Greatest Common Factor How To Divide Marbles Into Equal Groups

Understanding the Problem

In this mathematical problem, Amina and Salman have a collection of marbles. Amina possesses 30 marbles, while Salman has 45 marbles. Their objective is to organize these marbles into equal groups, ensuring that they do not mix the marbles belonging to each person. The core question here revolves around determining the greatest number of marbles that can be accommodated in a single group while adhering to the condition of equal distribution. This problem elegantly introduces the concept of the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD). The Highest Common Factor (HCF) is the largest number that divides two or more numbers without leaving a remainder. In simpler terms, it's the biggest number that fits perfectly into both 30 and 45. To solve this, we need to find the HCF of 30 and 45. The HCF will tell us the maximum number of marbles we can put in each group. Think of it like this: we want to find the biggest box we can use to pack both Amina's and Salman's marbles, with each box having the same number of marbles and no marbles left over. This is a practical application of HCF, showing how it helps in real-life situations where we need to divide things equally. Understanding this concept is crucial not only for solving this specific problem but also for grasping other mathematical concepts related to number theory and division. The problem also subtly touches upon the idea of factors and multiples, which are fundamental building blocks in mathematics. Factors are numbers that divide evenly into a given number, while multiples are numbers obtained by multiplying a given number by an integer. By finding the HCF, we are essentially identifying a common factor that is the largest possible, thereby ensuring the most efficient grouping of marbles.

Finding the Highest Common Factor (HCF)

To find the Highest Common Factor (HCF) of 30 and 45, we can use several methods, but let's focus on the prime factorization method, which is a straightforward and reliable approach. Prime factorization involves breaking down each number into its prime factors. Prime factors are prime numbers that, when multiplied together, give the original number. A prime number is a number greater than 1 that has only two factors: 1 and itself (e.g., 2, 3, 5, 7, 11). First, let's find the prime factors of 30. We can start by dividing 30 by the smallest prime number, which is 2. 30 divided by 2 is 15. Now, we look for the prime factors of 15. 15 is not divisible by 2, so we move to the next prime number, which is 3. 15 divided by 3 is 5. Both 3 and 5 are prime numbers, so we have found all the prime factors of 30. The prime factorization of 30 is 2 x 3 x 5. Next, let's find the prime factors of 45. 45 is not divisible by 2, so we try dividing it by 3. 45 divided by 3 is 15. We already know the prime factors of 15 from the previous step, which are 3 and 5. So, the prime factorization of 45 is 3 x 3 x 5, which can also be written as 3² x 5. Now that we have the prime factorizations of both numbers, we can identify the common prime factors. Both 30 and 45 share the prime factors 3 and 5. To find the HCF, we multiply the common prime factors with the lowest power they appear in either factorization. The common prime factor 3 appears once in the factorization of 30 (3¹) and twice in the factorization of 45 (3²). So, we take the lowest power, which is 3¹. The common prime factor 5 appears once in both factorizations (5¹), so we take 5¹. Therefore, the HCF of 30 and 45 is 3 x 5 = 15. This means the greatest number of marbles that can be in one group is 15.

Determining the Number of Groups

Now that we know the greatest number of marbles in one group is 15, we can determine how many groups Amina and Salman can make. To do this, we simply divide the number of marbles each person has by the HCF, which is 15. Amina has 30 marbles. To find out how many groups Amina can make, we divide 30 by 15: 30 / 15 = 2. So, Amina can make 2 groups. Each of these groups will contain 15 marbles. Salman has 45 marbles. To find out how many groups Salman can make, we divide 45 by 15: 45 / 15 = 3. So, Salman can make 3 groups. Each of these groups will also contain 15 marbles. In total, they can make 2 groups (from Amina) + 3 groups (from Salman) = 5 groups. However, the question asks for the number of groups they can make with 15 marbles each, without mixing the marbles. Therefore, Amina will have 2 groups of 15 marbles, and Salman will have 3 groups of 15 marbles. This demonstrates how the HCF helps us to divide quantities into equal groups. By finding the HCF, we ensured that each group has the maximum possible number of marbles while still maintaining equal group sizes and not mixing the marbles of Amina and Salman. This concept is useful in various real-life scenarios, such as dividing items into equal shares, organizing collections, or even planning events where resources need to be distributed evenly.

Therefore, they can make 2 groups of marbles from Amina and 3 groups of marbles from Salman, each group containing 15 marbles.

Solution

So they can make 2 groups of 15 marbles each from Amina's marbles and 3 groups of 15 marbles each from Salman's marbles.

Conclusion

In conclusion, this problem beautifully illustrates the practical application of the Highest Common Factor (HCF) in a real-world scenario. By understanding and calculating the HCF of 30 and 45, which is 15, we were able to determine the maximum number of marbles that could be placed in each group while ensuring equal distribution and avoiding any mixing of Amina's and Salman's marbles. This led us to the solution that Amina could form 2 groups of 15 marbles each, and Salman could form 3 groups of 15 marbles each. This problem not only reinforces the concept of HCF but also subtly introduces the related concepts of factors, multiples, and prime factorization, which are fundamental in number theory. The ability to break down numbers into their prime factors and identify common factors is a valuable skill in mathematics. Moreover, this problem highlights the importance of mathematical concepts in everyday situations. The ability to divide quantities equally is crucial in various contexts, from sharing resources among individuals to organizing items into manageable groups. The HCF provides a systematic approach to ensure fairness and efficiency in such scenarios. Furthermore, this problem encourages logical thinking and problem-solving skills. By carefully analyzing the information provided, identifying the core question, and applying the appropriate mathematical techniques, students can arrive at the correct solution. This process of problem-solving is essential not only in mathematics but also in other areas of life. In summary, this exercise serves as an excellent example of how mathematical concepts can be applied to solve real-world problems, making learning mathematics more engaging and relevant for students. It underscores the importance of understanding fundamental concepts such as HCF and their applications in everyday life, fostering a deeper appreciation for the subject.