A solution contains 30% alcohol. How much pure alcohol must be added to 10 liters of this solution to make it 50% alcohol?
["How to Increase Alcohol Concentration: Calculating How Much Pure Alcohol to Add to a 30% Alcohol Solution", "When working with alcohol-based solutions, adjusting the concentration is a common request—whether for safety, cost efficiency, or quality control. A typical scenario involves starting with 10 liters of a solution that contains 30% alcohol, and you want to increase its alcohol content to 50% by adding pure alcohol. This article walks through the step-by-step calculation to determine exactly how much pure alcohol to add.", "---", "### The Problem", "You begin with 10 liters of a solution that is 30% alcohol.\nYou want to increase the concentration to 50% by adding pure alcohol (100% alcohol).\nHow many liters of pure alcohol must be added?", "---", "### Step-by-Step Solution", "Let’s define:", "- Initial volume of solution: ( V = 10 ) liters\n- Initial alcohol concentration: ( 30% )\n- Final desired concentration: ( 50% )\n- Amount of pure alcohol to add: ( x ) liters (we want to find ( x ))", "---", "### Step 1: Determine Initial Amounts", "Amount of pure alcohol in the original solution:", "[\n\ ext{Initial alcohol} = 30% \ imes 10 = 0.30 \ imes 10 = 3 \ ext{ liters}\n]", "Amount of pure alcohol added:\n[\nx \ ext{ liters (since pure alcohol is 100%)\n]", "---", "### Step 2: Final Alcohol Amount", "After adding ( x ) liters of pure alcohol, the total volume becomes:", "[\n10 + x \ ext{ liters}\n]", "The total pure alcohol is:", "[\n3 + x \ ext{ liters}\n]", "Concentration after mixing:", "[\n\frac{3 + x}{10 + x} = 0.50 \quad \ ext{(desired concentration)}\n]", "---", "### Step 3: Set Up and Solve the Equation", "[\n\frac{3 + x}{10 + x} = 0.5\n]", "Multiply both sides by ( 10 + x ):", "[\n3 + x = 0.5(10 + x)\n]", "[\n3 + x = 5 + 0.5x\n]", "Subtract ( 0.5x ) from both sides:", "[\n3 + 0.5x = 5\n]", "Subtract 3 from both sides:", "[\n0.5x = 2\n]", "Divide by 0.5:", "[\nx = 4\n]", "---", "### Conclusion", "To increase 10 liters of a 30% alcohol solution to a 50% alcohol solution, you must add:", "[\n\boxed{4 \ ext{ liters of pure alcohol}\n]", "---", "### Summary", "- Start with 3 liters of alcohol in 10 liters of 30% solution\n- Add 4 liters of pure alcohol\n- Final volume: 14 liters\n- Final alcohol: ( 3 + 4 = 7 ) liters\n- Final concentration: ( \frac{7}{14} = 50% )", "Adding just 4 liters of pure alcohol reliably raises the concentration from 30% to 50% in a 10-liter solution.", "---", "### Why This Matters", "This calculation is essential in industries such as beverage production, pharmaceuticals, and chemical manufacturing, where precise control of alcohol concentration ensures product quality, regulatory compliance, and operational efficiency.", "---", "Keywords: alcohol concentration formula, how to increase alcohol percentage, adding pure alcohol to solution, 30% alcohol to 50% alcohol, pharmaceutical dilution calculation, alcohol mixture problem, chemical concentration adjustment."]
