Anna Prepares 36 Cookies, And 48 Candies. She Must Pack The Same Number Of Goodies In Each Box. What Is The Greatest Number Of Goods Anna Can Pack In Each Box? Please Help Me Solve Step By Step

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Introduction

In this problem, we are tasked with finding the greatest number of goodies that Anna can pack in each box. To do this, we need to find the greatest common divisor (GCD) of the total number of cookies and candies.

Step 1: Find the Total Number of Goodies

Anna prepares 36 cookies and 48 candies. To find the total number of goodies, we simply add the number of cookies and candies.

36 (cookies) + 48 (candies) = 84 goodies

Step 2: Find the Greatest Common Divisor (GCD)

To find the GCD of 36 and 48, we can use the Euclidean algorithm or list the factors of each number.

Factors of 36:

  • 1, 2, 3, 4, 6, 9, 12, 18, 36

Factors of 48:

  • 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Finding the GCD: The greatest common divisor of 36 and 48 is 12.

Step 3: Determine the Greatest Number of Goodies per Box

Since the GCD of 36 and 48 is 12, the greatest number of goodies that Anna can pack in each box is 12.

Conclusion

In this problem, we found the greatest number of goodies that Anna can pack in each box by finding the greatest common divisor of the total number of cookies and candies. The answer is 12 goodies per box.

Mathematical Proof

To prove that 12 is the greatest number of goodies per box, we can use the following reasoning:

  • If Anna packs 12 goodies in each box, she can pack a total of 84 / 12 = 7 boxes.
  • If Anna packs 11 goodies in each box, she can pack a total of 84 / 11 = 7.636 (not a whole number).
  • If Anna packs 10 goodies in each box, she can pack a total of 84 / 10 = 8.4 (not a whole number).

Therefore, 12 is the greatest number of goodies that Anna can pack in each box.

Real-World Application

This problem has real-world applications in packaging and distribution. For example, a company that produces cookies and candies may need to package them in boxes of a certain size. By finding the greatest common divisor of the total number of cookies and candies, they can determine the greatest number of goodies that can be packed in each box.

Additional Resources

For more information on greatest common divisors and their applications, see the following resources:

Introduction

In our previous article, we solved the problem of finding the greatest number of goodies that Anna can pack in each box. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the greatest common divisor (GCD) and why is it important?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. In the context of the problem, the GCD is important because it determines the greatest number of goodies that can be packed in each box.

Q: How do I find the GCD of two numbers?

A: There are several ways to find the GCD of two numbers. One way is to list the factors of each number and find the greatest common factor. Another way is to use the Euclidean algorithm, which is a step-by-step process for finding the GCD.

Q: What is the Euclidean algorithm and how does it work?

A: The Euclidean algorithm is a step-by-step process for finding the GCD of two numbers. It works by repeatedly dividing the larger number by the smaller number and taking the remainder. The process is repeated until the remainder is zero, at which point the GCD is the last non-zero remainder.

Q: Can I use a calculator to find the GCD?

A: Yes, you can use a calculator to find the GCD. Most calculators have a built-in function for finding the GCD, which can be accessed by pressing the "GCD" or "GCF" button.

Q: What if the GCD is not a whole number?

A: If the GCD is not a whole number, it means that the two numbers do not have a common divisor. In this case, the greatest number of goodies that can be packed in each box is not a whole number.

Q: Can I use the GCD to solve other problems?

A: Yes, the GCD can be used to solve other problems. For example, it can be used to find the least common multiple (LCM) of two numbers, which is the smallest number that is a multiple of both numbers.

Q: What is the least common multiple (LCM) and how is it related to the GCD?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. The LCM is related to the GCD in that the product of the GCD and the LCM is equal to the product of the two numbers.

Q: Can I use the LCM to solve problems related to packing and distribution?

A: Yes, the LCM can be used to solve problems related to packing and distribution. For example, it can be used to determine the greatest number of boxes that can be packed with a certain number of goodies.

Conclusion

In this article, we answered some frequently asked questions related to the great goodie packing problem. We hope that this article has provided you with a better understanding of the GCD and its applications in packing and distribution.

Additional Resources

For more information on the GCD and LCM, see the following resources: