Math Problem: Find The Number!

by TextBrain Team 31 views

Let's dive into this math problem together, guys! We need to figure out a mystery number based on the clues given. This is a classic algebraic puzzle, and we'll break it down step by step to make it super clear. So, grab your thinking caps, and let's get started!

Understanding the Problem

Okay, so the problem states: "If adding 8 to double one-third of a number equals 32, what is the number?" To solve this, we need to translate this sentence into a mathematical equation. This might sound intimidating, but it's just like translating from one language to another. We're going from English to Math!

First, let's identify the key parts:

  • "a number": This is our unknown, so we'll call it x. This is standard practice in algebra, where we use letters to represent values we don't yet know.
  • "one-third of a number": This means we're dividing our number x by 3, which we can write as x/3.
  • "double one-third of a number": We're taking the result from the previous step (x/3) and multiplying it by 2. This gives us 2(x/3), which simplifies to 2x/3.
  • "adding 8": We're adding 8 to the expression we just found, so we now have 2x/3 + 8.
  • "equals 32": This means our entire expression is equal to 32. So, 2x/3 + 8 = 32.

Now we have our equation! 2x/3 + 8 = 32. See? It wasn't so scary after all!

Solving the Equation

Now that we have the equation, the next step is to solve for x. This involves isolating x on one side of the equation. We'll do this by performing operations on both sides to maintain the balance. Think of it like a see-saw – whatever you do to one side, you have to do to the other to keep it level.

  1. Subtract 8 from both sides: This gets rid of the +8 on the left side. So, our equation becomes:

    • 2x/3 + 8 - 8 = 32 - 8
    • 2x/3 = 24
  2. Multiply both sides by 3: This gets rid of the denominator (the 3 in the fraction). We now have:

    • (2x/3) * 3 = 24 * 3
    • 2x = 72
  3. Divide both sides by 2: This isolates x and gives us our answer:

    • 2x / 2 = 72 / 2
    • x = 36

So, we've found our answer! x equals 36. That means the mystery number is 36.

Checking the Answer

It's always a good idea to check your answer to make sure it's correct. We can do this by plugging our solution (x = 36) back into the original equation and seeing if it holds true.

Our original equation was: 2x/3 + 8 = 32

Let's substitute x with 36:

  • 2(36)/3 + 8 = 32
  • 72/3 + 8 = 32
  • 24 + 8 = 32
  • 32 = 32

Yay! The equation holds true. This means our answer, x = 36, is correct. We nailed it!

Understanding the Concepts

This problem is a great example of how algebra can be used to solve real-world problems. We took a word problem and translated it into a mathematical equation, and then we used algebraic techniques to solve for the unknown. This is a fundamental skill in math and can be applied to all sorts of situations.

Let's recap some of the key concepts we used:

  • Variables: We used the variable x to represent the unknown number. Variables are essential in algebra because they allow us to work with quantities that we don't know yet.
  • Equations: An equation is a statement that two expressions are equal. In this case, our equation was 2x/3 + 8 = 32. Equations are the foundation of algebra and allow us to solve for unknowns.
  • Inverse Operations: We used inverse operations (subtraction, multiplication, and division) to isolate the variable x. Each operation "undoes" the previous one, allowing us to gradually solve for the unknown.
  • Order of Operations: While not explicitly a major factor in this problem, remembering the order of operations (PEMDAS/BODMAS) is crucial for more complex equations. This ensures you perform operations in the correct sequence (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).

Applying the Knowledge

Now that we've solved this problem, let's think about how we can apply these skills to other situations. Word problems can seem daunting at first, but by breaking them down into smaller steps, they become much more manageable. Here are some tips for tackling word problems:

  1. Read the problem carefully: Make sure you understand what the problem is asking. Identify the unknowns and the given information.
  2. Translate into an equation: This is the crucial step. Look for keywords that indicate mathematical operations (e.g., "sum," "difference," "product," "quotient").
  3. Solve the equation: Use algebraic techniques to isolate the unknown variable.
  4. Check your answer: Plug your solution back into the original equation to make sure it's correct.
  5. Think about the answer in context: Does your answer make sense in the real world? For example, if you're solving for the number of people, a negative answer wouldn't make sense.

Let's Try Another Example

To really solidify our understanding, let's try a similar problem:

Problem: If you subtract 5 from three times a number, the result is 16. What is the number?

Take a moment to try solving this on your own. Remember to follow the steps we discussed earlier:

  1. Identify the unknown: Let's call the number y this time (just to show we can use any letter!).
  2. Translate into an equation: "three times a number" is 3y, and "subtract 5" means 3y - 5. The problem states this equals 16, so our equation is 3y - 5 = 16.
  3. Solve the equation:
    • Add 5 to both sides: 3y = 21
    • Divide both sides by 3: y = 7

So, the number is 7.

  1. Check your answer: Plug 7 back into the original equation: 3(7) - 5 = 16. This simplifies to 21 - 5 = 16, which is true. Awesome!

Conclusion

We've successfully solved a tricky math problem by breaking it down into smaller, manageable steps. We translated a word problem into an algebraic equation, solved for the unknown, and even checked our answer. Remember, practice makes perfect! The more you work with these types of problems, the easier they become. Keep practicing, and you'll be a math whiz in no time! And remember, guys, math can be fun – especially when you crack the code!

So, to answer the original question, the correct answer is:

  • d) 36

We found it! You did great following along, and now you're one step closer to mastering algebra. Keep up the fantastic work!