`0.overline(142857 ) + 0.overline(285714 )` is equal to

Options

A. `10`
B. `2`
C. `3/7`
D. `1/3`

Answer

To solve the problem \(0.\overline{142857} + 0.\overline{285714}\), we will first convert each repeating decimal into a fraction.

### Step 1: Convert \(0.\overline{142857}\) to a fraction
Let \(x = 0.\overline{142857}\).

To eliminate the repeating part, multiply both sides by \(10^6\) (since the repeating part has 6 digits):

\[ 10^6 x = 142857.\overline{142857} \]

Now, subtract the original \(x\) from this equation:

\[ 10^6 x - x = 142857.\overline{142857} - 0.\overline{142857} \]

This simplifies to:

\[ 999999x = 142857 \]

Now, solve for \(x\):

\[ x = \frac{142857}{999999} \]

### Step 2: Simplify the fraction
Notice that \(999999 = 7 \times 142857\). Therefore:

\[ x = \frac{142857}{7 \times 142857} = \frac{1}{7} \]

So, \(0.\overline{142857} = \frac{1}{7}\).

### Step 3: Convert \(0.\overline{285714}\) to a fraction
Let \(y = 0.\overline{285714}\).

Similarly, multiply both sides by \(10^6\):

\[ 10^6 y = 285714.\overline{285714} \]

Subtract the original \(y\):

\[ 10^6 y - y = 285714.\overline{285714} - 0.\overline{285714} \]

This simplifies to:

\[ 999999y = 285714 \]

Now, solve for \(y\):

\[ y = \frac{285714}{999999} \]

### Step 4: Simplify the fraction
Again, notice that \(999999 = 7 \times 142857\) and \(285714 = 2 \times 142857\). Therefore:

\[ y = \frac{285714}{7 \times 142857} = \frac{2}{7} \]

So, \(0.\overline{285714} = \frac{2}{7}\).

### Step 5: Add the two fractions
Now we can add the two fractions:

\[ 0.\overline{142857} + 0.\overline{285714} = \frac{1}{7} + \frac{2}{7} \]

Combine the fractions:

\[ \frac{1 + 2}{7} = \frac{3}{7} \]

### Final Answer
Thus, \(0.\overline{142857} + 0.\overline{285714} = \frac{3}{7}\).

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