . . ( i i ) Subtracting equation i from equation i i , we get: 99 x = 131 x = 131 99 . Hence, the required answer is 131 99 .