Express 5 . 2 in the form p q , where p and q are both integers and q ≠ 0 .

The given decimal 5 . 2 ¯ is a recurring decimal as it is a decimal number with a repeated digit or section of repeated digits after a decimal point. Let x = 5 . 2 = 5 . 2222 . . . . - - - - - - 1 Multiply both sides with 10 , we get 10 x = 52 . 2 ¯ = 52 . 222 . . . . - - - - - - 2 Subtracting equation ( 1 ) from equation ( 2 ) , we get 10 x - x = 52 . 2 ¯ - 5 . 2 ¯ ⇒ 9 x = 47 ⇒ x = 47 9 (This is in the lowest form) Hence, the required answer is as shown above.

Express 5 . 2 in the form p q , where p and q are both integers and q ≠ 0 .

The given decimal 5 . 2 ¯ is a recurring decimal as it is a decimal number with a repeated digit or section of repeated digits after a decimal point. Let x = 5 . 2 = 5 . 2222 . . . . - - - - - - 1 Multiply both sides with 10 , we get 10 x = 52 . 2 ¯ = 52 . 222 . . . . - - - - - - 2 Subtracting equation ( 1 ) from equation ( 2 ) , we get 10 x - x = 52 . 2 ¯ - 5 . 2 ¯ ⇒ 9 x = 47 ⇒ x = 47 9 (This is in the lowest form) Hence, the required answer is as shown above.

Express 0 . 404040 . . . in the form p q , , where p and q are both integers and q ≠ 0 .

The given decimal 0 . 404040 . . . is a recurring decimal as it is a decimal number with a repeated digit or section of repeated digits after a decimal point. Let x = 0 . 404040 . . . . = 0 . 40 - - - - - - 1 Multiply both sides with 100 , we get 100 x = 40 . 404040 . . . = 40 . 40 - - - - - - 2 Subtracting equation ( 1 ) from equation ( 2 ) , we get 100 x - x = 40 . 40 - 0 . 40 ⇒ 90 x = 40 ⇒ x = 4 9 (This is in the lowest form) Hence, the required answer is as shown above.

Express 0 . 3 7 in the form p q , where p and q are both integers and q ≠ 0 .

Given, Let x = 0 . 3 7 (Mixed recurring decimal) i.e. x = 0 . 37777 . . . - - - - - - 1 There is one non-repeating digits after the decimal, multiplying equation 1 by 10 , to make it pure recurring decimal, we get 10 x = 3 . 7 = 3 . 7777 . . . - - - - - - 2 Further, multiplying equation ( 2 ) by 10 , we get 100 x = 37 . 7 = 37 . 777 . . . . - - - - - - 3 Subtracting equation 2 from equation ( 3 ) , we get 100 x - 10 x = 37 . 777 . . . - 3 . 777 . . . ⇒ 9 x = 37 . 7 - 3 . 7 ⇒ 9 x = 33 ⇒ x = 33 9 = 11 3 (This is in the lowest form) Hence, the required answer is as shown above.

Express 0 . 1 34 in the form p q , where p and q are both integers and q ≠ 0 .

Given, Let x = 0 . 1 34 (Mixed recurring decimal) i.e. x = 0 . 1343434 . . . - - - - - - 1 There is one non-repeating digits after the decimal, multiplying equation 1 by 10 , to make it pure recurring decimal, we get 10 x = 1 . 34 = 1 . 343434 . . . - - - - - - 2 Further, multiplying equation ( 2 ) by 100 , we get 1000 x = 134 . 34 = 134 . 343434 . . . - - - - - - 3 Subtracting equation 2 from equation ( 3 ) , we get 1000 x - 10 x = 134 . 343434 . . . - 1 . 343434 . . . ⇒ 990 x = 134 . 34 - 1 . 34 ⇒ 990 x = 133 ⇒ x = 133 990 (This is in the lowest form) Hence, the required answer is as shown above.

Express 0 . 001 in the form p q , where p and q are both integers and q ≠ 0 .

Given, Let x = 0 . 001 (recurring decimal) i.e. x = 0 . 001001001 . . . - - - - - - 1 Multiplying equation 1 by 1000 , to make it pure recurring decimal, we get 1000 x = 1 . 001 = 1 . 001001001 . . . - - - - 2 Subtracting equation 1 from equation ( 2 ) , we get 1000 x - x = 1 . 001001 . . . - 0 . 001001 . . . ⇒ 999 x = 1 . 001 - 0 . 001 ⇒ 999 x = 1 ⇒ x = 1 999 (This is in the lowest form) Hence, the required answer is as shown above.

Express 0 . 9999 . . . in the form p q . Are you surprised by your answer ?

The given decimal 0 . 999 . . . is a recurring decimal as it is a decimal number with a repeated digit or section of repeated digits after a decimal point. Let x = 0 . 999 . . . - - - - - - ( 1 ) There is one non-repeating digits after the decimal, multiplying equation 1 by 10 to make it pure recurring decimal, we get 10 x = 9 . 999 . . . - - - - - - ( 2 ) Subtraction equation 1 from 2 , we get 10 x - x = 9 . 999 . . . - 0 . 999 . . . ⇒ 9 x = 9 . 9 - 0 . 9 ⇒ 9 x = 9 ⇒ x = 9 9 = 1 Yes, I have surprised in my answer. Hence, the required answer is as shown above.