The value of `(0.34bar(67)+0.13bar33)` is :

Options

A. `0.48`
B. `0.48bar01`
C. `0.bar(48)`
D. `0.4bar8`

Answer

To solve the problem of finding the value of \(0.34\overline{67} + 0.13\overline{33}\), we will follow these steps:

### Step 1: Convert the recurring decimal \(0.34\overline{67}\) into a fraction.

Let \(x = 0.34\overline{67}\).

This can be expressed as:
\[ x = 0.34676767\ldots \]

To eliminate the repeating part, we multiply \(x\) by \(10000\) (since the repeating part has 4 digits):
\[ 10000x = 3467.676767\ldots \]

Next, we multiply \(x\) by \(100\) to shift the decimal point:
\[ 100x = 34.676767\ldots \]

### Step 2: Set up the equation to eliminate the repeating decimal.

Now we can subtract the second equation from the first:
\[ 10000x - 100x = 3467.676767\ldots - 34.676767\ldots \]
This simplifies to:
\[ 9900x = 3433 \]

### Step 3: Solve for \(x\).

Now, divide both sides by \(9900\):
\[ x = \frac{3433}{9900} \]

### Step 4: Convert the recurring decimal \(0.13\overline{33}\) into a fraction.

Let \(y = 0.13\overline{33}\).

This can be expressed as:
\[ y = 0.1333333\ldots \]

To eliminate the repeating part, we multiply \(y\) by \(100\):
\[ 100y = 13.33333\ldots \]

Now, we can also multiply \(y\) by \(10\) to shift the decimal point:
\[ 10y = 1.33333\ldots \]

### Step 5: Set up the equation to eliminate the repeating decimal.

Now we can subtract the second equation from the first:
\[ 100y - 10y = 13.33333\ldots - 1.33333\ldots \]
This simplifies to:
\[ 90y = 12 \]

### Step 6: Solve for \(y\).

Now, divide both sides by \(90\):
\[ y = \frac{12}{90} = \frac{2}{15} \]

### Step 7: Add the two fractions \(x\) and \(y\).

Now we have:
\[ x + y = \frac{3433}{9900} + \frac{2}{15} \]

To add these fractions, we need a common denominator. The least common multiple of \(9900\) and \(15\) is \(9900\).

Convert \(\frac{2}{15}\) to have a denominator of \(9900\):
\[ \frac{2}{15} = \frac{2 \times 660}{15 \times 660} = \frac{1320}{9900} \]

Now we can add:
\[ x + y = \frac{3433}{9900} + \frac{1320}{9900} = \frac{3433 + 1320}{9900} = \frac{4753}{9900} \]

### Step 8: Convert the fraction to decimal form.

Now, we can convert \(\frac{4753}{9900}\) to decimal:
\[ \frac{4753}{9900} \approx 0.4801001\ldots \]

### Final Answer:
Thus, the value of \(0.34\overline{67} + 0.13\overline{33}\) is approximately \(0.4801\).

---