Calculating The Weight Of Lucy's Gold Bracelet A Math Problem

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Introduction

Hey guys! Today, we're diving into a fascinating math problem that involves gold, jewelry, and a bit of calculation. Imagine Lucy, who decides to melt her 18-karat gold bracelet to create a stunning 20-karat gold chain. To make this happen, she needs an extra 36 grams of pure gold. The big question we're tackling is: How much did Lucy's original bracelet weigh? This problem is a fantastic blend of math and real-world applications, perfect for anyone who loves a good challenge. We'll break down the concepts of gold purity, karats, and how to set up the equations to solve this intriguing puzzle. So, grab your thinking caps, and let's get started!

Understanding Gold Purity and Karats

Before we jump into the calculations, let's get a solid understanding of what gold karats mean. When we talk about gold being 24 karats, we're referring to pure gold. This means 24 out of 24 parts are gold. However, pure gold is quite soft, making it less ideal for jewelry that needs to withstand daily wear and tear. That's why gold is often mixed with other metals like silver, copper, or zinc to increase its durability. When gold is mixed with other metals, the karat value decreases.

An 18-karat gold piece means 18 out of 24 parts are gold, while the remaining 6 parts are other metals. In terms of percentage, 18-karat gold is 75% pure (18/24 = 0.75). Similarly, 20-karat gold means 20 out of 24 parts are gold, making it approximately 83.33% pure (20/24 ≈ 0.8333). Understanding these percentages is crucial because they will help us calculate the actual amount of gold in Lucy's bracelet and the chain she wants to create. By knowing the purity levels, we can set up the equations needed to find the weight of the original bracelet. This concept of mixing gold with other metals to achieve the desired karat value is fundamental in jewelry making, and it's the key to solving our problem.

Setting Up the Equation

Now that we understand gold purity, let's set up the equation to solve this problem. This is where the math magic happens! Let's denote the weight of Lucy's original 18-karat gold bracelet as 'x' grams. Remember, 18-karat gold is 75% pure, so the amount of pure gold in the bracelet is 0.75x grams. Lucy melts this bracelet and adds 36 grams of pure gold to create a 20-karat gold chain. We need to figure out the total weight of the chain and the amount of pure gold in it.

The chain is 20-karat gold, which is approximately 83.33% pure. If we let the total weight of the chain be 'x + 36' grams (since she added 36 grams of pure gold), the amount of pure gold in the chain is 0.8333(x + 36) grams. Now, we can set up the equation. The pure gold from the bracelet plus the added gold equals the pure gold in the chain. This gives us the equation: 0.75x + 36 = 0.8333(x + 36). This equation is the heart of our solution. It represents the relationship between the initial gold in the bracelet, the added gold, and the final gold in the chain. Solving this equation will give us the value of 'x', which is the weight of Lucy's bracelet. So, let's move on to the next step and solve this equation to find our answer!

Solving the Equation Step-by-Step

Alright, let's dive into solving the equation we set up: 0.75x + 36 = 0.8333(x + 36). This might look a bit intimidating at first, but don't worry, we'll break it down step by step to make it super clear.

  1. Distribute the 0.8333: First, we need to distribute the 0.8333 across the terms inside the parentheses on the right side of the equation. This means multiplying 0.8333 by both 'x' and '36'.

    • 0.8333 * x = 0.8333x
    • 0.8333 * 36 ≈ 30

    So, the equation now looks like this: 0.75x + 36 = 0.8333x + 30

  2. Gather the 'x' terms: Next, we want to get all the terms with 'x' on one side of the equation. To do this, we can subtract 0.75x from both sides. This keeps the equation balanced while moving the 'x' terms where we want them.

    • 0.75x - 0.75x = 0
    • 0.8333x - 0.75x = 0.0833x

    Our equation is now: 36 = 0.0833x + 30

  3. Isolate the 'x' term: Now, we need to get the term with 'x' by itself. We can do this by subtracting 30 from both sides of the equation.

    • 36 - 30 = 6
    • 30 - 30 = 0

    So, the equation becomes: 6 = 0.0833x

  4. Solve for 'x': Finally, to solve for 'x', we need to divide both sides of the equation by 0.0833. This will give us the value of 'x', which is the weight of Lucy's bracelet.

    • 6 / 0.0833 ≈ 72

    Therefore, x ≈ 72

So, after all these steps, we've found that the weight of Lucy's original 18-karat gold bracelet was approximately 72 grams. See? We tackled that equation like pros!

Verifying the Solution

Awesome! We've calculated that Lucy's bracelet weighed approximately 72 grams. But before we celebrate, let's make sure our solution makes sense. It's always a good idea to double-check our work, you know, just to be absolutely certain we've nailed it. To verify our solution, we'll plug the value of x (72 grams) back into our original equation and see if both sides balance out. Remember our equation: 0.75x + 36 = 0.8333(x + 36)

  1. Calculate the pure gold in the bracelet:

      1. 75 * 72 = 54 grams of pure gold

  2. Calculate the total weight of the chain:

    • 72 (bracelet) + 36 (added gold) = 108 grams

  3. Calculate the pure gold in the chain:

      1. 8333 * 108 ≈ 90 grams of pure gold

  4. Check if both sides of the equation balance:

    • Left side: 54 (pure gold from bracelet) + 36 (added pure gold) = 90 grams
    • Right side: 90 grams (pure gold in chain)

Voila! Both sides of the equation balance out perfectly. This confirms that our solution of 72 grams is indeed correct. By verifying our solution, we've not only ensured accuracy but also gained confidence in our problem-solving skills. Always remember, guys, verifying your answer is a crucial step in any math problem!

Conclusion

Alright, guys! We've reached the end of our golden adventure, and what a journey it has been! We started with a simple question: How much did Lucy's 18-karat gold bracelet weigh if she melted it down, added 36 grams of pure gold, and created a 20-karat chain? By understanding the concepts of gold purity, karats, and setting up the correct equation, we were able to solve this problem step by step. We found that Lucy's bracelet weighed approximately 72 grams. We tackled the equation with confidence, verifying our solution to ensure accuracy. This problem not only sharpened our math skills but also showed us how math concepts can be applied in real-world scenarios, like jewelry making.

Remember, guys, every math problem is like a puzzle waiting to be solved. With a bit of patience, a clear understanding of the concepts, and a systematic approach, you can conquer any challenge. Whether it's calculating gold purity or figuring out your budget, the skills you learn in math are incredibly valuable. So, keep practicing, keep exploring, and most importantly, keep enjoying the process of learning. Who knows what other exciting math problems we'll tackle next time? Until then, keep shining bright like gold!