Math Riddle: Finding The Number After Incorrect Division

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Hey guys! Let's dive into a fun math riddle that involves a bit of fractional arithmetic and some clever thinking. This is the kind of problem that makes math feel like a puzzle, and who doesn't love a good puzzle? We'll break down the problem step by step, so you can see exactly how to solve it. So grab your thinking caps, and let's get started!

Understanding the Problem

Okay, so here's the riddle: A girl was asked to multiply a number by 7/8, but instead, she divided the number by 7/8. This mistake led to a result that was 15 more than what she should have gotten. The big question is: what was the original number? Sounds tricky, right? But don't worry, we'll make it super clear. To really nail this, we need to focus on what the problem is telling us and translate it into math terms. The key is to understand the relationship between multiplying and dividing by a fraction, and how that difference leads to the 15-unit discrepancy. We'll use some algebra to represent the unknown number, which will help us set up an equation and find our solution. Remember, word problems might seem daunting, but they become much easier once you break them down into smaller, manageable parts. We're essentially detectives here, piecing together the clues to find the hidden number!

Breaking Down the Math

Let’s start by defining our terms. We’ll call the unknown number "x". The girl was supposed to multiply x by 7/8, which would give us (7/8)x. But, she divided x by 7/8. Remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 7/8 is 8/7, so she actually calculated (8/7)x. The problem tells us that her incorrect answer, (8/7)x, was 15 more than the correct answer, (7/8)x. We can write this as an equation: (8/7)x = (7/8)x + 15. Now we have a clear algebraic equation, which is our roadmap to finding the value of x. Think of it like this: we've translated the word problem into a math sentence. And once we have a sentence, we can use our math skills to solve for the missing piece. The next step is to isolate x, which means getting all the x terms on one side of the equation and the constants on the other. This is where our algebra skills really shine!

Setting Up the Equation

Now, let's dive deeper into setting up the equation. We know the correct operation was multiplication by 7/8, resulting in (7/8)x. The incorrect operation, division by 7/8, is the same as multiplying by 8/7, giving us (8/7)x. The problem states the incorrect result is 15 more than the correct result. This translates directly into the equation: (8/7)x = (7/8)x + 15. To solve for x, we need to get all the x terms on one side. So, we subtract (7/8)x from both sides: (8/7)x - (7/8)x = 15. This step is crucial because it isolates the variable x, bringing us closer to the solution. Now we need to combine the x terms. This involves finding a common denominator for the fractions 8/7 and 7/8, so we can subtract them. Remember, the common denominator allows us to perform the subtraction and simplify the equation further. Once we've combined the x terms, we'll have a single fraction multiplied by x, which equals 15. Then, we can easily solve for x by multiplying both sides by the reciprocal of that fraction.

Solving the Equation

Alright, let's get down to solving this equation! We have (8/7)x - (7/8)x = 15. To subtract these fractions, we need a common denominator. The least common multiple of 7 and 8 is 56. So, we convert the fractions: (8/7)x becomes (64/56)x, and (7/8)x becomes (49/56)x. Now our equation looks like this: (64/56)x - (49/56)x = 15. Subtracting the fractions gives us (15/56)x = 15. See how much simpler it's getting? Now, to isolate x, we need to get rid of the fraction (15/56). We do this by multiplying both sides of the equation by the reciprocal of (15/56), which is (56/15). So, we have x = 15 * (56/15). The 15s cancel out, leaving us with x = 56. And there you have it! We've found the value of x, which is the original number. Isn't it satisfying when the pieces all fall into place like that? But we're not done yet; we should always check our answer to make sure it makes sense in the context of the original problem.

Step-by-Step Solution

  1. Represent the unknown number: Let the number be "x".
  2. Write the correct operation: The girl should have calculated (7/8) * x.
  3. Write the incorrect operation: She calculated x / (7/8), which is the same as (8/7) * x.
  4. Formulate the equation: The incorrect result is 15 more than the correct result, so (8/7)x = (7/8)x + 15.
  5. Subtract (7/8)x from both sides: (8/7)x - (7/8)x = 15.
  6. Find a common denominator: The common denominator for 7 and 8 is 56.
  7. Convert fractions: (64/56)x - (49/56)x = 15.
  8. Subtract the fractions: (15/56)x = 15.
  9. Multiply both sides by 56/15: x = 15 * (56/15).
  10. Simplify: x = 56.

Checking the Solution

Okay, we've found that the number is 56, but let's make sure it actually works. This is a crucial step in problem-solving – always double-check! If the girl had multiplied 56 by 7/8, she would have gotten (7/8) * 56 = 49. But, she divided 56 by 7/8, which is the same as multiplying 56 by 8/7. So, she calculated (8/7) * 56 = 64. Now, let's see if the difference between the incorrect answer and the correct answer is 15. 64 - 49 = 15. Bingo! It checks out. This confirms that our solution, x = 56, is correct. It's always a good feeling when your answer lines up perfectly with the problem's conditions. Plus, checking your work helps prevent those silly mistakes that can sometimes slip through. So, remember guys, never skip the check! It's the final piece of the puzzle.

Verifying the Answer

To verify our answer, let's plug 56 back into the original problem. The correct calculation should have been (7/8) * 56. This equals (7 * 56) / 8 = 392 / 8 = 49. The incorrect calculation was 56 divided by 7/8, which is the same as 56 * (8/7). This equals (56 * 8) / 7 = 448 / 7 = 64. The difference between the incorrect result (64) and the correct result (49) should be 15. So, 64 - 49 = 15. This confirms that our answer is correct. Always remember, guys, that verifying your solution is a crucial step in problem-solving. It not only ensures accuracy but also solidifies your understanding of the problem.

Conclusion

So, there you have it! The original number was 56. We solved this math riddle by translating the words into an algebraic equation, finding a common denominator, and isolating the variable. And most importantly, we checked our answer to make sure it made sense. Remember, guys, math problems are just puzzles waiting to be solved. By breaking them down into smaller steps and using the right tools (like algebra!), you can tackle even the trickiest challenges. Math isn't about memorizing formulas; it's about understanding the relationships between numbers and using logic to find solutions. So keep practicing, keep puzzling, and most importantly, have fun with it! Every problem you solve is a step forward in building your problem-solving skills. And who knows, maybe you'll be the one writing the next math riddle for us to solve! Keep up the awesome work, everyone!

Key Takeaways

This math riddle illustrates a few important concepts. Firstly, it highlights the relationship between multiplication and division, especially when dealing with fractions. Dividing by a fraction is the same as multiplying by its reciprocal. Secondly, it emphasizes the power of algebra in solving word problems. By representing the unknown quantity with a variable, we can set up an equation that captures the problem's conditions. Thirdly, it underscores the importance of verifying your solution. Plugging your answer back into the original problem is a foolproof way to ensure accuracy. And finally, it reminds us that problem-solving is a step-by-step process. Breaking down a complex problem into smaller, manageable steps makes it much easier to tackle. So, remember these key takeaways, guys, and you'll be well-equipped to handle any math riddle that comes your way. Keep practicing, stay curious, and enjoy the journey of learning!