The value of `(0.17bar(59)+0.30 bar(41))` is ____

Options

A. `0.48 bar(02)`
B. `0.48bar(01)`
C. `0.47bar(01)`
D. `0.48bar(11)`

Answer

To solve the expression \(0.17\overline{59} + 0.30\overline{41}\), we will follow these steps:

### Step 1: Convert the repeating decimals to fractions
First, we need to convert \(0.17\overline{59}\) and \(0.30\overline{41}\) into fractions.

**For \(0.17\overline{59}\):**
Let \(x = 0.17\overline{59}\).

To eliminate the repeating part, we can multiply by 10000 (since "59" has 2 digits):
\[ 10000x = 1759.\overline{59} \]
Now, we also multiply by 100 (to shift the decimal point):
\[ 100x = 17.59\overline{59} \]
Now we can subtract these two equations:
\[ 10000x - 100x = 1759.\overline{59} - 17.59\overline{59} \]
\[ 9900x = 1741 \]
\[ x = \frac{1741}{9900} \]

**For \(0.30\overline{41}\):**
Let \(y = 0.30\overline{41}\).

Again, we multiply by 10000:
\[ 10000y = 3041.\overline{41} \]
And by 100:
\[ 100y = 30.41\overline{41} \]
Subtracting these two equations:
\[ 10000y - 100y = 3041.\overline{41} - 30.41\overline{41} \]
\[ 9900y = 3011.6 \]
\[ y = \frac{3011.6}{9900} \]

### Step 2: Add the two fractions
Now we add the two fractions:
\[ 0.17\overline{59} + 0.30\overline{41} = \frac{1741}{9900} + \frac{3011.6}{9900} \]
Finding a common denominator:
\[ = \frac{1741 + 3011.6}{9900} \]
Calculating the numerator:
\[ = \frac{4752.6}{9900} \]

### Step 3: Convert back to decimal
Now we convert \(\frac{4752.6}{9900}\) back to decimal:
\[ = 0.4801\overline{01} \]

### Final Answer
Thus, the value of \(0.17\overline{59} + 0.30\overline{41}\) is \(0.4801\overline{01}\).