How Much Water Must Be Added to Achieve a Desired Water Content in a Mixture?
Abstract: This article examines the mathematics behind solving a common mixture problem, specifically how adding water to a mixture of milk and water changes its composition. By understanding the ratios and using simple algebra, one can determine how much water needs to be added to achieve a target water content.
Understanding the Initial Composition of the Mixture
In a 60 liter mixture of milk and water, we find that 10 liters are water. This can be expressed as a percentage, where 10 liters account for 10% of the total mixture (10/60 * 100 16.67%). The remaining 50 liters (83.33%) are milk.
Step-by-Step Calculation
Let's solve the initial problem where we need to determine how many liters of water must be added to make the water content 20%.
Let W denote the required amount in liters of water that must be added.
The new mixture will have a total volume of (60 W) liters. The new amount of water will be (10 W) liters.
The equation reflecting this is:
frac{10 W}'sright) / (60 W) 20/100
Simplifying this equation:
10 W 0.20(60 W)
10 W 12 0.20W
Rearranging terms:
10 - 12 0.20W - W
-2 -0.80W
W 2 / 0.80
W 2.5
Therefore, 2.5 liters of water must be added to achieve the new water content.
Exploring Other Scenarios
For a different scenario, let's consider a 16-liter mixture containing 10 liters of water. Using similar steps as above:
W for the new water content where 20% of the mixture is water:
frac{10 W}'sright)} / (16 W) 20/100
Simplifying this equation:
10 W 0.20(16 W)
10 W 3.2 0.20W
Rearranging terms:
10 - 3.2 0.20W - W
6.8 -0.80W
W 6.8 / 0.80
W 8.5
Therefore, 8.5 liters of water must be added to the 16-liter mixture to make the water content 20%.
Generalizing the Solution
Let's generalize the process using different mixture volumes and compositions. Consider a 60-liter mixture where the initial amount of water is 10 liters, making up 16.67% of the mixture.
The initial amount of water is:
10 60 * 16.67% 10 liters
The initial amount of milk is:
60 - 10 50 liters
Let x be the amount of water to be added. The new total volume of the mixture will be (60 x) liters. The new amount of water will be (10 x) liters. We want the new amount of water to be 20% of the new total volume:
(10 x) / (60 x) 20/100
Simplifying this equation:
10 x 0.20(60 x)
10 x 12 0.2
Rearranging terms:
10 - 12 0.2 - x
-2 -0.8
x 2 / 0.80
x 2.5
Thus, 2.5 liters of water must be added to achieve the desired water content in the mixture.
Conclusion
The process to solve mixture problems involving water content is systematic and straightforward. By understanding the initial composition and using the correct algebraic equations, one can solve for the amount of water needed to achieve a desired percentage in the mixture. This method can be applied in various scenarios to solve similar problems.