\frac{5}{4} = \frac{15}{x} \implies 5x = 60 \implies x = 12

["Understanding the Equation: How (\frac{5}{4} = \frac{15}{x} \implies 5x = 60 \implies x = 12)", "Solving equations step-by-step is a fundamental skill in algebra, and mastering the process helps build confidence in handling fractions and proportions. This article explores a common but powerful method used to solve the equation:", "[\n\frac{5}{4} = \frac{15}{x}\n]", "We’ll break down the solution clearly into three stages—cross-multiplication, algebraic rearrangement, and final simplification—to show exactly how we go from the original equation (\frac{5}{4} = \frac{15}{x}) to the answer (x = 12).", "---", "### Step 1: Cross-Multiplication – Eliminating Fractions", "The first step when solving proportions like (\frac{a}{b} = \frac{c}{d}) is to eliminate the fractions by cross-multiplying. This means multiplying the numerator of the first fraction by the denominator of the second, and vice versa:", "[\n\frac{5}{4} = \frac{15}{x} \implies 5 \cdot x = 4 \cdot 15\n]", "This gives us:", "[\n5x = 60\n]", "This transformation simplifies the original equation into a basic linear form, making it easier to solve for (x).", "---", "### Step 2: Solving for (x)", "Now that we have (5x = 60), we isolate (x) by dividing both sides of the equation by 5:", "[\nx = \frac{60}{5} = 12\n]", "Thus, the solution is:", "[\nx = 12\n]", "This result confirms that when the fraction (\frac{15}{x}) equals (\frac{5}{4}), then (x) must be 12 to maintain equality.", "---", "### Why This Method Works (A Quick Algebraic Insight)", "Cross-multiplying is justified by the principle of proportionality:\nIf (\frac{a}{b} = \frac{c}{d}), then (a \cdot d = b \cdot c). Applying this to our equation preserves equality and allows straightforward algebraic manipulation. This technique is especially effective for simple rational equations like the one above, avoiding the need for more complex steps.", "---", "### Practical Applications of Solving Proportions", "Understanding equations like (\frac{5}{4} = \frac{15}{x}) is more than just academic — this skill applies in real-world scenarios such as:", "- Scaling recipes up or down\n- Calculating unit rates and costs\n- Converting measurements in science and engineering\n- Solving for unknowns in physics problems", "Mastering these algebraic techniques strengthens problem-solving abilities across STEM fields.", "---", "### Summary", "To solve (\frac{5}{4} = \frac{15}{x}):", "1. Cross-multiply: (5x = 60)\n2. Divide both sides by 5: (x = 12)\n3. Verify by substituting (x = 12) back into the original equation:\n (\frac{5}{4} = \frac{15}{12} = \frac{5}{4}) ✓", "This clear, step-by-step approach makes proportional reasoning accessible, empowering learners to tackle similar problems confidently.", "---", "Key Takeaways:", "- Cross-multiply to eliminate fractions\n- Isolate the variable by balancing both sides of the equation\n- Always verify the solution by plugging it back into the original equation", "With consistent practice, solving equations like (\frac{5}{4} = \frac{15}{x}) becomes second nature — a key building block in algebra and beyond.", "---", "Related Topics:", "- Rational equations\n- Cross-multiplication explained\n- Solving for variables in proportions\n- Step-by-step algebra worksheets", "Feel free to explore these resources to strengthen your algebra skills!"]