Fraction Of Cargo Unloaded In The Fourth Hour

by ADMIN 46 views

Hey guys! Let's tackle this math problem together. It's all about figuring out what fraction of a cargo shipment was unloaded during the fourth hour, given the fractions unloaded in the first three hours. It's like piecing together a puzzle, but with fractions instead of picture pieces. So, grab your thinking caps, and let’s dive right in!

Understanding the Problem

So, here's the deal: we know that in the first hour, 5/18 of the cargo was unloaded. During the second hour, they managed to unload 25% of the cargo. The third hour saw 1/6 of the cargo being taken off. Our mission is to find out what fraction of the total cargo was unloaded in that final, fourth hour. To crack this, we're going to have to do a little bit of fraction manipulation and some basic arithmetic. It's all about figuring out what's left after the first three hours have done their thing. We need to determine the remainder of the cargo after the initial unloading spree. This problem is a classic example of how fractions and percentages pop up in real-world scenarios, like logistics and shipping. By solving this, we're not just doing math; we're understanding how to manage and track resources efficiently!

Step-by-Step Solution

First things first, we need to make sure all our fractions and percentages are speaking the same language. Right now, we have fractions and a percentage, which is like trying to have a conversation with someone who only speaks emoji. Let's convert that percentage into a fraction so we can work with everything consistently. Remember, 25% is the same as 25/100. We can simplify this fraction by dividing both the numerator and the denominator by 25, which gives us 1/4. Now we're talking the same language! Next up, we need to figure out the total fraction of cargo unloaded in the first three hours. This means adding up all those fractions we now have: 5/18 + 1/4 + 1/6. But hold on, we can't just add these fractions as they are; they need a common denominator. The least common multiple of 18, 4, and 6 is 36. So, let's convert each fraction to have a denominator of 36:

  • 5/18 = (5 * 2) / (18 * 2) = 10/36
  • 1/4 = (1 * 9) / (4 * 9) = 9/36
  • 1/6 = (1 * 6) / (6 * 6) = 6/36

Now we can add them up: 10/36 + 9/36 + 6/36 = 25/36. So, in the first three hours, 25/36 of the cargo was unloaded. To find out how much was unloaded in the fourth hour, we need to subtract this fraction from the whole, which is represented as 1/1 or 36/36. Therefore, the fraction of cargo unloaded in the fourth hour is 36/36 - 25/36 = 11/36. Bingo! We've found our answer.

Alternative Approach: Using Decimals

Alright, math enthusiasts, here's another way to slice this problem! Instead of sticking strictly to fractions, we can convert everything to decimals. This can sometimes simplify the calculations and make the problem a bit more intuitive for some of you. First, let's convert the given fractions and the percentage into decimals:

  • 5/18 is approximately 0.2778
  • 25%, which we know is 1/4, is 0.25
  • 1/6 is approximately 0.1667

Now, we add these decimals together to find the total proportion of the cargo unloaded in the first three hours: 0.2778 + 0.25 + 0.1667 = 0.6945. To find the proportion unloaded in the fourth hour, we subtract this sum from 1 (since the whole cargo is represented as 1 in decimal form): 1 - 0.6945 = 0.3055. Now, let's convert this decimal back into a fraction to match our previous answer. 0.3055 is approximately 3055/10000. If we simplify this fraction (which can be a bit tricky), we'll find that it's very close to 11/36. The slight difference is due to rounding errors when converting the fractions to decimals. This approach gives us a way to verify our original answer and shows how different mathematical tools can be used to solve the same problem!

Common Mistakes to Avoid

Alright, let's talk about some common slip-ups that people often make when tackling problems like this. Knowing these pitfalls can save you from making unnecessary errors and help you ace similar questions in the future.

  1. Forgetting to Convert to a Common Denominator: This is a classic! You can't add or subtract fractions unless they have the same denominator. It's like trying to add apples and oranges without converting them to a common unit, like "pieces of fruit". Always make sure your fractions are on the same footing before you start adding or subtracting.
  2. Misunderstanding Percentages: Percentages can be tricky if you don't remember what they represent. Always remember that a percentage is just a fraction out of 100. So, 25% is 25/100, not just 25. Failing to convert percentages to fractions or decimals correctly can throw off your entire calculation.
  3. Rounding Errors: If you're using the decimal approach, rounding errors can creep in, especially if you round too early in the process. Try to keep as many decimal places as possible during your calculations and only round at the very end to get the most accurate result.
  4. Not Double-Checking Your Work: It's always a good idea to double-check your calculations, especially when dealing with multiple steps. A small mistake early on can snowball into a big error later. Take a few extra moments to review your work and make sure everything adds up (literally!).
  5. Ignoring the Question's Context: Always make sure you understand what the question is asking. In this case, we were looking for the fraction of cargo unloaded in the fourth hour, not the total unloaded in all four hours. Misunderstanding the question can lead you to solve for the wrong thing entirely.

By keeping these common mistakes in mind, you'll be well-equipped to tackle similar problems with confidence and accuracy!

Real-World Applications

So, you might be wondering, "Where would I ever use this stuff in real life?" Well, believe it or not, understanding fractions and percentages is super useful in a ton of everyday situations. Let's explore some real-world applications where these skills come in handy.

  1. Logistics and Shipping: This is the most obvious one, given our problem! In logistics, knowing how to calculate fractions and percentages is crucial for tracking inventory, managing shipments, and optimizing delivery routes. For example, a shipping company might need to calculate what fraction of a container has been unloaded at each stop, or what percentage of a truck's capacity is being used.
  2. Cooking and Baking: Recipes often use fractions to specify ingredient amounts. Knowing how to adjust these fractions is essential when scaling a recipe up or down. For example, if a recipe calls for 1/2 cup of flour and you want to double the recipe, you'll need to know that 1/2 + 1/2 = 1 cup.
  3. Personal Finance: Fractions and percentages are used all the time in personal finance. Calculating interest rates, figuring out discounts, and budgeting your expenses all involve working with these concepts. For example, if an item is 20% off, you need to know how to calculate the sale price.
  4. Construction and Engineering: These fields rely heavily on precise measurements, which often involve fractions. Architects and engineers need to be able to work with fractions to design structures and ensure that everything fits together correctly.
  5. Sales and Retail: Retailers use percentages to calculate markups, discounts, and sales tax. Understanding these calculations can help you make informed purchasing decisions and avoid being overcharged.

By mastering fractions and percentages, you're not just learning math; you're gaining valuable skills that can help you navigate the world more effectively. So, keep practicing, and you'll be amazed at how often these concepts come up in your daily life!

Conclusion

Alright, we've reached the end of our mathematical journey! We successfully figured out that 11/36 of the cargo was unloaded during the fourth hour. Whether you prefer sticking to fractions or diving into decimals, you've got the tools to solve similar problems. Remember, math isn't just about getting the right answer; it's about understanding the process and building skills that you can use in all sorts of situations. So, keep practicing, stay curious, and don't be afraid to tackle those tricky problems head-on. You've got this!