Next, $ \frac{27}{170} + \frac{1}{26} = \frac{27 \cdot 26 + 170}{170 \cdot 26} = \frac{702 + 170}{4420} = \frac{872}{4420} = \frac{218}{1105} $

["Understanding the Addition of Fractions: $ \frac{27}{170} + \frac{1}{26} = \frac{218}{1105} $", "Calculating the sum of fractions often seems tricky, but mastering the process makes it simple and rewarding. Today, we break down the fraction addition $ \frac{27}{170} + \frac{1}{26} $ step by step to clarify how to combine unlike denominators and simplify the result — in this case, arriving at $ \frac{218}{1105} $.", "---", "### Why Finding a Common Denominator Matters", "When adding fractions with different denominators, we must first find a common denominator — a shared multiple of both denominators. This allows us to combine numerators correctly while preserving the value of each fraction.", "The denominators here are 170 and 26.", "### Step 1: Calculate the Least Common Multiple (LCM) of 170 and 26", "First, factor both numbers:", "- $ 170 = 2 \ imes 5 \ imes 17 $\n- $ 26 = 2 \ imes 13 $", "The LCM takes the highest power of each prime:", "- $ \ ext{LCM} = 2 \ imes 5 \ imes 13 \ imes 17 = 4420 $", "Alternatively, since full LCM is not needed for verification, note that $ 170 \ imes 26 = 4420 $, which serves as a valid common denominator.", "### Step 2: Rewrite Each Fraction with the Common Denominator", "We convert $ \frac{27}{170} $ and $ \frac{1}{26} $ to equivalent fractions with denominator 4420:", "- For $ \frac{27}{170} $:\n $ \frac{27}{170} = \frac{27 \ imes 26}{170 \ imes 26} = \frac{702}{4420} $", "- For $ \frac{1}{26} $:\n $ \frac{1}{26} = \frac{1 \ imes 170}{26 \ imes 170} = \frac{170}{4420} $", "### Step 3: Add the Numerators", "Now, add the numerators while keeping the common denominator:", "$$\n\frac{702 + 170}{4420} = \frac{872}{4420}\n$$", "### Step 4: Simplify the Result", "Next, reduce $ \frac{872}{4420} $ to its lowest terms.", "Find the GCD of 872 and 4420. Using the Euclidean algorithm:", "- $ 4420 \div 872 = 5 $ remainder $ 200 $\n- $ 872 \div 200 = 4 $ remainder $ 72 $\n- $ 200 \div 72 = 2 $ remainder $ 56 $\n- $ 72 \div 56 = 1 $ remainder $ 16 $\n- $ 56 \div 16 = 3 $ remainder $ 8 $\n- $ 16 \div 8 = 2 $ remainder $ 0 $", "So GCD = 8.", "Now divide numerator and denominator by 8:", "$$\n\frac{872 \div 8}{4420 \div 8} = \frac{109}{552}\n$$", "Wait — here’s a correction: Earlier simplification claimed $ \frac{218}{1105} $, but this does not match $ \frac{109}{552} $. Let’s recheck.", "But notice: $ \frac{872}{4420} = \frac{872 \div 4}{4420 \div 4} = \frac{218}{1105} $, which matches the original result!", "Why?", "Because $ 872 \div 4 = 218 $, and $ 4420 \div 4 = 1105 $ — because $ 4 \ imes 1105 = 4420 $. So $ \frac{872}{4420} = \frac{218}{1105} $ is indeed correct.", "Thus, our earlier simplification was correct but mislabeled — simplifying $ \frac{872}{4420} $ by 4 gives $ \frac{218}{1105} $, which is fully reduced because:", "- $ \gcd(218, 1105) = 1 $:\n $ 1105 \div 218 \approx 5.07 $, $ 218 \ imes 5 = 1090 $, remainder 15\n $ 218 \div 15 = 14, remainder,8 $ — continuing shows no common factors.", "Final simplified form: $ \frac{218}{1105} $", "---", "### Why This Example Matters for Learners", "Understanding each step — finding LCM, adjusting denominators, adding numerators, and simplifying — builds foundational math fluency. These skills are essential for algebra, data analysis, and real-world problem solving.", "Whether you're solving equations, calculating ratios, or reviewing your homework, mastering fraction addition is a key step toward math confidence.", "---", "Summary", "- $ \frac{27}{170} + \frac{1}{26} = \frac{27 \cdot 26 + 170}{170 \cdot 26} $\n- $ = \frac{702 + 170}{4420} = \frac{872}{4420} $\n- Simplified by dividing numerator and denominator by 4:\n $ \frac{872 \div 4}{4420 \div 4} = \frac{218}{1105} $", "Final answer:\n$$\n\frac{27}{170} + \frac{1}{26} = \frac{218}{1105}\n$$", "Master these steps and turn tricky fraction sums into confident calculations!"]