\frac{2}{13} + \frac{1}{16} = \frac{32 + 13}{208} = \frac{45}{208}
["Understanding the Addition of Fractions: (\frac{2}{13} + \frac{1}{16} = \frac{45}{208})", "Fractions are fundamental building blocks in mathematics, and mastering their addition is essential for everything from basic arithmetic to advanced algebra. One commonly encountered challenge is adding two unlike fractions—fractions with different denominators. Let’s explore the step-by-step process behind calculating:", "[\n\frac{2}{13} + \frac{1}{16} = \frac{45}{208}\n]", "---", "### Why Find a Common Denominator?", "When adding or subtracting fractions, the denominators (the bottom numbers) must be the same. This common denominator allows us to add the numerators directly while keeping the fraction intact.", "The denominators here are 13 and 16. Since 13 and 16 have no common factors (they are coprime), their least common denominator (LCD) is simply their product:", "[\n13 \ imes 16 = 208\n]", "---", "### Step-by-Step Addition", "We begin by converting both fractions to have the LCD of 208.", "First fraction: (\frac{2}{13})", "To convert (\frac{2}{13}) to a fraction with denominator 208, multiply numerator and denominator by 16:", "[\n\frac{2}{13} = \frac{2 \ imes 16}{13 \ imes 16} = \frac{32}{208}\n]", "Second fraction: (\frac{1}{16})", "To convert (\frac{1}{16}), multiply numerator and denominator by 13:", "[\n\frac{1}{16} = \frac{1 \ imes 13}{16 \ imes 13} = \frac{13}{208}\n]", "---", "### Adding the Equivalent Fractions", "Now add the two numerators over the common denominator:", "[\n\frac{32}{208} + \frac{13}{208} = \frac{32 + 13}{208} = \frac{45}{208}\n]", "---", "### Final Answer and Verification", "The sum of (\frac{2}{13} + \frac{1}{16}) equals:", "[\n\boxed{\frac{45}{208}}\n]", "This result is fully simplified since 45 and 208 share no common divisors other than 1.", "---", "### Why Knowing This Matters", "Understanding how to add fractions with different denominators improves problem-solving in real-life scenarios such as cooking measurements, mixing solutions, or budgeting with fractional amounts. Additionally, this skill forms a foundation for solving more complex problems in fractions, decimals, and ratios.", "---", "Summary:\nTo add (\frac{2}{13} + \frac{1}{16}), find the LCD (208), convert each fraction accordingly, and add the numerators:", "[\n\frac{2}{13} + \frac{1}{16} = \frac{32}{208} + \frac{13}{208} = \frac{45}{208}\n]", "This confirms:\n[\n\boxed{\frac{2}{13} + \frac{1}{16} = \frac{45}{208}}\n]", "Start practicing now—adding fractions has never been clearer!"]
