Sugar Repackaging Problem: Packets, Quantity, And Remainder

Hey guys! Let's break down this math problem about repackaging sugar. We've got a big bag of sugar, and we need to figure out how many smaller packets we can make, how much sugar ends up in those packets, and what's left over. It's a practical problem, like something you might encounter in a kitchen or a small business. So, grab your thinking caps, and let's get started!

Understanding the Sugar Repackaging Scenario

So, here's the deal: We start with 12 kg of sugar in a bag. The packer's job is to divide this sugar into smaller packets, each containing 5/6 kg. The main goal here is to figure out how many of these smaller packets we can completely fill, and then we need to calculate the total amount of sugar that goes into these packets. Finally, we'll determine how much sugar is left over in the original bag after filling all those smaller packets. This involves some basic division and multiplication, so let's make sure we understand each step clearly.

Breaking Down the Problem

To really get a handle on this, let's break down the problem into smaller, manageable parts. First, we need to figure out how many packets we can fill. This is a division problem, where we divide the total amount of sugar by the amount in each packet. Once we know the number of packets, we can easily calculate the total amount of sugar in those packets by multiplying the number of packets by the weight of each packet. And finally, to find out how much sugar is left over, we subtract the total amount of sugar in the packets from the original amount of sugar in the bag. Each of these steps is pretty straightforward, but it's important to follow them in the right order to get the correct answer. We'll walk through each one carefully, so don't worry if it seems a bit confusing at first. By the end, you'll be a pro at solving these kinds of problems!

1) How Many Packets Were Filled?

Okay, so the first thing we need to figure out is how many packets of sugar we can fill. Remember, we have 12 kg of sugar, and each packet holds 5/6 kg. To find out how many packets we can make, we need to divide the total amount of sugar (12 kg) by the amount of sugar per packet (5/6 kg). This might sound a little tricky, but it's just a matter of knowing how to divide fractions. So, let's dive into the math!

Dividing to Find the Number of Packets

To divide by a fraction, we actually multiply by its reciprocal. The reciprocal of 5/6 is 6/5. So, instead of dividing 12 by 5/6, we're going to multiply 12 by 6/5. This makes the calculation much easier. We can write this as 12 * (6/5). Now, remember that 12 can be written as a fraction as well: 12/1. So, our calculation becomes (12/1) * (6/5). To multiply fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, (12 * 6) / (1 * 5) = 72/5. Now, we have an improper fraction, which means the numerator is bigger than the denominator. To make it easier to understand, we need to convert this to a mixed number.

Converting Improper Fractions to Mixed Numbers

To convert 72/5 to a mixed number, we need to divide 72 by 5. 5 goes into 72 fourteen times (14 * 5 = 70), with a remainder of 2. This means that 72/5 is the same as 14 and 2/5. So, we can fill 14 whole packets of sugar, but we have a little bit left over – not enough to fill another whole packet. Since the question asks how many packets were filled, we only count the complete packets. The fraction 2/5 represents the remaining sugar, which isn't enough to fill a full packet. So, the answer to the first question is 14 packets. We've successfully figured out how many full packets we can make from our 12 kg of sugar. Now, let's move on to the next part of the problem: figuring out how much sugar is in all those packets.

2) How Much Sugar is in All the Packets?

Now that we know we filled 14 packets, and each packet contains 5/6 kg of sugar, we need to calculate the total amount of sugar in these packets. This is a multiplication problem. We're multiplying the number of packets by the amount of sugar in each packet. So, we'll multiply 14 by 5/6. This will give us the total weight of the sugar that was used to fill the packets.

Multiplying to Find the Total Sugar in Packets

To multiply 14 by 5/6, we can write 14 as a fraction: 14/1. Then, we multiply the fractions: (14/1) * (5/6). Just like before, we multiply the numerators and the denominators. So, (14 * 5) / (1 * 6) = 70/6. Now we have another improper fraction. To make it easier to understand, we need to convert it to a mixed number. This will tell us exactly how many kilograms of sugar are in all the filled packets.

Simplifying the Result

To convert 70/6 to a mixed number, we divide 70 by 6. 6 goes into 70 eleven times (11 * 6 = 66), with a remainder of 4. This means that 70/6 is the same as 11 and 4/6. So, there are 11 and 4/6 kilograms of sugar in all the filled packets. But, we can simplify the fraction 4/6. Both 4 and 6 can be divided by 2. So, 4/6 simplifies to 2/3. Therefore, the total amount of sugar in all the packets is 11 and 2/3 kg. We've now figured out how much sugar went into the packets. Only one question left: how much sugar remains in the bag?

3) How Much Sugar Remains in the Bag?

Alright, we're almost there! We started with 12 kg of sugar, and we used 11 and 2/3 kg to fill the packets. Now, to find out how much sugar is left in the bag, we need to subtract the amount of sugar used from the original amount. So, we'll subtract 11 and 2/3 from 12. This will give us the final answer: the amount of sugar remaining.

Subtracting to Find the Remaining Sugar

To subtract 11 and 2/3 from 12, it's helpful to rewrite 12 as a mixed number. We can think of 12 as 11 and 3/3, because 3/3 is equal to 1. So, our subtraction problem becomes (11 and 3/3) - (11 and 2/3). Now we can subtract the whole numbers and the fractions separately. 11 minus 11 is 0, and 3/3 minus 2/3 is 1/3. So, the answer is 1/3 kg. This means that there is 1/3 kg of sugar remaining in the bag. We've successfully solved the entire problem! We figured out how many packets were filled, how much sugar was in those packets, and how much sugar was left over. Great job, guys!

Final Recap of the Sugar Repackaging Solution

So, let's quickly recap what we've found out. We started with 12 kg of sugar and packed it into smaller packets of 5/6 kg each. First, we calculated that we could fill 14 packets completely. Then, we figured out that these 14 packets contained a total of 11 and 2/3 kg of sugar. And finally, we determined that there was 1/3 kg of sugar remaining in the original bag. This problem might have seemed a bit complicated at first, but by breaking it down into smaller steps, we were able to solve it easily. Remember, that's often the key to tackling any math problem: break it down, step by step, and you'll get there! You guys rock!