Dividing Candies: How Many Ways Can Mikael Share?

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Hey guys! Let's dive into this fun math problem about Mikael and her bag of candies. She's got 84 delicious pieces and wants to share them with her friends. But there are a few rules: she doesn't want to make more than 20 bags, and she definitely wants to share with at least 3 friends. So, how many different ways can she divide up the candy? Let's figure it out together!

Understanding the Problem

First, let's break down what we know. The core of our problem lies in dividing the 84 candies equally. This means we need to find the factors of 84. A factor is simply a number that divides evenly into another number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because 12 can be divided by each of these numbers without any remainder. So, to help Mikael share her candies, we need to identify all the numbers that divide 84 perfectly.

To do this, we will systematically go through numbers and check if they divide 84. Start with 1, which always divides any number, and continue checking each number up to the square root of 84 (which is a little over 9) because factors usually come in pairs. If we find that a number is a factor, we also know its corresponding pair is a factor. This significantly cuts down our work. Remember, the goal here is to ensure every friend gets the same number of candies, which is super important for fair sharing and happy friends! Also, it is important to remember that Mikael wants to share the candies with at least three friends and does not want to use more than 20 bags. These additional constraints help us narrow down the possibilities and identify the most viable options for Mikael.

Finding the Factors of 84

Let's roll up our sleeves and find those factors! We'll go through the numbers one by one, like detectives on a candy-sharing case:

  • 1: Of course, 1 divides 84 (84 / 1 = 84). So, 1 and 84 are a factor pair.
  • 2: 84 is an even number, so it's divisible by 2 (84 / 2 = 42). That gives us the pair 2 and 42.
  • 3: The sum of the digits of 84 (8 + 4 = 12) is divisible by 3, so 84 is also divisible by 3 (84 / 3 = 28). Our pair here is 3 and 28.
  • 4: 84 / 4 = 21, so 4 and 21 are a pair.
  • 5: 84 doesn't end in 0 or 5, so it's not divisible by 5.
  • 6: 84 is divisible by both 2 and 3, so it's also divisible by 6 (84 / 6 = 14). This gives us the pair 6 and 14.
  • 7: 84 / 7 = 12, so 7 and 12 are a factor pair.
  • 8: 84 is not divisible by 8.
  • 9: 84 is not divisible by 9.

So, we've found all the factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84. That's quite a few ways to divide up the candies!

Applying the Constraints

Now, let's bring in those rules Mikael has. Remember, she wants to make no more than 20 bags and share with at least 3 friends. This means we can eliminate some of our factors.

The factors that represent the number of bags Mikael can make are: 1, 2, 3, 4, 6, 7, 12, 14. We exclude factors greater than 20 because Mikael doesn't want to make more than 20 bags. We also need to consider that she wants to share with at least 3 friends. This eliminates the possibilities of 1 and 2 bags, as she needs to distribute candies to a minimum of three friends. Now, let's consider each remaining possibility:

  • 3 bags: Mikael can put 84 / 3 = 28 candies in each bag.
  • 4 bags: Mikael can put 84 / 4 = 21 candies in each bag.
  • 6 bags: Mikael can put 84 / 6 = 14 candies in each bag.
  • 7 bags: Mikael can put 84 / 7 = 12 candies in each bag.
  • 12 bags: Mikael can put 84 / 12 = 7 candies in each bag.
  • 14 bags: Mikael can put 84 / 14 = 6 candies in each bag.

These are all the possible ways Mikael can divide the candies while following her rules. Each option ensures that every bag contains an equal number of candies, which is the essence of fair sharing!

Listing the Possible Ways

Alright, let's put it all together and list the ways Mikael can divide her candies:

  1. 3 bags: 28 candies per bag
  2. 4 bags: 21 candies per bag
  3. 6 bags: 14 candies per bag
  4. 7 bags: 12 candies per bag
  5. 12 bags: 7 candies per bag
  6. 14 bags: 6 candies per bag

So, Mikael has 6 different ways to divide her 84 candies, making sure each of her friends gets a fair share and staying within her self-imposed bag limit. That’s some sweet math in action!

Why This Matters

This isn't just about candies, guys! This problem shows how math, specifically factors and division, plays a role in everyday situations. Think about it: sharing snacks, organizing items, or even planning events – they all involve dividing things equally. Understanding factors helps us make fair and efficient decisions. Plus, it's a great way to make sure everyone gets their fair share of candy!

By working through this problem, we've not only helped Mikael figure out her candy situation, but we've also flexed our math muscles and seen how these concepts can be applied in the real world. Math isn't just numbers on a page; it's a tool that helps us solve problems and make sense of the world around us. So, next time you're sharing something, remember the factors and make sure the distribution is sweet and fair!

Conclusion

So, there you have it! Mikael has six awesome ways to divide her 84 candies among her friends, making sure everyone gets an equal treat. We used factors and a little bit of logic to solve this candy conundrum. Remember, math is all around us, making everyday tasks a little sweeter!

This problem illustrates the practical application of mathematical concepts in everyday situations. Factors and division are not just abstract ideas; they are tools that we use to solve real-world problems. Whether it's sharing candies, organizing items, or planning events, the ability to divide things equally is essential for fairness and efficiency. By understanding and applying these concepts, we can make informed decisions and ensure that everyone gets their fair share. So, next time you encounter a situation that requires division, remember the lessons learned from Mikael's candy-sharing adventure and approach it with confidence.

In conclusion, the problem of dividing candies highlights the importance of math in our daily lives. It demonstrates how factors and division can be used to solve practical problems and make fair decisions. By breaking down the problem, identifying the relevant factors, and applying the given constraints, we were able to determine all the possible ways Mikael could divide her candies. This exercise not only helps Mikael share her treats equitably but also reinforces the value of mathematical thinking in everyday situations. So, keep exploring the world of math, and you'll be amazed at how it can help you solve problems and make sense of the world around you.