Fraction Calculation: Find The Remainder!

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Fraction Calculation: Find the Remainder!

Hey guys! Let's dive into a fraction problem that might seem a bit daunting at first glance, but I promise we'll break it down and make it super easy to understand. We're going to figure out how much is left over when we compare two different sets of fraction operations. So, grab your thinking caps, and let's get started!

Understanding the Question

The question we're tackling is: How much is left over from 6/5 of 8/9 of 3/4 to be equal to half of 3/5 of 4/15 of 5/8? And we have some multiple-choice answers to choose from: a) 1/4, b) 1/2, c) 3/4, d) 3/8, and e) 3/5. This looks like a mouthful, but don't worry, we'll take it step by step. The core of this problem lies in understanding how to multiply fractions and then finding the difference between two resulting fractions. Remember, fractions represent parts of a whole, and these calculations help us see how these parts relate to each other. Let's begin by simplifying each side of the problem separately.

Step-by-Step Breakdown

Part 1: Calculating 6/5 of 8/9 of 3/4

First, we need to calculate "6/5 of 8/9 of 3/4". In mathematics, "of" usually means multiplication. So, we need to multiply these three fractions together:

(6/5) * (8/9) * (3/4)

To multiply fractions, we simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together:

Numerator: 6 * 8 * 3 = 144 Denominator: 5 * 9 * 4 = 180

So, the fraction we get is 144/180. Now, let's simplify this fraction. Both 144 and 180 are divisible by 36. Dividing both the numerator and the denominator by 36, we get:

144 ÷ 36 = 4 180 ÷ 36 = 5

Thus, 6/5 of 8/9 of 3/4 simplifies to 4/5. This means we've taken a portion of a portion of a portion, and what we're left with after these successive multiplications is 4/5. This fraction, 4/5, is our starting point. We still need to figure out what we're comparing it to before we can figure out what's "left over".

Part 2: Calculating Half of 3/5 of 4/15 of 5/8

Next, we need to calculate "half of 3/5 of 4/15 of 5/8". Again, "of" means multiplication. Also, "half of" something means multiplying by 1/2. So, we have:

(1/2) * (3/5) * (4/15) * (5/8)

Let's multiply all the numerators and denominators together:

Numerator: 1 * 3 * 4 * 5 = 60 Denominator: 2 * 5 * 15 * 8 = 1200

So, the fraction is 60/1200. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 60:

60 ÷ 60 = 1 1200 ÷ 60 = 20

Thus, half of 3/5 of 4/15 of 5/8 simplifies to 1/20. We have now simplified both parts of the original question. We know that one part equals 4/5, and the other part equals 1/20. Now all we need to do is find the difference to figure out how much is "left over."

Finding the Difference

The question asks how much is left over from 4/5 to be equal to 1/20. This means we need to subtract 1/20 from 4/5:

4/5 - 1/20

To subtract fractions, we need a common denominator. The least common denominator (LCD) of 5 and 20 is 20. So, we need to convert 4/5 to an equivalent fraction with a denominator of 20. To do this, we multiply both the numerator and the denominator of 4/5 by 4:

(4 * 4) / (5 * 4) = 16/20

Now we can subtract:

16/20 - 1/20 = (16 - 1) / 20 = 15/20

Finally, we simplify 15/20 by dividing both the numerator and the denominator by their greatest common divisor, which is 5:

15 ÷ 5 = 3 20 ÷ 5 = 4

So, 15/20 simplifies to 3/4.

The Answer

Therefore, the amount left over from 6/5 of 8/9 of 3/4 to be equal to half of 3/5 of 4/15 of 5/8 is 3/4. Looking at our multiple-choice answers, the correct answer is:

c) 3/4

So, there you have it! The answer is 3/4. By breaking down the problem into smaller, manageable steps, we were able to easily solve what initially seemed like a complex fraction problem. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a fraction master in no time!

Tips for Solving Fraction Problems

  • Read Carefully: Always read the problem carefully to understand what it's asking.
  • Break It Down: Break the problem down into smaller, more manageable steps.
  • Simplify: Simplify fractions whenever possible to make calculations easier.
  • Find Common Denominators: When adding or subtracting fractions, always find a common denominator.
  • Double-Check: Always double-check your work to avoid errors.

Why are Fractions Important?

Understanding fractions is crucial, guys, not just for math class but also for real-life situations! Think about cooking, where you often need to halve or quarter recipes. Or consider measuring ingredients for a science experiment. Even splitting a pizza fairly among friends involves fractions! These mathematical concepts show up all over the place in everyday scenarios, which is why grasping them is so valuable. The practical applications of fractions are endless. They also build a foundation for more advanced mathematical concepts that are essential in various fields like engineering, finance, and computer science.

Conclusion

Solving fraction problems like this one can seem tricky at first, but with a systematic approach, it becomes much easier. Remember to break down the problem into smaller parts, simplify when possible, and always double-check your work. With practice, you'll become more comfortable and confident in handling fractions. Fractions are a fundamental part of mathematics, and mastering them opens doors to more complex concepts and real-world applications. Keep practicing, and you'll be a fraction pro in no time! Thanks for joining me, and I hope this explanation helps! Keep up the great work, and remember, math can be fun!