Introduction

Sarah, a dedicated student, wanted to fundraise for her class and decided to sell pies. By cleverly breaking down the problem, we can demonstrate a step-by-step process to solve for the unknown quantity of pies Sarah initially had. This problem not only tests understanding of fractions but also algebraic thinking, making it a great tool for teaching problem-solving techniques.

The Problem

Sarah sold pies to raise funds for her class. She sold 5/8 of the pies herself and gave her friend, Alic, to sell 1/3 of the remainder. Sarah had 12 pies left. How many pies did Sarah have at first?

To solve this, we break down the problem into smaller, manageable steps and use algebra to find the initial number of pies Sarah owned.

Solution

Let's denote the number of pies Sarah had at first by X.

Step 1: Calculate the pies sold by Sarah

First, Sarah sold 5/8 of the pies herself. The number of pies she sold is (5/8)X.

Step 2: Calculate the remainder of the pies

The number of pies remaining after Sarah sold 5/8 of them is X - (5/8)X, which simplifies to (3/8)X.

Step 3: Calculate the pies Alice sold

Alic, her friend, sold 1/3 of the remainder. So, the number of pies Alicia sold is (1/3)(3/8)X, which simplifies to (1/8)X.

Step 4: Calculate the final number of pies left

The number of pies left after Alicia sold her share is the remainder minus the pies Alicia sold, which is (3/8)X - (1/8)X, or (1/4)X. We know that the number of pies left is 12, so:

(1/4)X 12

Step 5: Solve for X

Multiplying both sides of the equation by 4, we get:

X 12 * 4

X 48

Conclusion

Sarah initially had 48 pies.

Verification

To verify, let's perform the calculations step by step:

Step 1: Sarah sold (5/8) * 48 pies, which is 30 pies.

So, the remaining pies after Sarah sold her share is 48 - 30 18.

Step 2: Alicia sold (1/3) * 18 pies, which is 6 pies.

So, the remaining pies after Alicia sold her share is 18 - 6 12, which matches the problem statement.

Additional Insights and Applications

This problem is an excellent tool for teaching concepts such as fractions, algebraic equations, and problem-solving strategies. It encourages students to think critically and work through multi-step problems. A similar approach can be used to solve various mathematical puzzles and real-world problems involving fractions and algebra.

Conclusion

Sarah initially had 48 pies. Through careful analysis and step-by-step calculations, we arrived at this solution, demonstrating the versatility and application of mathematical concepts in problem-solving scenarios.