Solve the Fraction Addition: 1/4 + 3/9 Step-by-Step

To solve the problem of adding the fractions $\frac{1}{4}$ and $\frac{3}{9}$, we will first find a common denominator and then perform the addition:

Step 1: Finding a Common Denominator
The denominators are 4 and 9. The easiest way to find a common denominator is to multiply these two numbers. Hence, $4 \times 9 = 36$ gives us a common denominator of 36.

Step 2: Convert Each Fraction
Convert $\frac{1}{4}$ to a fraction with denominator 36. To do this, multiply the numerator and denominator by 9 (since $4 \times 9 = 36$):
$\frac{1}{4} = \frac{1 \times 9}{4 \times 9} = \frac{9}{36}$

Next, convert $\frac{3}{9}$ to a fraction with denominator 36. Multiply the numerator and denominator by 4 (since $9 \times 4 = 36$):
$\frac{3}{9} = \frac{3 \times 4}{9 \times 4} = \frac{12}{36}$

Step 3: Add the Fractions
Now add the two fractions:
$\frac{9}{36} + \frac{12}{36} = \frac{9+12}{36} = \frac{21}{36}$

Step 4: Simplify the Result (if necessary)
The fraction $\frac{21}{36}$ can be simplified by finding the greatest common divisor (GCD) of 21 and 36, which is 3. However, in the current situation with the answer choices provided, $\frac{21}{36}$ matches one of the options directly without further simplification, ensuring it meets the expected answer format.

Therefore, the sum of $\frac{1}{4} + \frac{3}{9}$ is $\frac{21}{36}$, which corresponds to choice $1$.

Thus, the correct answer is $\frac{21}{36}$.