Buko Juice Math Problem Solving The Difference In Purchases

In this article, we will explore a mathematical problem involving the calculation of differences in buko juice purchases. The problem revolves around Hanz, who bought a certain amount of buko juice last week and a different amount this week. Our goal is to determine exactly how much more buko juice he purchased this week compared to the previous week. This is a common type of problem that involves working with mixed numbers and fractions, requiring us to apply fundamental arithmetic operations such as subtraction and finding common denominators. By breaking down the problem into smaller, manageable steps, we can clearly understand the process and arrive at the correct solution. This exercise is not only a test of mathematical skills but also an opportunity to enhance our understanding of real-world applications of fractions and mixed numbers. So, let’s dive in and solve this buko juice puzzle together!

Understanding the Problem

Before we begin crunching numbers, it’s crucial to fully understand the problem statement. Hanz bought $5 \frac{5}{9}$ liters of buko juice last week and $6 \frac{1}{3}$ liters this week. The core question we need to answer is: How much more buko juice did Hanz purchase this week than last week? This immediately indicates that we need to find the difference between the two quantities. The quantities are given in mixed numbers, which means we’ll need to handle both whole numbers and fractions. A clear understanding of what’s being asked is the first step towards solving any mathematical problem accurately.

To successfully solve this, we will be focusing our efforts on mixed fractions. We are dealing with two mixed fractions, $5 \frac{5}{9}$ and $6 \frac{1}{3}$, representing the amount of buko juice Hanz bought last week and this week, respectively. Mixed fractions can sometimes appear daunting, but they are simply a combination of a whole number and a proper fraction. To work with them effectively in calculations, such as subtraction, we often need to convert them into improper fractions. This involves multiplying the whole number by the denominator of the fraction and then adding the numerator. The result becomes the new numerator, while the denominator remains the same. Mastering the conversion between mixed and improper fractions is a fundamental skill in arithmetic and is essential for solving problems like this one. By converting to improper fractions, we make the subsequent subtraction process much easier and less prone to errors. This conversion simplifies our task and prepares us for the next step in finding the difference in buko juice purchases.

Converting Mixed Numbers to Improper Fractions

The next critical step in solving our buko juice problem is to convert the mixed numbers into improper fractions. This conversion is essential because it simplifies the subtraction process. We have two mixed numbers to convert: $5 \frac{5}{9}$ and $6 \frac{1}{3}$.

Let's start with $5 \frac{5}{9}$. To convert this into an improper fraction, we multiply the whole number (5) by the denominator (9) and then add the numerator (5). This gives us $(5 imes 9) + 5 = 45 + 5 = 50$. So, the new numerator is 50, and the denominator remains the same, which is 9. Therefore, $5 \frac{5}{9}$ is equivalent to $\frac{50}{9}$.

Now, let's convert $6 \frac{1}{3}$ into an improper fraction. We multiply the whole number (6) by the denominator (3) and then add the numerator (1). This gives us $(6 imes 3) + 1 = 18 + 1 = 19$. The new numerator is 19, and the denominator remains 3. Thus, $6 \frac{1}{3}$ is equivalent to $\frac{19}{3}$.

By converting both mixed numbers into improper fractions, we have transformed the problem into a more manageable form. We now have $\frac{50}{9}$ and $\frac{19}{3}$, which are much easier to subtract. This conversion is a key technique in dealing with mixed numbers in arithmetic problems, allowing us to perform operations like subtraction and addition more efficiently. With the numbers in this form, we are well-prepared to move on to the next step: finding a common denominator and subtracting the fractions.

Finding a Common Denominator

Before we can subtract the two fractions, $\frac{50}{9}$ and $\frac{19}{3}$, we need to ensure they have a common denominator. This is a fundamental rule of fraction arithmetic: you cannot directly add or subtract fractions unless they share the same denominator. The common denominator is a number that both denominators can divide into evenly. In this case, our denominators are 9 and 3.

To find the least common denominator (LCD), we look for the smallest multiple that both 9 and 3 share. The multiples of 3 are 3, 6, 9, 12, and so on. The multiples of 9 are 9, 18, 27, and so on. We can see that the smallest number that appears in both lists is 9. Therefore, 9 is our least common denominator.

Now, we need to convert $\frac{19}{3}$ into an equivalent fraction with a denominator of 9. To do this, we ask ourselves: What do we need to multiply 3 by to get 9? The answer is 3. So, we multiply both the numerator and the denominator of $\frac{19}{3}$ by 3. This gives us $\frac{19 imes 3}{3 imes 3} = \frac{57}{9}$.

Now we have two fractions with the same denominator: $\frac{50}{9}$ and $\frac{57}{9}$. With a common denominator in place, we are ready to subtract the fractions and find the difference in the amount of buko juice Hanz purchased. This step of finding the common denominator is crucial for accurate calculations and ensures that we are comparing like quantities.

Subtracting the Fractions

Now that we have our fractions with a common denominator, $\frac{50}{9}$ and $\frac{57}{9}$, we can proceed with the subtraction. To find out how much more buko juice Hanz bought this week than last week, we subtract the amount he bought last week ($\frac{50}{9}$) from the amount he bought this week ($\frac{57}{9}$).

The subtraction of fractions with a common denominator is straightforward. We simply subtract the numerators and keep the denominator the same. In this case, we have $\frac{57}{9} - \frac{50}{9}$.

Subtracting the numerators, we get $57 - 50 = 7$. The denominator remains 9. So, the result of the subtraction is $\frac{7}{9}$.

This means that Hanz bought $\frac{7}{9}$ liters more buko juice this week than last week. The subtraction step is the heart of solving the problem, allowing us to quantify the difference between the two quantities. By ensuring we had a common denominator, we could perform this subtraction accurately. The result, $\frac{7}{9}$ liters, represents the answer to our problem. However, it's always good practice to review the answer in the context of the original question and consider whether it can be simplified or expressed in a different form if necessary. In this case, $\frac{7}{9}$ is already in its simplest form, so we can confidently state that Hanz bought $\frac{7}{9}$ liters more buko juice this week.

Converting Back to a Mixed Number (If Necessary)

In our case, the result of the subtraction, $\frac{7}{9}$, is a proper fraction, meaning the numerator (7) is less than the denominator (9). Therefore, there is no need to convert it back into a mixed number. A mixed number consists of a whole number and a proper fraction, and it’s typically used when the improper fraction (where the numerator is greater than or equal to the denominator) can be simplified to include a whole number component.

However, it's a useful skill to know how to convert an improper fraction back to a mixed number, as it's often required to present answers in the most appropriate form. Let’s briefly review this process. If we had an improper fraction, say $\frac{11}{4}$, we would divide the numerator (11) by the denominator (4). This gives us 2 with a remainder of 3. The whole number part of the mixed number is the quotient (2), the numerator of the fractional part is the remainder (3), and the denominator remains the same (4). So, $\frac{11}{4}$ would be converted to $2 \frac{3}{4}$.

In the context of our buko juice problem, since we obtained a proper fraction $\frac{7}{9}$ as our answer, converting back to a mixed number is not necessary. This fraction is already in its simplest and most understandable form for the problem at hand. It clearly tells us that Hanz bought seven-ninths of a liter more buko juice this week than last week.

Final Answer and Conclusion

After carefully working through the problem, converting mixed numbers to improper fractions, finding a common denominator, and subtracting the fractions, we have arrived at our final answer. Hanz purchased $\frac{7}{9}$ liters more buko juice this week than last week.

Therefore, none of the provided options (A. $\frac{1}{3}$, B. $\frac{1}{5}$, C. $\frac{2}{3}$, D. $\frac{2}{5}$) is correct. The correct answer is $\frac{7}{9}$ liters.

This exercise highlights the importance of several key mathematical skills. First, it demonstrates the need to understand and accurately convert mixed numbers to improper fractions and vice versa. Second, it emphasizes the critical step of finding a common denominator when adding or subtracting fractions. Finally, it reinforces the basic principles of fraction subtraction. By mastering these concepts, we can confidently tackle similar problems and apply these skills to real-world situations.

In conclusion, solving this buko juice problem has not only provided us with a numerical answer but has also reinforced our understanding of fundamental arithmetic operations involving fractions. The process of breaking down the problem into manageable steps, from converting mixed numbers to subtracting fractions with a common denominator, is a valuable approach that can be applied to various mathematical challenges. We hope this step-by-step guide has been helpful in clarifying the process and enhancing your problem-solving abilities. Remember, practice is key to mastering these skills, so keep exploring and tackling new problems!