If `a = 0.18181818….and b = 0.303003003…….` then `(a +b)` is :

Options

A. a rational no.
B. perfect number
C. an irrational no.
D. both (b) and (c)

Answer

To solve the problem, we need to first express the repeating decimals \( a \) and \( b \) in a more manageable form, then add them together.

### Step 1: Convert \( a \) into a fraction
The number \( a = 0.18181818...\) is a repeating decimal. We can express it as a fraction.

Let:
\[ x = 0.18181818... \]

Multiply both sides by 100 (since the decimal repeats every two digits):
\[ 100x = 18.18181818... \]

Now, subtract the first equation from the second:
\[ 100x - x = 18.18181818... - 0.18181818... \]
\[ 99x = 18 \]

Now, solve for \( x \):
\[ x = \frac{18}{99} \]

Simplifying this fraction:
\[ x = \frac{2}{11} \]

Thus, we have:
\[ a = \frac{2}{11} \]

### Step 2: Convert \( b \) into a fraction
Now, let's convert \( b = 0.303003003...\) into a fraction.

Let:
\[ y = 0.303003003... \]

Multiply both sides by 1000 (since the decimal repeats every three digits):
\[ 1000y = 303.003003... \]

Now, subtract the first equation from the second:
\[ 1000y - y = 303.003003... - 0.303003003... \]
\[ 999y = 303 \]

Now, solve for \( y \):
\[ y = \frac{303}{999} \]

Simplifying this fraction:
\[ y = \frac{101}{333} \]

Thus, we have:
\[ b = \frac{101}{333} \]

### Step 3: Add \( a \) and \( b \)
Now we need to add \( a \) and \( b \):
\[ a + b = \frac{2}{11} + \frac{101}{333} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 11 and 333 is 3663.

Convert \( \frac{2}{11} \) to have a denominator of 3663:
\[ \frac{2}{11} = \frac{2 \times 333}{11 \times 333} = \frac{666}{3663} \]

Convert \( \frac{101}{333} \) to have a denominator of 3663:
\[ \frac{101}{333} = \frac{101 \times 11}{333 \times 11} = \frac{1111}{3663} \]

Now, we can add the two fractions:
\[ a + b = \frac{666}{3663} + \frac{1111}{3663} = \frac{666 + 1111}{3663} = \frac{1777}{3663} \]

### Step 4: Conclusion
Thus, the final answer for \( a + b \) is:
\[ a + b = \frac{1777}{3663} \]