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Alligation alternate problems are a common type of mathematical word problem used to find the ratio in which two or more solutions or quantities should be mixed to achieve a desired concentration or amount. Using conversion factors effectively is essential to solving these problems accurately and efficiently.
Understanding Alligation Alternate Problems
Alligation alternate problems typically involve mixing solutions of different concentrations or quantities. The goal is to determine the correct ratio or amount of each solution needed to achieve a target concentration or total volume.
The Role of Conversion Factors
Conversion factors are ratios or fractions that convert one unit or measurement into another. In alligation problems, they help standardize units, convert percentages to decimal form, and relate quantities to their concentrations.
Common Conversion Factors Used
- Percentage to decimal: 1% = 0.01
- Volume units: liters to milliliters (1 L = 1000 mL)
- Concentration units: grams per liter to grams per milliliter
Applying Conversion Factors Step-by-Step
1. Identify the known quantities and their units.
2. Convert all measurements to consistent units using appropriate conversion factors.
3. Express concentrations or ratios in a comparable form, often as decimals or fractions.
4. Set up the alligation grid or algebraic equation using the converted values.
Example Problem
Suppose you have two solutions: one 20% saline solution and another 50% saline solution. You want to prepare 10 liters of a 30% saline solution. How much of each solution should you mix?
Step 1: Convert percentages to decimals
20% = 0.20, 50% = 0.50, 30% = 0.30
Step 2: Set up the alligation
The difference between each solution’s concentration and the target:
- 50% solution difference: 0.50 – 0.30 = 0.20
- 20% solution difference: 0.30 – 0.20 = 0.10
The ratio of solutions needed:
- 20% solution: 0.20
- 50% solution: 0.10
Using these ratios, the total parts are 0.20 + 0.10 = 0.30. To find the quantities:
Amount of 20% solution: (0.20 / 0.30) × 10 L = 6.67 L
Amount of 50% solution: (0.10 / 0.30) × 10 L = 3.33 L
Conclusion
Using conversion factors effectively streamlines the process of solving alligation alternate problems. By converting all measurements to consistent units and expressing concentrations as decimals, students and teachers can approach these problems with greater confidence and accuracy.