Fractions and Geometry: A Clever Cake Problem Solved Step by Step

Imagine a scenario where Jason, a cake enthusiast, has made a magnificent cake and left it out, only to notice his sister sneakily taking a piece. This problem involves the use of fractions and geometric reasoning to determine the exact fraction of the remaining cake that is not iced. Let's break down the problem to understand the solution better.

Understanding the Cake

The problem begins with Jason iced 3/4 of a cake. Let's consider the cake divided into 24 equal pieces for simplicity. This means that 3/4 of the cake is 18 pieces, leaving 6 pieces uniced.

Step 1: Identifying the Iced and Uniced Sections

Jason iced 18 out of 24 pieces. His sister took 1/6 of the cake, which is 4 pieces, but only 2/3 of one of those pieces have icing on it. This means she took 2.66 pieces, but for simplicity, we will consider it as 2 1/3 of another piece.

Step 2: Recalculating the Uniced Fraction

The piece her sister took contained both iced and uniced sections. We need to determine how much of the taken piece is uniced. The total fraction of the cake taken by her sister is 1/6, which is equivalent to 4 pieces out of 24.

Out of these 4 pieces, 2/3 of one piece, which is 2 pieces, are iced. This leaves 1 1/3 of a piece uniced, which is 1/6 of a piece uniced. Therefore, the total amount of uniced cake is:

1/6 (taken by sister) - 1/18 (iced part of the taken piece) 1/18 uniced.

Step 3: Calculating the Remaining Pieced

Initially, the whole cake was 24 pieces. Jason left 18 iced pieces, and his sister took 1/6, which is 4 pieces with 2 uniced. We need to calculate the fraction of the remaining cake that is uniced:

24 - 18 - 1/3 (from her sister) - 1/18 (uniced part from her sister's piece) 7/36.

Geometric Interpretation

Visualizing the cake as a line divided into 24 pieces can help us understand the problem better. Jason iced 18 pieces, and his sister took 4, leaving 20 pieces. Of the 4 pieces taken, 2 were iced and 2 uniced. Therefore, the total uniced cake is 18 (unchanged) 2 (uniced from her sister's piece) - 1/18 (from her sister's piece).

Exploring Another Perspective

Another way to solve this problem is to calculate it step by step:

1/6 of the cake equals 2/3 of the piece her sister took, which is 1/9 of the cake. So 1/9 is uniced by her sister. Therefore, the total uniced cake is:

1/9 uniced 1/18 (from her sister's piece) 1/18.

Now, subtract this from the total cake:

1 - 3/4 (iced by Jason) - 1/18 (from sister) 1/4 - 1/18 9/36 - 2/36 7/36.

Conclusion

Therefore, the fraction of the remaining cake that is not iced is 7/36. This problem required a combination of fractions and geometric understanding to solve. By breaking down the problem into smaller steps and using visualization techniques, we can solve these types of problems efficiently.