The difference of `5.bar76 and 2.bar3` is

Options

A. `2.bar54`
B. `3.bar73`
C. `3.bar46`
D. `3.bar43`

Answer

To find the difference between \(5.\overline{76}\) and \(2.\overline{3}\), we will follow these steps:

### Step 1: Convert the repeating decimals to fractions
1. **Convert \(5.\overline{76}\)**:
- Let \(x = 5.\overline{76}\).
- Multiply by 100 (because the repeating part has 2 digits):
\[ 100x = 576.\overline{76} \]
- Subtract the original equation from this:
\[ 100x - x = 576.\overline{76} - 5.\overline{76} \]
\[ 99x = 571 \]
- Therefore,
\[ x = \frac{571}{99} \]

2. **Convert \(2.\overline{3}\)**:
- Let \(y = 2.\overline{3}\).
- Multiply by 10 (because the repeating part has 1 digit):
\[ 10y = 23.\overline{3} \]
- Subtract the original equation from this:
\[ 10y - y = 23.\overline{3} - 2.\overline{3} \]
\[ 9y = 21 \]
- Therefore,
\[ y = \frac{21}{9} = \frac{7}{3} \]

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

### Step 3: Find a common denominator
The common denominator for \(99\) and \(3\) is \(99\). Convert \(\frac{7}{3}\) to have a denominator of \(99\):
\[ \frac{7}{3} = \frac{7 \times 33}{3 \times 33} = \frac{231}{99} \]

### Step 4: Subtract the fractions
Now we can subtract:
\[ \frac{571}{99} - \frac{231}{99} = \frac{571 - 231}{99} = \frac{340}{99} \]

### Step 5: Convert the result to a mixed number
Now we convert \(\frac{340}{99}\) to a mixed number:
- Divide \(340\) by \(99\):
- \(340 \div 99 = 3\) (whole number part)
- Remainder: \(340 - 297 = 43\)
- Therefore,
\[ \frac{340}{99} = 3 \frac{43}{99} \]

### Step 6: Write the final answer
The final answer in decimal form is:
\[ 3.\overline{43} \]

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

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