Solving Real-World Problems with Algebra: A Cupcake Conundrum

As an SEO expert working at Google, it's essential to understand how to tackle real-world problems using mathematical logic. This article will walk you through a practical problem involving the distribution and sale of cupcakes, demonstrating how clear thinking and algebra can reveal the answer. Let's break down the problem and use some basic algebra to find out the total number of cupcakes Mrs. Lala baked.

The Problem

Mrs. Lala baked some cupcakes. 3/7 of them were vanilla, and the rest were chocolate. She sold 1/4 of the vanilla and 5/8 of the chocolate. After these sales, she had 450 cupcakes left. We need to determine how many cupcakes she initially baked.

The Solution

Let's define X as the total number of cupcakes initially baked by Mrs. Lala.

Step 1: Determine the Amount Sold

Vanilla Cupcakes: 3/7 of the total cupcakes were vanilla. Therefore, the number of vanilla cupcakes sold is (3/7)X * 1/4 (3/28)X.

Chocolate Cupcakes: The remaining 4/7 of the cupcakes were chocolate. Therefore, the number of chocolate cupcakes sold is (4/7)X * 5/8 (20/56)X (10/28)X.

Step 2: Subtract the Sold Amounts from the Total

The total number of cupcakes sold is (3/28)X (10/28)X (13/28)X.

Therefore, the number of cupcakes left is X - (13/28)X (15/28)X.

According to the problem, after selling some of the cupcakes, Mrs. Lala had 450 cupcakes left. So, we set up the equation:

(15/28)X 450

Step 3: Solve the Equation

To find X, we need to isolate it on one side of the equation:

X 450 * (28/15)

Multiplying the numbers:

X 90 * 28 / 3 30 * 28 840

Therefore, the total number of cupcakes Mrs. Lala baked is 840.

Verification

To verify, we can check the distribution of the cupcakes:

Vanilla: 3/7 of 840 360 cupcakes, and she sold 1/4 of them, so 90 were sold. This leaves 270 vanilla cupcakes. Chocolate: 4/7 of 840 480 cupcakes, and she sold 5/8 of them, so 300 were sold. This leaves 180 chocolate cupcakes.

The total number of cupcakes left (270 180) is 450, which confirms our solution.

Conclusion

This problem demonstrates how algebra can be used to solve real-world problems. By breaking down the problem into manageable parts and systematically solving equations, we can find the answer clearly and accurately. The key is to clearly define the unknowns and use logical steps to reach a solution.

For more such problems and solutions, visit our Algebra Problem Solver and Real-World Problems Analysis sections.