The difference of `5.bar(76) and 2.bar(3)` is

Options

A. `2.bar(54)`
B. `3.bar(73)`
C. `3.bar(46)`
D. `3.bar(43)`

Answer

To find the difference between \( 5.\overline{76} \) and \( 2.\overline{3} \), we will first convert these repeating decimals into fractions.

### Step 1: Convert \( 5.\overline{76} \) to a fraction
Let \( x = 5.\overline{76} \). This means:
\[ x = 5.76767676... \]

To eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating part has 2 digits):
\[ 100x = 576.767676... \]

Now, we can set up the equation:
\[ 100x - x = 576.767676... - 5.767676... \]
\[ 99x = 571 \]

Now, solving for \( x \):
\[ x = \frac{571}{99} \]

### Step 2: Convert \( 2.\overline{3} \) to a fraction
Let \( y = 2.\overline{3} \). This means:
\[ y = 2.333333... \]

To eliminate the repeating part, we can multiply \( y \) by 10:
\[ 10y = 23.33333... \]

Now, we can set up the equation:
\[ 10y - y = 23.33333... - 2.33333... \]
\[ 9y = 21 \]

Now, solving for \( y \):
\[ y = \frac{21}{9} = \frac{7}{3} \]

### Step 3: Find the difference \( 5.\overline{76} - 2.\overline{3} \)
Now we need to subtract the two fractions:
\[ 5.\overline{76} - 2.\overline{3} = \frac{571}{99} - \frac{7}{3} \]

To perform the subtraction, we need a common denominator. The least common multiple of 99 and 3 is 99.

Convert \( \frac{7}{3} \) to have a denominator of 99:
\[ \frac{7}{3} = \frac{7 \times 33}{3 \times 33} = \frac{231}{99} \]

Now we can subtract:
\[ \frac{571}{99} - \frac{231}{99} = \frac{571 - 231}{99} = \frac{340}{99} \]

### Step 4: Simplify the fraction if possible
Now we check if \( \frac{340}{99} \) can be simplified. The numerator and denominator have no common factors, so this is in its simplest form.

### Step 5: Convert \( \frac{340}{99} \) back to a decimal
To convert \( \frac{340}{99} \) back to a decimal, we can perform long division:
- 340 divided by 99 gives approximately 3.434343...

Thus, the difference \( 5.\overline{76} - 2.\overline{3} = 3.\overline{43} \).

### Final Answer
The difference of \( 5.\overline{76} \) and \( 2.\overline{3} \) is \( 3.\overline{43} \).

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