Evaluate Division And Solve Real World Math Problems

Let's dive into the world of fractions and division, guys! We're going to tackle the problem of evaluating $w \\div z$ when $w=\\frac{6}{7}$ and $z=3$. This might seem tricky at first, but I promise, it's super manageable when we break it down. So, let's get started!

At the heart of this problem is understanding how to divide a fraction by a whole number. Remember, division is essentially the inverse operation of multiplication. When we divide by a number, it's the same as multiplying by its reciprocal. A reciprocal is simply what you multiply a number by to get 1. For example, the reciprocal of 3 is $\\frac{1}{3}$ because $3 \\times \\frac{1}{3} = 1$. This is the golden rule that will help us conquer this problem. Now, armed with this knowledge, let's jump into the specifics.

Our mission is to evaluate $\\frac{6}{7} \\div 3$. Applying our reciprocal trick, we can rewrite this division problem as a multiplication problem. Instead of dividing by 3, we'll multiply by the reciprocal of 3, which is $\\frac{1}{3}$. So, our problem now becomes $\\frac{6}{7} \\times \\frac{1}{3}$. This is a much friendlier operation, right? Multiplying fractions is straightforward: you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. In our case, we multiply 6 by 1 to get 6, and 7 by 3 to get 21. This gives us the fraction $\\frac{6}{21}$. But hold on, we're not quite done yet! It's crucial to always simplify fractions to their lowest terms. This means finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it. Looking at 6 and 21, we can see that their GCF is 3. Dividing both 6 and 21 by 3 gives us $\\frac{2}{7}$. So, the final result of evaluating $w \\div z$ when $w=\\frac{6}{7}$ and $z=3$ is $\\frac{2}{7}$. See? It wasn't so scary after all! This whole process really highlights the connection between division and multiplication and the importance of simplifying fractions. Now that we've successfully navigated this problem, you're better equipped to handle similar challenges in the future. Keep practicing, and you'll become a fraction division master in no time!

Let's shift gears and tackle a real-world problem involving speed, distance, and time, guys! Robin's planning a trip to her friend's house, which is a cool 170 miles away, and she wants to get there in $2\\frac{1}{2}$ hours. The big question is: what average speed does she need to maintain? This is a classic problem that uses the fundamental relationship between speed, distance, and time, so it's super practical to understand.

The core concept we need to remember is the formula that connects these three amigos: speed = distance / time. This formula tells us that speed is calculated by dividing the total distance traveled by the time it takes to travel that distance. It's a simple yet powerful relationship that forms the basis for solving many travel-related problems. To apply this formula effectively, it's essential to ensure that all units are consistent. In our case, the distance is given in miles and the time is given in hours, which aligns perfectly with the standard unit for speed, miles per hour (mph). So, we're all set to plug in the values and get cracking! Before we dive into the calculation, let's take a closer look at the time given, which is $2\\frac{1}{2}$ hours. This is a mixed number, and it's often easier to work with improper fractions in calculations. To convert $2\\frac{1}{2}$ to an improper fraction, we multiply the whole number part (2) by the denominator (2) and add the numerator (1), then put the result over the original denominator. So, $2\\frac{1}{2}$ becomes $\\frac{(2 \\times 2) + 1}{2} = \\frac{5}{2}$ hours. Now we have all the pieces we need to solve for Robin's average speed. We know the distance is 170 miles and the time is $\\frac{5}{2}$ hours. Plugging these values into our formula, we get: speed = 170 miles / $\\frac{5}{2}$ hours. Remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\\frac{5}{2}$ is $\\frac{2}{5}$. So, our equation becomes: speed = 170 miles * $\\frac{2}{5}$ hours. Now we can multiply 170 by $\\frac{2}{5}$, which gives us speed = $\\frac{340}{5}$ mph. Finally, we divide 340 by 5 to get the average speed, which is 68 mph. So, Robin needs to drive at an average speed of 68 mph to reach her friend's house in $2\\frac{1}{2}$ hours. This problem demonstrates how mathematical concepts can be applied to everyday situations, like planning a road trip. Understanding the relationship between speed, distance, and time is not only useful for math problems but also for making informed decisions in real life. Whether you're calculating your commute time or planning a cross-country adventure, these skills will come in handy.

Now, let's switch gears again and dive into a sewing project, guys! This time, we're dealing with the task of figuring out how many choir robes can be made from a certain amount of fabric. This problem combines basic arithmetic with a practical application, making it super relevant for anyone who enjoys sewing or crafting. So, let's unravel the details and see how we can solve it.

To tackle this problem effectively, we need to know a few key pieces of information. First, we need to know the total amount of fabric available. This is our starting point, the resource we're working with. Second, we need to know how much fabric is required to make one choir robe. This is the unit cost, the amount of fabric needed per item. Once we have these two pieces of information, we can use division to determine how many robes can be made. The fundamental concept here is that we're dividing the total available fabric by the fabric required per robe. This will give us the number of robes that can be produced. Let's put this into a mathematical framework. Suppose we have X amount of fabric in total, and each robe requires Y amount of fabric. Then, the number of robes we can make is given by X / Y. This simple equation is the key to solving our problem. However, there's a crucial detail we need to consider: fabric measurements. Fabric is often measured in yards, feet, or inches. It's essential to ensure that all measurements are in the same units before performing any calculations. If, for instance, the total fabric is given in yards and the fabric required per robe is given in feet, we need to convert one of the measurements so that they are both in the same unit. This conversion step is vital for accurate results. Remember, there are 3 feet in a yard, so we can easily convert between these units by multiplying or dividing by 3. Similarly, there are 12 inches in a foot and 36 inches in a yard. Choosing the right unit for our calculations can make the problem much simpler. For example, if we're dealing with small amounts of fabric, it might be easier to work in inches. If we're dealing with larger amounts, yards might be more convenient. Let's illustrate this with a hypothetical scenario. Suppose we have 20 yards of fabric, and each choir robe requires 2.5 yards of fabric. To find out how many robes we can make, we simply divide the total fabric by the fabric per robe: 20 yards / 2.5 yards per robe = 8 robes. In this case, the units are already consistent (yards), so we can proceed directly with the division. Now, let's consider a slightly more complex scenario. Suppose we have 15 yards of fabric, and each robe requires 6 feet of fabric. Before we can divide, we need to convert the feet to yards. Since there are 3 feet in a yard, 6 feet is equal to 6 / 3 = 2 yards. Now we can divide: 15 yards / 2 yards per robe = 7.5 robes. But wait, can we make half a robe? No, we can't! In this practical context, we need to round down to the nearest whole number. So, we can make 7 complete choir robes. This highlights an important aspect of real-world problem-solving: we need to consider the context and make sure our answers make sense. In this case, we can't make a fraction of a robe, so we round down to the nearest whole number. This sewing problem demonstrates how math is intertwined with everyday activities. Whether you're planning a sewing project, cooking a meal, or building a bookshelf, math skills are essential for accurate measurements and calculations. By understanding these concepts, you can tackle a wide range of practical challenges with confidence.