Sakura's Softball Purchase Determining Bats Bought

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Sakura, a dedicated student preparing for her PE class, embarks on a shopping trip to equip herself with the necessary gear. Her shopping list includes softballs, essential for practice, and bats, the tools to make contact and send those softballs soaring. Each softball comes at a cost of $3, while each bat commands a price of $35. As a savvy shopper, Sakura understands the importance of balance, opting to purchase twice as many softballs as bats. With her items selected, Sakura heads to the checkout, where the total bill, including $8 in tax, amounts to $131. This scenario presents us with an intriguing mathematical puzzle: How can we determine the number of bats Sakura purchased?

Decoding the Problem

To solve this problem, we'll need to translate the given information into a mathematical equation. This equation will serve as a roadmap, guiding us through the steps required to find the solution. Let's break down the information piece by piece:

  • Cost of softballs: Each softball costs $3, and Sakura buys twice as many softballs as bats. If we let 'x' represent the number of bats, then the number of softballs is 2x. The total cost of the softballs is therefore 3 * (2x) = 6x dollars.
  • Cost of bats: Each bat costs $35, and Sakura buys 'x' bats. The total cost of the bats is 35x dollars.
  • Total cost before tax: The total cost of the softballs and bats before tax is 6x + 35x = 41x dollars.
  • Total cost including tax: The total cost, including the $8 tax, is $131. This means the cost of the softballs and bats before tax is $131 - $8 = $123.

Constructing the Equation

Now we can put all the pieces together to form our equation. We know that the cost of the softballs and bats before tax (41x) is equal to $123. This gives us the equation:

41x = 123

This equation encapsulates the relationships between the number of bats, the cost of the equipment, and the total expenditure. It's the key to unlocking the solution to our problem.

Solving for x

To find the number of bats (x), we need to isolate 'x' on one side of the equation. We can do this by dividing both sides of the equation by 41:

x = 123 / 41

x = 3

Therefore, Sakura bought 3 bats.

Verifying the Solution

To ensure our solution is correct, let's plug the value of x back into the original problem and see if it satisfies all the conditions.

  • Number of bats: Sakura bought 3 bats.
  • Number of softballs: Sakura bought twice as many softballs as bats, so she bought 2 * 3 = 6 softballs.
  • Cost of softballs: The cost of 6 softballs at $3 each is 6 * $3 = $18.
  • Cost of bats: The cost of 3 bats at $35 each is 3 * $35 = $105.
  • Total cost before tax: The total cost of the softballs and bats is $18 + $105 = $123.
  • Total cost including tax: Adding the $8 tax, the total cost is $123 + $8 = $131.

Our solution satisfies all the conditions of the problem, confirming that Sakura indeed bought 3 bats.

The Power of Equations

This problem highlights the power of equations in solving real-world scenarios. By translating the given information into a mathematical equation, we can systematically unravel the relationships between different variables and arrive at a precise solution. Equations are not just abstract symbols; they are tools that help us make sense of the world around us.

Identifying the Correct Equation

The question asks us to identify the equation that would be used to find the number of bats Sakura buys. Based on our analysis, the correct equation is:

41x = 123

This equation directly relates the number of bats (x) to the total cost of the softballs and bats before tax. It's the most concise and efficient way to represent the problem mathematically.

Refining the Equation for Clarity

While the equation 41x = 123 is mathematically correct, we can refine it slightly to make it even clearer and more intuitive. Let's go back to our breakdown of the costs:

  • Cost of softballs: 6x
  • Cost of bats: 35x
  • Total cost before tax: 6x + 35x

Instead of combining the terms 6x and 35x into 41x, we can leave them separate to emphasize the individual contributions of the softballs and bats to the total cost. This gives us the equation:

6x + 35x = 123

This equation is equivalent to 41x = 123, but it provides a more detailed view of the cost structure. It clearly shows the cost of the softballs (6x) and the cost of the bats (35x) adding up to the total cost before tax ($123). This can be particularly helpful for students who are learning to translate word problems into mathematical equations.

Emphasizing Key Concepts

This problem reinforces several key mathematical concepts:

  • Variable representation: Using a variable (x) to represent an unknown quantity.
  • Translating word problems: Converting real-world scenarios into mathematical expressions and equations.
  • Equation solving: Applying algebraic techniques to isolate the variable and find its value.
  • Verification: Checking the solution to ensure it satisfies the conditions of the problem.

By mastering these concepts, students can develop their problem-solving skills and confidently tackle a wide range of mathematical challenges.

Real-World Applications

The skills used to solve this problem are applicable to many real-world situations. For example, we might use similar techniques to:

  • Calculate the cost of a shopping trip with multiple items.
  • Determine the quantities of different ingredients needed for a recipe.
  • Analyze financial data and make informed decisions.

Mathematics is not just an academic subject; it's a powerful tool that can help us navigate the complexities of everyday life.

Conclusion

Sakura's purchase of softballs and bats provides a valuable opportunity to practice our mathematical problem-solving skills. By carefully analyzing the given information, constructing an appropriate equation, and solving for the unknown variable, we can successfully determine the number of bats she bought. This problem underscores the importance of translating word problems into mathematical expressions, a crucial skill for success in mathematics and beyond.

Whether we choose the concise equation 41x = 123 or the more detailed equation 6x + 35x = 123, the underlying principles remain the same. The key is to represent the relationships between the variables accurately and apply the rules of algebra to find the solution. By mastering these techniques, we can unlock the power of mathematics to solve a wide range of real-world problems.

Further Exploration

To further enhance your understanding of this type of problem, consider exploring variations and extensions. For example:

  • What if Sakura had a coupon for a certain percentage off the total cost?
  • What if the tax rate was different?
  • What if Sakura had a budget constraint and wanted to maximize the number of softballs and bats she could buy?

By tackling these challenges, you can deepen your understanding of the underlying mathematical concepts and develop your problem-solving abilities even further.

In conclusion, Sakura's softball and bat purchase serves as a compelling example of how mathematics can be used to solve everyday problems. By breaking down the problem into smaller parts, representing the information mathematically, and applying the appropriate techniques, we can arrive at a solution and gain a deeper appreciation for the power and versatility of mathematics. The equation 41x = 123, or its more detailed form 6x + 35x = 123, serves as the key to unlocking the answer, highlighting the importance of translating real-world scenarios into mathematical expressions. This exercise not only strengthens our problem-solving skills but also demonstrates the relevance of mathematics in our daily lives.