Solution: Let total days be $ x $. Ratio $ \frac{3}{8} = \frac{15}{x} $. Cross-multiply: $ 3x = 120 $. Divide: $ x = 40 $. \boxed{40}

["How to Solve Proportions Using Cross-Multiplication: A Step-by-Step Guide", "Understanding how to solve proportions is a fundamental math skill useful in many real-world applications—from scaling recipes to calculating rates and understanding maps. One of the most efficient ways to solve a proportion like ( \frac{3}{8} = \frac{15}{x} ) is through cross-multiplication. In this article, we’ll walk you through the solution step-by-step and explain why this method works.", "---", "### What is a Proportion?", "A proportion is an equation that states two ratios are equal. For example:", "[\n\frac{a}{b} = \frac{c}{d}\n]", "This expression means that the fraction ( \frac{a}{b} ) has the same value as ( \frac{c}{d} ). Proportions allow us to find unknown values when three out of the four terms are known.", "---", "### Step-by-Step Solution: Using Cross-Multiplication", "Let’s solve the proportion:", "[\n\frac{3}{8} = \frac{15}{x}\n]", "Step 1: Cross-multiply\nMultiply the numerator of the first fraction by the denominator of the second, and set it equal to the numerator of the second times the denominator of the first:", "[\n3 \ imes x = 8 \ imes 15\n]", "Step 2: Simplify the right side\nCalculate ( 8 \ imes 15 ):", "[\n3x = 120\n]", "Step 3: Solve for ( x )\nDivide both sides by 3:", "[\nx = \frac{120}{3} = 40\n]", "---", "### Final Answer", "[\n\boxed{40}\n]", "So, the total number of days ( x ) is 40.", "---", "### Why This Method Works", "Cross-multiplication is based on the principle that if ( \frac{a}{b} = \frac{c}{d} ), then ( a \ imes d = b \ imes c ). This method avoids complex fraction arithmetic and provides a clear, quick path to solving for the unknown.", "---", "### Real-World Applications", "- Scaling Recipes: If 3 cups feed 8 people, how much for 15 servings?\n- Map Scaling: If 3 cm equals 8 km in reality, how many km for 15 cm?\n- Conversion: If 3 hours equals ( x ) minutes and 15 minutes = 8 hours, solve for ( x ).", "---", "### Summary", "Solving proportions using cross-multiplication is fast, logical, and widely applicable. By following just a few simple steps—set up the ratio, cross-multiply, and simplify—you can find unknown quantities easily and accurately. Use this technique confidently in your math practice and everyday problem-solving!", "---", "Keywords: proportion equation, cross-multiply method, solving for x, ratio problem, math solution, real-world math, equivalent ratios, algebra basics", "Meta Description: Learn how to solve proportions using cross-multiplication with a step-by-step example. Discover why solving ( \frac{3}{8} = \frac{15}{x} ) gives ( x = 40 ) using simple math principles."]