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Alligation alternate is a useful mathematical technique used to solve mixture problems efficiently. It helps in finding the ratio in which two or more ingredients should be mixed to achieve a desired concentration or concentration level. Practicing problems with solutions enhances understanding and speeds up problem-solving skills.
Understanding Alligation Alternate
The alligation alternate method involves comparing the differences between the given concentrations and the desired concentration to determine the ratio in which the components should be mixed. It simplifies complex mixture problems by focusing on differences rather than direct calculations.
Practice Problems with Solutions
Problem 1
How much of a 40% alcohol solution should be mixed with a 70% alcohol solution to get 100 liters of a 50% alcohol solution?
Solution:
Let the quantity of 40% solution be x liters, and the 70% solution be y liters.
According to the problem:
- x + y = 100
- 0.40x + 0.70y = 0.50 * 100 = 50
Using alligation alternate, the difference between the concentrations and the desired concentration is:
- 70% – 50% = 20
- 50% – 40% = 10
Ratio of 40% to 70% solution = 20 : 10 = 2 : 1
Therefore, the 40% solution should be mixed in the ratio 2 parts, and the 70% solution in the ratio 1 part.
To find the actual quantities:
Since total mixture is 100 liters:
Let 2 parts correspond to x liters, then 1 part corresponds to y liters, with 3 parts total.
Thus, x = (2/3) * 100 ≈ 66.67 liters of 40% solution
And y = (1/3) * 100 ≈ 33.33 liters of 70% solution.
Problem 2
A chemist has two solutions: one is 30% acid, and the other is 60% acid. How much of each should be mixed to obtain 80 liters of a 45% acid solution?
Solution:
Let the amount of 30% solution be x liters, and the 60% solution be y liters.
Set up the equations:
- x + y = 80
- 0.30x + 0.60y = 0.45 * 80 = 36
Differences for alligation:
- 60% – 45% = 15
- 45% – 30% = 15
Ratio of 30% to 60% solutions = 15 : 15 = 1 : 1
Both solutions are mixed in equal parts:
x = y = 80 / 2 = 40 liters each.
Key Tips for Solving Alligation Problems
– Always identify the concentrations and the total quantity required.
– Use the difference method to find the ratio.
– Convert ratios into actual quantities based on total volume.
Conclusion
Practicing alligation problems with solutions helps develop quick mental calculation skills and a clear understanding of mixture concepts. Regular practice ensures mastery of the technique and prepares students for competitive exams and real-world applications.