Solving Mixture Problems: Half and Half Water and Syrup
When dealing with mixture problems, it is essential to understand the ratios involved and how to manipulate these ratios to achieve a specific composition. This article explores a classic example where a vessel contains a mixture of water and syrup in certain proportions. We will walk through the steps to solve the problem of determining how much of the liquid should be drawn out and replaced by water to make the mixture half water and half syrup.
The Problem
Consider a vessel that is filled with a mixture consisting of 5 parts water and 7 parts syrup. The challenge is to determine how much of this liquid should be replaced with water to achieve a 1:1 ratio (half water and half syrup).
Solution Steps
Step 1: Determine the Proportions
First, find the total parts of the original mixture.
Total Parts: 5 parts water 7 parts syrup 12 parts
The proportion of each component is calculated as follows:
Proportion of Water: 5 / 12
Proportion of Syrup: 7 / 12
Step 2: Set Up the Equation
Let's denote the amount of liquid to be replaced as x parts. When this amount is replaced with water, the amount of water and syrup in the mixture changes:
New Amount of Water: 5 - x parts
New Amount of Syrup: 7 - x parts
For the mixture to be half water and half syrup, the new amounts of water and syrup should be equal:
Equation: (5 - x) / (7 - x) 1 / 1
Solving the equation:
5 - x 7 - x
2x 2
x 1
Therefore, 1 part of the original mixture should be drawn out and replaced with water to achieve a 1:1 ratio of water to syrup.
Alternative Method
Alternatively, you can use the equations for syrup and water directly:
Equation for Syrup: 7 - (7/12)x (1/2) * 12
Solving for x:
7 - (7/12)x 6
(7/12)x 1
x 12/7
Equation for Water: 5 - (5/12)x (1/2) * 12
Solving for x using the same method:
5 - (5/12)x 6
(5/12)x -1
x 12/7
Both methods confirm that 1.714 (approximately 1.71) parts of the mixture need to be removed and replaced with water to achieve the desired proportion.
Real-World Application
Mixture problems such as these are crucial in various fields, including chemistry, cooking, and even industrial processes. Understanding the underlying principles can help in calculating the correct proportions needed to achieve specific mixtures.
In this example, we started with a 12:14 ratio of syrup to water. By removing 1 part and replacing it with water, we managed to balance the components to achieve an equal ratio. This method can be applied to other similar problems by following the steps outlined above: determine the initial proportions, set up the equation for the desired ratio, and solve for the unknown variable.
Conclusion
Mixture problems can be solved by carefully using proportions and solving equations. Whether it is adjusting a recipe or balancing a scientific solution, these methods provide a practical and logical approach to achieving the desired composition.
By understanding and applying these concepts, you can tackle a wide variety of similar problems with confidence. If you have any more questions or need further assistance, feel free to ask!