Calculate \( 0 . \dot{2} \dot{3}+0 . \dot{2} \) Give your answer as a fraction in its lowest terms.

Let's begin by denoting the repeating decimals in a more manageable form. Let \( x = 0.\dot{2}\dot{3} \) (which means \( x = 0.232323\ldots \)), and let \( y = 0.\dot{2} \) (which means \( y = 0.2222\ldots \)). **Step 1: Convert \( y = 0.\dot{2} \) into a fraction.** To convert \( y \) into a fraction: \[ y = 0.2222\dots \] Multiplying both sides by 10, we have: \[ 10y = 2.2222\dots \] Subtracting the original \( y \) from this equation: \[ 10y - y = 2.2222\dots - 0.2222\dots \implies 9y = 2 \] Thus, \[ y = \frac{2}{9}. \] **Step 2: Convert \( x = 0.\dot{2}\dot{3} \) into a fraction.** To convert \( x \) into a fraction: \[ x = 0.232323\dots \] Multiplying both sides by 100 (because the repeating block is 2 digits long): \[ 100x = 23.232323\dots \] Next, we subtract the original \( x \): \[ 100x - x = 23.232323\dots - 0.232323\dots \implies 99x = 23 \] Thus, \[ x = \frac{23}{99}. \] **Step 3: Calculate \( x + y \)** Now we need to compute \( 0.\dot{2}\dot{3} + 0.\dot{2} = x + y \): \[ x + y = \frac{23}{99} + \frac{2}{9}. \] To add these fractions, we need a common denominator. The least common multiple of 99 and 9 is 99. We will convert \( \frac{2}{9} \) into a fraction with a denominator of 99: \[ \frac{2}{9} = \frac{2 \times 11}{9 \times 11} = \frac{22}{99}. \] Now we can add the two fractions: \[ x + y = \frac{23}{99} + \frac{22}{99} = \frac{23 + 22}{99} = \frac{45}{99}. \] **Step 4: Simplifying the fraction** Next, we can simplify \( \frac{45}{99} \): \[ \frac{45 \div 9}{99 \div 9} = \frac{5}{11}. \] Thus, the final answer is: \[ \boxed{\frac{5}{11}}. \]