Shoes Before: Solve Sarah's Footwear Math Problem!

Hey there, math enthusiasts! Today, we're diving into a fun little problem about Sarah and her ever-growing shoe collection. This isn't just any math problem; it’s a real-world scenario that helps us flex our algebraic muscles. So, lace up your thinking shoes, and let’s get started!

Understanding the Shoe Equation

At the heart of our problem is a classic algebraic puzzle. The question we're tackling is: How many pairs of shoes did Sarah have seven months ago, given that she currently owns thirty pairs, a number that's nine more than three times the pairs she had seven months prior? This might sound like a mouthful, but let's break it down piece by piece. The key here is to translate the words into a mathematical equation. We'll use 'x' to represent the unknown number of pairs of shoes Sarah had seven months ago. This is a common practice in algebra, where variables stand in for values we need to find. It's like giving the unknown a name so we can work with it more easily. Now, let's dissect the information given in the problem. We know that Sarah's current shoe count (30 pairs) is related to her past shoe count ('x') in a specific way. The problem states that her current number of shoes is “nine more than three times” the number she had seven months ago. This is a crucial piece of information because it gives us the structure of our equation. "Three times" the number of shoes she had seven months ago can be written as 3 * x, or simply 3x. The phrase "nine more than" tells us that we need to add 9 to this quantity. So, "nine more than three times the number of shoes" becomes 3x + 9. Finally, we know that this entire expression is equal to the number of shoes Sarah has today, which is 30 pairs. Putting it all together, we get the equation 3x + 9 = 30. This equation is the foundation for solving our problem. It neatly summarizes the relationship between Sarah's past and present shoe collections. Once we solve for 'x', we'll have our answer. This process of translating words into equations is a fundamental skill in algebra and problem-solving in general. It allows us to take complex scenarios and represent them in a clear, concise way that we can then manipulate and solve. So, with our equation in hand, we're ready to move on to the next step: solving for 'x'.

Solving for the Unknown

Now comes the fun part: solving the equation! We've established that 3x + 9 = 30. Our goal is to isolate 'x' on one side of the equation to find its value. This involves using inverse operations to undo the operations that are being performed on 'x'. The first step in isolating 'x' is to get rid of the +9 on the left side of the equation. To do this, we perform the inverse operation, which is subtraction. We subtract 9 from both sides of the equation. It's crucial to do this on both sides to maintain the balance of the equation. If we only subtracted 9 from one side, the equation would no longer be true. So, subtracting 9 from both sides, we get: 3x + 9 - 9 = 30 - 9. This simplifies to 3x = 21. We're one step closer to finding 'x'! Now, we have 3x = 21. This means that 3 multiplied by 'x' equals 21. To isolate 'x', we need to undo this multiplication. The inverse operation of multiplication is division. So, we divide both sides of the equation by 3. Again, it's essential to perform the same operation on both sides to keep the equation balanced. Dividing both sides by 3, we get: 3x / 3 = 21 / 3. This simplifies to x = 7. Eureka! We've found the value of 'x'. This means that Sarah had 7 pairs of shoes seven months ago. It's always a good idea to check our answer to make sure it makes sense in the context of the original problem. If Sarah had 7 pairs of shoes seven months ago, then three times that amount would be 3 * 7 = 21 pairs. Nine more than that would be 21 + 9 = 30 pairs, which is the number of shoes Sarah has today. So, our answer checks out! We've successfully solved for 'x' and answered the question. This process of using inverse operations to isolate a variable is a cornerstone of algebra, and it's a skill that can be applied to a wide range of problems.

Checking Our Shoe Math

Before we declare victory, let's make absolutely sure our answer is correct. We've calculated that Sarah had 7 pairs of shoes seven months ago. Now, we'll plug this number back into the original problem to see if it fits. The problem stated that Sarah has thirty pairs of shoes today, which is nine more than three times the amount she had seven months ago. If our answer is correct, then three times the number of shoes she had seven months ago, plus nine, should equal thirty. Let's do the math. Three times the number of shoes she had seven months ago (7 pairs) is 3 * 7 = 21 pairs. Nine more than that is 21 + 9 = 30 pairs. This matches the information given in the problem! This check confirms that our solution is correct. It's always a good practice to verify your answers, especially in math problems. This not only ensures accuracy but also helps you understand the problem and the solution more deeply. By plugging our answer back into the original equation, we've essentially reversed the problem-solving process, which can be a powerful way to solidify your understanding. This step also highlights the importance of careful reading and attention to detail in math problems. We needed to make sure we were using the correct relationships and operations as described in the problem. A small mistake in setting up the equation or performing the calculations could lead to a wrong answer. Checking our work helps us catch these errors and correct them. So, we can confidently say that Sarah had 7 pairs of shoes seven months ago. We've not only solved the problem but also verified our solution, demonstrating a thorough understanding of the concepts involved.

Real-World Shoe Scenarios

This problem about Sarah's shoe collection might seem like just a mathematical exercise, but it actually reflects the kind of thinking we use in everyday life. Math isn't just about numbers and equations; it's about problem-solving, logical reasoning, and critical thinking. The ability to translate a real-world situation into a mathematical model, like we did with Sarah's shoes, is a valuable skill in many areas of life. For example, imagine you're trying to budget your finances. You might have a certain amount of money coming in each month, and you need to figure out how much you can spend on different things while still saving enough for your goals. This involves setting up equations and inequalities, just like we did with Sarah's shoes, to represent your income, expenses, and savings targets. Or, let's say you're planning a road trip. You need to calculate how much gas you'll need, how long it will take to drive, and how much it will cost. These calculations involve using formulas and equations to relate distance, speed, time, and cost. Even something as simple as doubling a recipe involves mathematical thinking. You need to adjust the quantities of all the ingredients proportionally, which is a form of algebra. The beauty of math is that it provides a framework for solving problems in a systematic and logical way. By breaking down a problem into smaller parts, identifying the relevant information, and setting up equations or models, we can find solutions that might not be immediately obvious. So, the next time you encounter a problem, whether it's a math problem or a real-life situation, remember the steps we took to solve Sarah's shoe problem: understand the problem, translate it into a mathematical model, solve the equation, and check your answer. These skills will serve you well in all aspects of life.

Conclusion: The Mystery of the Missing Shoes

We've successfully solved the mystery of Sarah's shoes! By carefully translating the problem into an algebraic equation and using inverse operations, we determined that Sarah had 7 pairs of shoes seven months ago. This exercise not only sharpened our math skills but also highlighted how mathematical thinking can be applied to everyday situations. Remember, math is more than just numbers; it's a powerful tool for problem-solving and critical thinking.

If you're eager to explore more mathematical puzzles and deepen your understanding of algebra, I encourage you to check out resources like Khan Academy's Algebra section. Happy problem-solving!