Question

convert following recurring decimals into Fractions:2.324(where bar is placed above 24)and 0.81(where bar is above 81)​

Answer 1

Given :

• For understanding, consider number in brackets as recurring digits
• 2.3(24)      
• 0.(81)

To find:

• Fraction form of the recursive numbers

Solution:

Steps to be followed:

1. Let, the recursive number be x
2. Multiply the equation by 10 so u get 10 x on one side.
3. Split the other recursive side such that you get, 10 x = number + x
4. Hence, you get 9 x = number
5. Hence x = number / 9
6. Multiply and divide by 10 so the number is free from decimal places.

• Let 2.3(24) = x
• 10 x = 23.2424
• 10 x = 20.91816 + 2.3(24)
• 10 x = 20.91816 + x
• x = 20.91816/9
• x = 2091816/900000
• Let 0.(81) = x
• 10 x = 8.1818
• 10 x = 7.363637 + 0.(81)
• 10 x = 7.363637 + x
• x = 7.3636363637/9
• x = 736363637/900000000

Answer:

The fraction form of the given numbers is,

• 0.23242424 = 2091816/900000
• 0.818181 = 736363637/900000000  

Answer 2

$1) 2.3\overline{24} = \frac{2301}{990}\\\\2) 0.\overline{81} = \frac{81}{99}$

Step-by-step explanation:

1) Let x = $2.3\overline {24}$

Numerator = 324 - 3 = 321

Denominator = 2 nines (as there are two recurring digits) followed by 1 zero( as there is one non-recurring digit) = 990

Hence, $2.3\overline {24} = 2\frac{321}{990} = \frac{2301}{990}$

2) Let x = $0.\overline{81}$

Numerator = 81 - 0 = 81

Denominator = 2 nines (as there are two recurring digits)

Hence, $0.\overline{81} = \frac{81}{99}$