`2bar.75+3bar.78=?`

Options

A. `1bar.03`
B. `1bar.53`
C. `5bar.53`
D. `bar5.54`

Answer

To solve the problem \(2.\overline{75} + 3.\overline{78}\), we first need to understand what the repeating decimals mean.

### Step 1: Convert the repeating decimals to fractions

1. **Convert \(2.\overline{75}\) to a fraction:**
Let \(x = 2.\overline{75}\).
Then, \(100x = 275.\overline{75}\) (multiplying by 100 shifts the decimal point two places to the right).
Subtracting the first equation from the second gives:
\[ 100x - x = 275.\overline{75} - 2.\overline{75} \]
\[ 99x = 273 \]
\[ x = \frac{273}{99} \]
Simplifying \(\frac{273}{99}\) gives:
\[ x = \frac{91}{33} \]

2. **Convert \(3.\overline{78}\) to a fraction:**
Let \(y = 3.\overline{78}\).
Then, \(100y = 378.\overline{78}\).
Subtracting gives:
\[ 100y - y = 378.\overline{78} - 3.\overline{78} \]
\[ 99y = 375 \]
\[ y = \frac{375}{99} \]
Simplifying \(\frac{375}{99}\) gives:
\[ y = \frac{125}{33} \]

### Step 2: Add the two fractions

Now we need to add \(x\) and \(y\):
\[ x + y = \frac{91}{33} + \frac{125}{33} = \frac{91 + 125}{33} = \frac{216}{33} \]

### Step 3: Simplify the result

Now, simplify \(\frac{216}{33}\):
\[ \frac{216}{33} = \frac{72}{11} \]

### Step 4: Convert back to decimal

Now convert \(\frac{72}{11}\) back to a decimal:
\[ 72 \div 11 = 6.545454\ldots \]
This is \(6.\overline{54}\).

### Final Answer

Thus, the final answer is:
\[ 2.\overline{75} + 3.\overline{78} = 6.\overline{54} \]

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