Understanding the Problem

The question asks us to find the value of the given expression involving recurring decimals: \(0.\bar 4 + 0.5 \bar 9 - 0.4 \overline{23}\). To solve this, we first need to convert each recurring decimal into a simple fraction. Then we can perform the arithmetic operations (addition and subtraction) on these fractions.

Converting Recurring Decimals to Fractions

Let's convert each term into a fraction:

1. \(0.\bar 4\): This is a simple recurring decimal with one digit repeating. \[0.\bar 4 = 0.444... = \frac{4}{9}\]
2. \(0.5 \bar 9\): This decimal has a non-repeating part (0.5) and a repeating part (9). We can write \(0.5 \bar 9 = 0.5 + 0.0\bar 9\). \(0.5 = \frac{5}{10}\). For \(0.0\bar 9\): Let \(x = 0.0\bar 9\). Then \(10x = 0.\bar 9 = \frac{9}{9} = 1\). So \(x = \frac{1}{10}\). Therefore, \(0.5 \bar 9 = \frac{5}{10} + \frac{1}{10} = \frac{6}{10} = \frac{3}{5}\). Alternatively, using the general rule: \(0.5\bar{9} = \frac{59 - 5}{90} = \frac{54}{90} = \frac{6 \times 9}{10 \times 9} = \frac{6}{10} = \frac{3}{5}\).
3. \(0.4 \overline{23}\): This decimal has a non-repeating part (0.4) and a repeating part (23). We can write \(0.4 \overline{23} = 0.4 + 0.0\overline{23}\). \(0.4 = \frac{4}{10}\). For \(0.0\overline{23}\): Let \(y = 0.0\overline{23}\). Then \(10y = 0.\overline{23}\) and \(1000y = 23.\overline{23}\). \(1000y - 10y = 23.\overline{23} - 0.\overline{23}\) \(990y = 23\) \(y = \frac{23}{990}\). Therefore, \(0.4 \overline{23} = \frac{4}{10} + \frac{23}{990}\). To add these fractions, find a common denominator, which is 990. \(\frac{4}{10} = \frac{4 \times 99}{10 \times 99} = \frac{396}{990}\). So, \(0.4 \overline{23} = \frac{396}{990} + \frac{23}{990} = \frac{396 + 23}{990} = \frac{419}{990}\). Alternatively, using the general rule: \(0.4\overline{23} = \frac{423 - 4}{990} = \frac{419}{990}\).

Performing the Calculation

Now substitute the fraction values back into the original expression:

To add and subtract these fractions, we need a common denominator. The least common multiple (LCM) of 9, 5, and 990 is 990.

Convert each fraction to have a denominator of 990:

• \(\frac{4}{9} = \frac{4 \times 110}{9 \times 110} = \frac{440}{990}\)
• \(\frac{3}{5} = \frac{3 \times 198}{5 \times 198} = \frac{594}{990}\)
• \(\frac{419}{990}\) remains the same.

Now perform the addition and subtraction:

Simplifying the Resulting Fraction

The fraction \(\frac{615}{990}\) can be simplified. Both the numerator and denominator are divisible by 5.

Both 123 and 198 are divisible by 3 (since the sum of digits 1+2+3=6 and 1+9+8=18 are divisible by 3).

The fraction \(\frac{41}{66}\) is in its simplest form.

Converting Fraction back to Decimal

To compare with the options, we convert the fraction \(\frac{41}{66}\) back into a decimal by performing long division.

Dividing 41 by 66:

   0.62121...
66|41.00000
   -39 6
   ----
    1 40
   -1 32
   ----
       80
      -66
      ---
       140
      -132
      ----
         80
        -66
        ---
         14...

The remainders are 14, 8, 14, 8, ... and the digits in the quotient after the first digit 6 are 2, 1, 2, 1, ...

So, the decimal representation is \(0.6212121... = 0.6\overline{21}\).

Comparing with Options

Comparing the calculated value \(0.6\overline{21}\) with the given options:

The calculated value matches Option 1.

Conclusion

The value of the expression \(0.\bar 4 + 0.5 \bar 9 - 0.4 \overline{23}\) is \(0.6\overline{21}\).