Solving Mixture Problems: Hazelnuts, Walnuts, And Raisins

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Solving the Mixture Puzzle: Finding the Hazelnuts!

Hey guys! Let's dive into a fun math problem involving a tasty mixture of walnuts, hazelnuts, and raisins. We've got a scenario where someone, let's call them Sude, has created a delightful mix weighing 425 grams. The challenge? To figure out the exact amount of hazelnuts in the mix. Sounds like a delicious puzzle, right?

Understanding the Problem:

First things first, let's break down what we know. Sude combined walnuts, hazelnuts, and raisins, and the total weight of the mixture is 425 grams. We're also given some ratios that describe the relationship between the amounts of each ingredient. These ratios are the key to unlocking the solution. We know that the ratio of walnuts to hazelnuts is 2/3, and the ratio of hazelnuts to raisins is 2/5. These ratios tell us how the ingredients are proportioned within the mixture. Our goal is to use these ratios and the total weight of the mixture to determine how many grams of hazelnuts are in the mix. This kind of problem is common in mathematics and is often used in real-world situations, such as in cooking or creating recipes. Understanding these ratios is crucial for solving the problem effectively. So, let's get started and find out how we can calculate the hazelnut quantity!

To solve this, we'll use the ratios provided and the total weight to calculate the amount of hazelnuts. Let's convert the ratios into equations to make the solution clearer and more understandable. We can assign variables to each component: let 'w' represent the weight of walnuts, 'h' represent the weight of hazelnuts, and 'r' represent the weight of raisins. Based on the information given, we can set up the following equations:

  • w / h = 2/3 (The ratio of walnuts to hazelnuts is 2/3)
  • h / r = 2/5 (The ratio of hazelnuts to raisins is 2/5)
  • w + h + r = 425 (The total weight of the mixture is 425 grams)

Our objective now is to solve these equations to find the value of 'h,' which represents the weight of the hazelnuts. These ratios represent the relationship between each ingredient in the mix. The total weight of the mixture helps us scale these ratios to the actual quantities. By understanding these ratios, we can accurately determine the amount of each ingredient in the entire mix. Let's use the information we have to calculate the quantity of each ingredient and then determine the quantity of hazelnuts.

Setting Up the Equations and Solving for Hazelnuts

Alright, let's get our hands dirty and start solving this! We know that the ratio of walnuts to hazelnuts is 2/3, which means for every 2 parts of walnuts, there are 3 parts of hazelnuts. Also, the ratio of hazelnuts to raisins is 2/5. Let's work with these ratios to find the quantity of hazelnuts.

First, we need to find a common ground to work with these ratios. A great way to do this is by making sure our 'h' (hazelnuts) value is consistent across both ratios. We can express the ratio of walnuts to hazelnuts as w = (2/3)h. Similarly, we can express the ratio of raisins to hazelnuts as r = (5/2)h. This way, we have all the other ingredients related to the amount of hazelnuts. Now we can substitute these values in the total weight equation which is: w + h + r = 425. Substituting our expressions for 'w' and 'r' we get: (2/3)h + h + (5/2)h = 425.

To simplify the equation, we need to find a common denominator for the fractions. The least common multiple of 2 and 3 is 6. So, we convert all the fractions to have a denominator of 6. Our equation now looks like this: (4/6)h + (6/6)h + (15/6)h = 425. Adding up the fractions gives us (25/6)h = 425. Now, we can solve for 'h' by multiplying both sides of the equation by 6/25. This will isolate 'h' on one side, giving us h = 425 * (6/25).

Calculating this out, h = (425 * 6) / 25 = 2550 / 25 = 102. So, the amount of hazelnuts in the mixture is 102 grams! That's how we calculate it. Let's summarize the steps in detail. We started with the ratios provided and converted these ratios into equations involving walnuts, hazelnuts, and raisins. By doing so, we were able to relate all quantities to the quantity of hazelnuts. Then we substituted these expressions into the total weight equation. This created an equation with only the hazelnut variable that could be solved. In the end, we solved for hazelnuts and successfully found the amount of hazelnuts.

Detailed Step-by-Step Solution

Let's break down the solution into easy-to-follow steps to make sure we understand it completely.

  1. Define Variables:

    • Let 'w' = weight of walnuts.
    • Let 'h' = weight of hazelnuts.
    • Let 'r' = weight of raisins.

  2. Write Down the Given Ratios:

    • w / h = 2/3.
    • h / r = 2/5.

  3. Express Walnuts and Raisins in Terms of Hazelnuts:

    • From w / h = 2/3, we get w = (2/3)h.
    • From h / r = 2/5, we get r = (5/2)h.

  4. Set Up the Total Weight Equation:

    • w + h + r = 425.

  5. Substitute the Expressions for Walnuts and Raisins:

    • (2/3)h + h + (5/2)h = 425.

  6. Find a Common Denominator and Combine Terms:

    • (4/6)h + (6/6)h + (15/6)h = 425.
    • (25/6)h = 425.

  7. Solve for Hazelnuts (h):

    • h = 425 * (6/25).
    • h = 102 grams.

Conclusion: The Hazelnut Harvest

So, there you have it, folks! The mixture contains 102 grams of hazelnuts. We successfully navigated the ratios, converted them into equations, and solved for the unknown. It shows how useful mathematical ratios are to solve real-world problems. This type of problem-solving is often used in various fields, from cooking to chemistry. Think of it like a delicious recipe where we figured out one of the key ingredients. Now we know exactly how much hazelnut goodness is in Sude's tasty mix! Remember, these kinds of problems are common. With a little practice, you'll be able to solve these mixture problems quickly and accurately.

This problem has taught us the importance of understanding and converting ratios into equations. We learned how to manipulate equations and use them to find the unknown variables. These skills are useful not just in mathematics, but also in everyday life when dealing with proportions and mixtures. And next time, when you have a mixture problem, remember the steps we took, and you'll be well on your way to cracking the code and finding the solution. This whole process reinforces the idea that math is not just about numbers, but about logic and problem-solving!