Mixing Milk And Water Ratios A Step By Step Guide
Hey guys! Ever found yourself scratching your head over mixture problems, especially those involving milk and water ratios? You're not alone! These types of questions can seem tricky, but with a systematic approach, they become super manageable. In this article, we're going to break down a classic problem step-by-step, ensuring you not only understand the solution but also the underlying concepts. Let's dive in!
Understanding the Problem
Before we jump into calculations, let's clearly understand the problem we're tackling. The question states: Two vessels contain water and milk in the ratios of 3:4 and 5:4, respectively. These mixtures are then mixed in the ratio of 1:4. What will be the ratio of water and milk in the resulting mixture?
This is a classic mixture problem, and to solve it effectively, we need to break it down into smaller, digestible parts. We'll look at each vessel individually, determine the quantities of water and milk, and then combine them according to the given mixing ratio. So, buckle up, and let's get started!
Step 1: Analyzing the First Vessel
Our first step involves dissecting the contents of the first vessel. The first vessel has water and milk in the ratio of 3:4. What does this actually mean? Well, it tells us that for every 3 parts of water, there are 4 parts of milk. To make things concrete, let's assume the vessel contains a total of 7 liters (since 3 + 4 = 7). This assumption simplifies our calculations without affecting the final ratio.
Now, how much of this 7 liters is water, and how much is milk? To find this out, we divide the total volume (7 liters) proportionally according to the ratio. The fraction of water is 3 parts out of the total 7 parts, which is 3/7. Similarly, the fraction of milk is 4 parts out of 7, which is 4/7. So, in the first vessel, we have:
- Water: (3/7) * 7 liters = 3 liters
- Milk: (4/7) * 7 liters = 4 liters
This gives us a clear picture of the composition of the first vessel. Now, we're ready to move on to the second vessel and repeat the process.
Step 2: Analyzing the Second Vessel
Next up, let's break down the composition of the second vessel. The second vessel contains water and milk in the ratio of 5:4. Similar to our approach with the first vessel, this means that for every 5 parts of water, there are 4 parts of milk. Again, to keep things simple, let's assume this vessel also contains a total of 9 liters (5 + 4 = 9). This assumption is crucial for our calculations, allowing us to work with concrete quantities.
Now, let's figure out the individual quantities of water and milk in this vessel. The fraction of water is 5 parts out of the total 9 parts, which is 5/9. The fraction of milk is 4 parts out of 9, which is 4/9. Therefore, in the second vessel, we have:
- Water: (5/9) * 9 liters = 5 liters
- Milk: (4/9) * 9 liters = 4 liters
Great! We now know the exact amounts of water and milk in the second vessel. With both vessels analyzed, we're one step closer to finding the final ratio. Let's move on to the crucial step of mixing these vessels.
Step 3: Mixing the Vessels in the Given Ratio
This is where the problem gets a bit more interesting. The contents of the two vessels are mixed in the ratio of 1:4. This means that for every 1 part of the mixture from the first vessel, we add 4 parts from the second vessel. To make this clearer, let's say we take 1 liter from the first vessel and 4 liters from the second vessel. This specific ratio is key to solving the problem correctly.
Now, we need to calculate how much water and milk we're taking from each vessel according to this mixing ratio. From the first vessel (1 liter), we have:
- Water: (3/7) * 1 liter = 3/7 liters
- Milk: (4/7) * 1 liter = 4/7 liters
From the second vessel (4 liters), we have:
- Water: (5/9) * 4 liters = 20/9 liters
- Milk: (4/9) * 4 liters = 16/9 liters
These calculations give us the precise amounts of water and milk being contributed from each vessel. The next step is to combine these quantities to find the total amounts in the final mixture.
Step 4: Calculating the Total Water and Milk
Now that we know how much water and milk we're getting from each vessel, it's time to add them up. We need to find the total amount of water and the total amount of milk in the final mixture. This will give us the numbers we need to determine the final ratio.
Let's start with the water. We have 3/7 liters of water from the first vessel and 20/9 liters from the second vessel. Adding these together:
Total Water = (3/7) + (20/9)
To add these fractions, we need a common denominator, which is 63. So, we convert the fractions:
Total Water = (3/7) * (9/9) + (20/9) * (7/7) = (27/63) + (140/63) = 167/63 liters
Now, let's calculate the total amount of milk. We have 4/7 liters of milk from the first vessel and 16/9 liters from the second vessel. Adding these up:
Total Milk = (4/7) + (16/9)
Again, we use the common denominator of 63:
Total Milk = (4/7) * (9/9) + (16/9) * (7/7) = (36/63) + (112/63) = 148/63 liters
So, in our final mixture, we have 167/63 liters of water and 148/63 liters of milk. We're almost there! The last step is to express these quantities as a ratio.
Step 5: Determining the Final Ratio
We've calculated the total amounts of water and milk in the final mixture. Now, the crucial final step is to express these amounts as a ratio. We have 167/63 liters of water and 148/63 liters of milk. To find the ratio, we simply put these two quantities in a ratio:
Ratio of Water to Milk = (167/63) : (148/63)
Notice that both fractions have the same denominator (63). This makes our job easier because we can cancel out the denominators:
Final Ratio = 167 : 148
And there you have it! The resulting mixture has water and milk in the ratio of 167:148. This matches option (c) in the given choices. You've successfully solved a classic mixture problem!
Conclusion
Mixture problems can seem daunting at first, but by breaking them down into smaller steps, they become much more manageable. We started by understanding the problem, then analyzed each vessel individually, mixed the contents in the given ratio, calculated the total amounts of water and milk, and finally, determined the final ratio. Remember, the key is to stay organized and methodical in your approach.
So, next time you encounter a similar problem, don't sweat it! Just follow these steps, and you'll be mixing milk and water ratios like a pro in no time. Keep practicing, and you'll master these problems in no time. Happy mixing, guys!