Assertion (A): The fraction value of `3.bar(12)=309/99`, so the makes `3.bar(12)` a rational number.
Reason(R):Non terminating and recurring decimals are also irrational numbers

Options

A. Both A and R are true and R is the correct explanation for A .
B. Both A and Rare true and R is not correct explanation for A.
C. A is true but R is false.
D. A is false but R is true.

Answer

To solve the question, we need to analyze both the assertion and the reason provided.

### Step 1: Understand the Assertion
The assertion states that the fraction value of \( 3.\overline{12} \) is equal to \( \frac{309}{99} \), which makes \( 3.\overline{12} \) a rational number.

### Step 2: Convert \( 3.\overline{12} \) to a Fraction
Let \( x = 3.\overline{12} \).

To eliminate the repeating decimal, we can multiply \( x \) by 100 (since the repeating part has two digits):

\[ 100x = 312.\overline{12} \]

### Step 3: Set Up the Equation
Now, we can set up the equation:

\[ 100x = 312.\overline{12} \]
\[ x = 3.\overline{12} \]

Next, we subtract the first equation from the second:

\[ 100x - x = 312.\overline{12} - 3.\overline{12} \]

This simplifies to:

\[ 99x = 312 - 3 \]
\[ 99x = 309 \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \):

\[ x = \frac{309}{99} \]

### Step 5: Conclusion about the Assertion
Since \( 3.\overline{12} \) can be expressed as a fraction \( \frac{309}{99} \), the assertion is true. Therefore, \( 3.\overline{12} \) is indeed a rational number.

### Step 6: Analyze the Reason
The reason states that non-terminating and recurring decimals are irrational numbers. This statement is false because non-terminating and recurring decimals are rational numbers, not irrational.

### Final Conclusion
- Assertion (A) is true: \( 3.\overline{12} \) is a rational number.
- Reason (R) is false: Non-terminating and recurring decimals are rational, not irrational.

### Answer
The correct option is that assertion (A) is true and reason (R) is false.

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