Girls And Boys In A Class: A Math Problem Solved

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Hey guys, let's dive into a cool math problem! We've got a classroom packed with 33 students, a mix of girls and boys. The problem gives us a little twist: the difference between twice the number of girls and the number of boys is 12. Our mission? To figure out exactly how many girls and boys are in the class. Ready to solve this math puzzle?

Understanding the Problem

Alright, before we jump into calculations, let's break down what the problem is telling us. We know a few key things:

  • Total Students: There are 33 students in the class. This means that the total number of girls plus the total number of boys equals 33.
  • The Difference: The problem mentions a difference. Specifically, it tells us that if you double the number of girls and then subtract the number of boys, you get 12. This is the crucial piece of information that helps us set up our equations.

So, we have two pieces of information. Total number of students and the difference between twice the number of girls and the number of boys.

Setting up the Equations

Now comes the fun part – turning the words into math. We can use variables to represent the unknowns. Let's use:

  • g for the number of girls
  • b for the number of boys

Based on the information above, we can create two equations:

  1. g + b = 33 (The total number of students is the sum of girls and boys)
  2. 2g - b = 12 (Twice the number of girls minus the number of boys equals 12)

These equations are the key to unlocking the solution. You can see how we took the word problem and turned it into mathematical expressions. The first equation represents the total count, and the second translates the difference between the girls and boys.

Solving the Equations

Now, let's solve these equations. There are a few ways to do this, but let's use a method called elimination. This approach is perfect for this system of equations.

  1. Adding the Equations: If we add the two equations together, the b terms will cancel each other out:

    (g + b) + (2g - b) = 33 + 12 This simplifies to: 3g = 45

  2. Solving for g: To find the value of g (the number of girls), divide both sides of the equation by 3:

    3g / 3 = 45 / 3 This gives us: g = 15

    So, we know there are 15 girls in the class.

  3. Solving for b: Now that we know g = 15, we can plug this value back into one of the original equations. Let's use the first equation: g + b = 33

    15 + b = 33

    Subtract 15 from both sides:

    b = 33 - 15 This gives us: b = 18

    Therefore, there are 18 boys in the class.

So, we've solved the problem! There are 15 girls and 18 boys in the class. Wasn't that fun?

Verification

Always a good idea to verify, right? Let's double-check our answers to make sure they make sense. We know there are 15 girls and 18 boys. Let's look back at our initial problem setup.

  • Total Students: 15 girls + 18 boys = 33 students. ✅ This checks out!
  • The Difference: Twice the number of girls (2 * 15 = 30) minus the number of boys (18) equals 12. ✅ This also checks out!

So our answers are correct. It's always a good practice to verify the results. Because it gives a better understanding of the problem itself.

Alternate Solution Methods

There's more than one way to solve a math problem, and that's the beauty of it! Let's look at another approach to solve this problem. Understanding different methods can enhance your overall problem-solving skills. Here’s a quick overview.

Substitution Method

We could have used the substitution method. In this method, you solve one of the equations for one variable and then substitute that expression into the other equation. It’s a great technique to use when one of the equations is easily solved for a variable.

  1. Isolate a Variable: From the first equation (g + b = 33), we can easily isolate b: b = 33 - g
  2. Substitute: Substitute this expression for b in the second equation: 2g - (33 - g) = 12.
  3. Solve for g: Simplify and solve for g: 2g - 33 + g = 12 → 3g = 45 → g = 15
  4. Solve for b: Now substitute g = 15 back into b = 33 - g: b = 33 - 15 = 18

You get the same answer. The steps are slightly different, but the outcome is identical. Choosing the right method often depends on the structure of the equations.

Graphical Method

If you're into visuals, you could also graph these equations. Each equation represents a line on a graph, and the solution is where the lines intersect. It provides a great visual representation of the solution.

Comparison of Methods

Each method has its advantages. Elimination is often efficient when the coefficients of one variable are opposites or can easily be made opposites. Substitution is excellent when you can easily isolate a variable. Graphical methods provide a visual understanding, which can be very helpful for some learners.

Why This Math Problem Matters

You might be wondering,