Cup Puzzle: Find The Number Of 7-Inch Cups!

Have you ever encountered a seemingly simple math problem that makes you pause and think? This cup puzzle is one of those! Let's dive into a fun mathematical problem involving cups of different heights. We'll break it down step by step, making it super easy to understand and solve. Math can be engaging, and this puzzle proves just that! So, grab your thinking cap, and let’s get started on solving this intriguing puzzle together. Remember, the key is to approach it methodically, and you’ll find the solution in no time. Let's turn this puzzle into a triumph! Are you ready to unlock the mystery of the cups?

Understanding the Cup Conundrum

In this mathematical puzzle, we are presented with a scenario involving cups of two different heights. To restate, we have a total of nine cups neatly arranged in a kitchen cabinet. The crucial detail here is that these cups are not all the same size. Some cups stand at a height of 4 inches, while others are taller, measuring 7 inches. This variation in height is the core of our puzzle. Adding another layer to the challenge, we know the total combined height of all nine cups. If you were to stack these cups one on top of the other, they would reach a total height of 45 inches. This is the key piece of information that we will use to solve our puzzle. The question we aim to answer is specific: how many of the taller, 7-inch cups are present in the kitchen cabinet? This puzzle isn't just about numbers; it's about using logic and a bit of algebra to unravel a real-world problem. It's a perfect example of how math can be both practical and engaging. Keep in mind, problem-solving is a skill that sharpens with practice, and this cup puzzle offers a fantastic opportunity to hone those skills. So, let's move forward and explore the strategies we can use to find the solution! Remember, every puzzle has a solution, and we're on the right track to find it.

Setting Up the Equations

To effectively solve this cup puzzle, we need to translate the word problem into mathematical equations. This is a crucial step in tackling many mathematical problems, as it allows us to represent the given information in a structured and solvable form. First, let's define our variables. Let's use 'x' to represent the number of 4-inch cups. This means that every time we see 'x' in our equations, we're referring to the quantity of the smaller cups. Similarly, let's use 'y' to represent the number of 7-inch cups. The variable 'y' will always stand for the quantity of the taller cups. Now that we have our variables defined, we can start forming the equations. We know there are a total of 9 cups. This gives us our first equation: x + y = 9. This equation simply states that the number of 4-inch cups (x) plus the number of 7-inch cups (y) equals the total number of cups, which is 9. Next, we need to consider the total height. We know that the 4-inch cups contribute 4 inches each to the total height, and the 7-inch cups contribute 7 inches each. The combined height of all the cups is 45 inches. This gives us our second equation: 4x + 7y = 45. This equation represents the total height contributed by the cups. It states that 4 times the number of 4-inch cups (4x) plus 7 times the number of 7-inch cups (7y) equals the total height of 45 inches. With these two equations, we have a system of equations that we can solve to find the values of x and y. Solving these equations will reveal the number of 4-inch and 7-inch cups, ultimately answering our puzzle. This step of setting up equations is fundamental in mathematics, and mastering it will help you solve a wide array of problems.

Solving the System of Equations

Now that we have our two equations, x + y = 9 and 4x + 7y = 45, it's time to solve this system and find the values of x and y. There are several methods we could use, such as substitution or elimination. For this puzzle, let's use the substitution method. The first step in the substitution method is to solve one of the equations for one variable in terms of the other. Looking at our equations, it seems easiest to solve the first equation, x + y = 9, for x. Subtracting y from both sides, we get: x = 9 - y. Now we have an expression for x in terms of y. The next step is to substitute this expression for x into our second equation. So, wherever we see x in the second equation, 4x + 7y = 45, we'll replace it with (9 - y). This gives us: 4(9 - y) + 7y = 45. Now we have a single equation with just one variable, y, which we can solve. Let's simplify the equation. First, distribute the 4: 36 - 4y + 7y = 45. Next, combine the y terms: 36 + 3y = 45. Now, subtract 36 from both sides: 3y = 45 - 36, which simplifies to 3y = 9. Finally, divide both sides by 3 to solve for y: y = 9 / 3, which gives us y = 3. So, we've found that y, the number of 7-inch cups, is 3. But we're not quite done yet! We still need to find the value of x, the number of 4-inch cups. We can use our expression x = 9 - y to find x. Substitute the value of y we just found: x = 9 - 3, which gives us x = 6. Therefore, we have 6 cups that are 4 inches tall and 3 cups that are 7 inches tall. The question asks for the number of 7-inch cups, which we've determined is 3. Solving systems of equations is a vital skill in mathematics, and it’s rewarding to see how it helps us unravel puzzles like this one.

The Solution: Number of 7-Inch Cups

After meticulously setting up our equations and solving them, we've arrived at the solution to our cup puzzle! Remember, our goal was to determine the number of 7-inch cups present in the kitchen cabinet. Through the process of substitution, we successfully found that y = 3. Since we defined 'y' as the number of 7-inch cups, this means there are 3 cups that stand 7 inches tall. To recap, we started by understanding the problem, defining variables, and creating a system of equations. This system consisted of two equations: x + y = 9 (representing the total number of cups) and 4x + 7y = 45 (representing the total height of the cups). We then used the substitution method to solve for y. We substituted (9 - y) for x in the second equation, simplified, and solved for y. This gave us y = 3, which directly answers our question. Therefore, the solution to the puzzle is that there are 3 cups that are 7 inches tall in the kitchen cabinet. This puzzle beautifully illustrates how mathematical concepts can be applied to everyday scenarios. It also highlights the importance of breaking down complex problems into smaller, manageable steps. By approaching the puzzle systematically, we were able to navigate the information, form equations, and ultimately find the solution. Puzzles like this not only challenge our mathematical skills but also enhance our problem-solving abilities, which are valuable in various aspects of life. So, congratulations on solving the cup conundrum! You've successfully used math to unravel this interesting puzzle.

Checking Our Answer

To ensure the accuracy of our solution, it’s always a good practice to check our answer. This step is crucial in problem-solving, as it helps us catch any potential errors and reinforces our understanding of the problem. We've determined that there are 3 cups that are 7 inches tall and 6 cups that are 4 inches tall. Let's verify if these numbers fit the conditions given in the problem. First, let's check if the total number of cups adds up correctly. We have 3 cups + 6 cups = 9 cups. This matches the information provided in the puzzle, so far so good! Next, we need to check if the total height of the cups is indeed 45 inches. The 3 cups that are 7 inches tall contribute 3 * 7 = 21 inches to the total height. The 6 cups that are 4 inches tall contribute 6 * 4 = 24 inches to the total height. Adding these together, we get 21 inches + 24 inches = 45 inches. This also matches the information given in the puzzle! Since both the total number of cups and the total height match the problem's conditions, we can confidently say that our solution is correct. This step-by-step verification process not only confirms our answer but also enhances our understanding of the problem's dynamics. Checking our work is a valuable habit to cultivate in mathematics, as it ensures accuracy and reinforces the problem-solving process. So, with the answer checked and confirmed, we can be sure that there are indeed 3 cups that are 7 inches tall in the kitchen cabinet. Congratulations on solving and verifying the solution to this engaging puzzle!

In conclusion, the cup puzzle presented a fun and engaging way to apply mathematical concepts. By understanding the problem, setting up equations, solving the system, and verifying our answer, we successfully determined that there are 3 cups that are 7 inches tall in the kitchen cabinet. This exercise highlights the importance of problem-solving skills and the practical applications of mathematics in everyday scenarios. Remember, tackling puzzles like these not only sharpens our mathematical abilities but also enhances our logical thinking and analytical skills. So, keep practicing, keep exploring, and keep enjoying the world of mathematics! For more interesting math puzzles and concepts, you can visit Math is Fun.