The value of `(0.34bar(68)+.17bar(32))` is

Options

A. 0.52 `bar(01)`
B. `0.bar(51)`
C. 0.5
D. `0.5bar(1)`

Answer

To solve the expression \(0.3468\overline{68} + 0.1732\overline{32}\), we will first convert the repeating decimals into fractions and then perform the addition.

### Step 1: Convert \(0.3468\overline{68}\) to a fraction.

Let \(x = 0.3468\overline{68}\).

To eliminate the repeating part, we can multiply by \(10000\) (to shift the decimal point four places) and then by \(100\) (to shift two more places for the repeat):

\[ 10000x = 3468.686868... \]
\[ 100x = 34.68\overline{68} \]

Now, subtract the second equation from the first:

\[ 10000x - 100x = 3468.686868... - 34.68\overline{68} \]
\[ 9900x = 3434 \]

Now, solve for \(x\):

\[ x = \frac{3434}{9900} \]

### Step 2: Convert \(0.1732\overline{32}\) to a fraction.

Let \(y = 0.1732\overline{32}\).

Similarly, we can multiply by \(10000\) and \(100\):

\[ 10000y = 1732.323232... \]
\[ 100y = 17.32\overline{32} \]

Now, subtract the second equation from the first:

\[ 10000y - 100y = 1732.323232... - 17.32\overline{32} \]
\[ 9900y = 1715 \]

Now, solve for \(y\):

\[ y = \frac{1715}{9900} \]

### Step 3: Add the two fractions.

Now we need to add the two fractions:

\[ 0.3468\overline{68} + 0.1732\overline{32} = \frac{3434}{9900} + \frac{1715}{9900} \]

Since they have the same denominator, we can add the numerators:

\[ = \frac{3434 + 1715}{9900} = \frac{5149}{9900} \]

### Step 4: Convert the result back to decimal form.

Now we can convert \(\frac{5149}{9900}\) back to decimal:

\[ 5149 \div 9900 \approx 0.52010101... \]

Thus, the value of \(0.3468\overline{68} + 0.1732\overline{32}\) is \(0.5201\overline{01}\).

### Final Answer:
The final answer is \(0.5201\overline{01}\).

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