Math Problems: Solve & Create Equations!

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Math Problems: Solve & Create Equations!

Hey guys! Let's dive into some cool math problems. We're going to create and solve equations, which is super useful for understanding how numbers work. Get ready to put on your thinking caps!

1. Creating and Solving a Problem from an Equation

Let’s start with the equation: [350 - (2x + 40)] : 11 = 10. Our mission is to craft a word problem that this equation perfectly solves. Think of real-life scenarios where you might need to subtract something, divide it up, and end up with a certain result. This is where your creativity comes into play! We need a scenario that makes sense and guides someone to set up the same equation to solve it.

Problem Creation:

Here’s a possible scenario: Imagine you're organizing a charity event. You start with a budget of $350. You need to allocate some funds for decorations and then divide the remaining amount equally among 11 different charity projects. The decoration cost involves a fixed expense of $40, plus an additional cost of $2 per participant (represented by 'x'). If each charity project receives $10, how many participants are there?

Setting up the Equation:

  • The total budget is $350.
  • The cost of decorations is $(2x + 40), where 'x' is the number of participants.
  • The amount left after paying for decorations is $(350 - (2x + 40)).
  • This remaining amount is divided among 11 charity projects.
  • Each project receives $10.

So, the equation that represents this problem is: [350 - (2x + 40)] : 11 = 10

Solving the Equation:

Alright, now let’s solve this equation step by step. Remember the order of operations (PEMDAS/BODMAS)? It’s key here!

  1. Isolate the Bracket: Multiply both sides by 11: 350 - (2x + 40) = 10 * 11 which simplifies to 350 - (2x + 40) = 110.
  2. Simplify Inside the Bracket: Distribute the negative sign: 350 - 2x - 40 = 110. Combine like terms: 310 - 2x = 110.
  3. Isolate the Term with 'x': Subtract 310 from both sides: -2x = 110 - 310 which simplifies to -2x = -200.
  4. Solve for 'x': Divide both sides by -2: x = -200 / -2 which simplifies to x = 100.

Therefore, there are 100 participants.

Verification:

Let's plug x = 100 back into our original equation to check if it’s correct:

[350 - (2 * 100 + 40)] : 11 = [350 - (200 + 40)] : 11 = [350 - 240] : 11 = 110 : 11 = 10

The equation holds true, so our answer is correct. There are indeed 100 participants.

Creating problems from equations not only reinforces your understanding of algebra but also makes math more relatable. You can apply this skill to various scenarios in your daily life. Think about budgeting, cooking, or even planning events. Math is everywhere!

2. Finding the Initial Number

Now, let's tackle the second problem. It involves reversing a series of operations to find an initial number. These kinds of problems are excellent for building logical thinking and understanding how mathematical operations affect numbers.

Problem Statement:

If we increase one-third of a number 5 times, reduce the product by 20, and then divide the result by 8, we get the number 20. Find the initial number.

Setting up the Equation:

Let's break this down step by step and translate it into an equation.

  • Let the initial number be 'y'.
  • One-third of the number is y/3.
  • Increasing it 5 times gives us 5 * (y/3) or 5y/3.
  • Reducing the product by 20 means we have (5y/3) - 20.
  • Dividing the result by 8 gives us [(5y/3) - 20] / 8.
  • The final result is 20.

So, the equation is: [(5y/3) - 20] / 8 = 20

Solving the Equation:

To find the initial number 'y', we need to reverse the operations in the correct order.

  1. Isolate the Bracket: Multiply both sides by 8: (5y/3) - 20 = 20 * 8 which simplifies to (5y/3) - 20 = 160.
  2. Isolate the Term with 'y': Add 20 to both sides: 5y/3 = 160 + 20 which simplifies to 5y/3 = 180.
  3. Solve for 'y': Multiply both sides by 3: 5y = 180 * 3 which simplifies to 5y = 540. Divide both sides by 5: y = 540 / 5 which simplifies to y = 108.

Therefore, the initial number is 108.

Verification:

Let's check if our answer is correct by plugging y = 108 back into the original equation:

[(5 * (108/3)) - 20] / 8 = [(5 * 36) - 20] / 8 = [180 - 20] / 8 = 160 / 8 = 20

The equation holds true, so our answer is correct. The initial number is indeed 108.

This type of problem is fundamental in algebra and helps build a strong foundation for more complex mathematical concepts. Understanding how to reverse operations is a critical skill in problem-solving.

Why This Matters

These exercises are not just about getting the right answer. They're about training your brain to think logically and systematically. The ability to break down a problem into smaller, manageable steps is invaluable, not just in math but in everyday life. So, keep practicing, keep challenging yourself, and remember that every problem is an opportunity to learn something new!

Keep up the great work, and you'll be a math whiz in no time! Remember, practice makes perfect, and understanding the process is just as important as getting the correct answer.