Concert Hall Audience: Find The Number Of Male Spectators

Let's dive into this interesting math problem about a concert hall audience! We're going to break it down step by step to figure out exactly how many men were in the audience. So, grab your thinking caps, guys, and let’s get started!

Understanding the Problem

First, let's make sure we understand the information given in the problem. This is super important for solving it correctly. Keywords here are fractions and the relationship between the number of women, men, and children in the audience. So, what do we know? We know that women make up 3/7 of the total audience. Then, out of the remaining audience (after accounting for the women), 3/4 are men. The tricky part is that there are 42 fewer children than men. Our mission is to find the exact number of male spectators. Remember, math problems like this often involve unraveling the layers of information to get to the solution. Don't worry, we'll tackle it together!

To effectively approach this problem, we need to formulate a strategy. Our strategy will involve several steps, including representing the unknown total number of spectators with a variable, expressing the number of women, men, and children in terms of this variable, and setting up an equation based on the given relationship between the number of children and men. This structured approach will help us to break down the problem into manageable parts, making it easier to solve. Let's start by defining our variable and expressing the number of women in terms of the total audience. This first step is crucial as it lays the foundation for the subsequent calculations. Remember, a clear and methodical approach is key to solving complex math problems. So, let's keep our focus sharp and move forward step by step.

Setting Up the Equation

Okay, let's get to the nitty-gritty of setting up our equation. This is where we'll translate the words of the problem into mathematical language. To start, we're going to use a variable to represent the total number of people in the audience. A classic choice here is 'x'. So, let x be the total number of spectators. Now, we need to express the number of women, men, and children in terms of x. Remember, 3/7 of the audience are women, so the number of women is (3/7)x. This is our first piece of the puzzle, and it's a pretty important one. Knowing how to represent parts of a whole with fractions and variables is a fundamental skill in algebra, and it's super useful in real-life situations too.

Next, we need to figure out the number of men. This involves a little more work. After accounting for the women, the remaining audience is 1 - 3/7 = 4/7 of the total. So, 4/7 of the audience is neither women nor children. Of this remaining portion, 3/4 are men. Therefore, the number of men is (3/4) * (4/7)x, which simplifies to (3/7)x. Hey, look at that! The number of men is the same fraction of the total audience as the number of women. This might seem surprising, but it's a key piece of information that will help us later. Now, let's move on to the children. We know there are 42 fewer children than men, so the number of children is (3/7)x - 42. See how we're building up our expressions? It's like constructing a building, one brick at a time. With expressions for women, men, and children in hand, we are now one step closer to setting up our final equation. Remember, the key is to carefully translate each piece of information from the problem into mathematical terms. And with that, let's move on to forming the actual equation.

Now that we have expressions for the number of women, men, and children, we can set up an equation to represent the total audience. This is where everything comes together! We know that the total audience (x) is the sum of the number of women, men, and children. So, we can write the equation as: x = (3/7)x + (3/7)x + ((3/7)x - 42). This equation is the heart of our problem, and solving it will give us the total number of spectators. Notice how we've combined all the information into one neat mathematical statement. It's like we've taken all the scattered pieces of a puzzle and fitted them together to form a complete picture.

Our next step is to simplify this equation. Combining like terms will make it easier to solve for x. So, let's get our algebraic skills ready and simplify this equation. Remember, the goal is to isolate x on one side of the equation. Once we've done that, we'll know the total number of spectators, and we'll be one step closer to finding the number of men. So, let's roll up our sleeves and dive into the simplification process!

Solving for the Unknown

Alright, let's roll up our sleeves and dive into solving the equation. This is where the algebra magic happens! Our equation is: x = (3/7)x + (3/7)x + (3/7)x - 42. The first step in simplifying this equation is to combine the terms with x on the right side. We have three (3/7)x terms, which add up to (3/7 + 3/7 + 3/7)x = (9/7)x. So, our equation now looks like this: x = (9/7)x - 42. We're making progress, guys! It's like we're slowly untangling a knot, and with each step, it becomes clearer.

Now, we want to get all the x terms on one side of the equation. To do this, we can subtract (9/7)x from both sides. This gives us: x - (9/7)x = -42. But wait, how do we subtract fractions from whole numbers? No problem! We can rewrite x as (7/7)x. So, the equation becomes (7/7)x - (9/7)x = -42. Now we can easily subtract the fractions: (7/7 - 9/7)x = (-2/7)x. So, our equation simplifies to (-2/7)x = -42. See how we're breaking it down step by step? Each operation brings us closer to isolating x.

To finally solve for x, we need to get rid of the (-2/7) coefficient. We can do this by multiplying both sides of the equation by the reciprocal of -2/7, which is -7/2. Remember, multiplying by the reciprocal is a handy trick for isolating a variable. So, we multiply both sides by -7/2: (-7/2) * (-2/7)x = -42 * (-7/2). On the left side, the fractions cancel out, leaving us with x. On the right side, we have -42 * (-7/2). We can simplify this by first dividing -42 by 2, which gives us -21. Then, we multiply -21 by -7, which gives us 147. So, x = 147. We've done it! We've solved for x, the total number of spectators. But hold on, we're not quite finished yet. The problem asks for the number of male spectators, not the total number of people.

Finding the Number of Men

Okay, now that we've cracked the code and found the total number of spectators (x = 147), it's time to zoom in on what the problem actually asks: the number of men. Remember, we already figured out an expression for the number of men in terms of x. Way back when, we found that the number of men is (3/7)x. So, to find the number of men, we just need to substitute our value of x into this expression.

So, the number of men is (3/7) * 147. How do we calculate this? Well, we can first divide 147 by 7, which gives us 21. Then, we multiply 21 by 3, which gives us 63. Therefore, there are 63 men in the audience. Hooray! We've found our answer. It's like we've reached the top of a mountain, and now we can look back and see all the steps we took to get here.

Final Answer

So, after all that mathematical detective work, we've arrived at our final answer. There are 63 male spectators in the concert hall. That's it! We've successfully solved the problem. But before we give ourselves a pat on the back, let's just take a moment to review what we've done. We started by understanding the problem, then we set up an equation, solved for the unknown, and finally, we used that information to find the number of men. Each step was crucial, and together, they led us to the solution. Remember, in math, it's not just about getting the right answer; it's also about understanding the process. And in this case, we've not only found the answer, but we've also strengthened our problem-solving skills. So, let's keep practicing and keep exploring the wonderful world of mathematics!