Hemavathi cuts a cake into two equal halves and then cuts one half into smaller pieces of equal size. Each of these small pieces weighs 15 grams. If she now has a total of 9 pieces of cake, how heavy was the original cake?

Introduction / Context:
This word problem is a simple application of fractions and multiplication in a real-life context. A cake is cut into two halves, one half is further cut into equal small pieces, and we are told how many pieces there are in total and the weight of each small piece. The task is to determine the total weight of the original cake from this information.

Given Data / Assumptions:
- The original cake is cut into two equal halves, so each half weighs half of the original weight.
- One half is cut into smaller equal pieces; each small piece weighs 15 grams.
- The other half remains as one large piece.
- In total, Hemavathi now has 9 pieces of cake: 1 large half and some number of small pieces.
- All pieces are assumed to have precise weights as stated (no wastage or crumbs lost).

Concept / Approach:
We first determine how many of the 9 pieces are small pieces and then use the weight of each small piece to find the weight of the half that was cut. Since that half is exactly half the original cake, doubling its weight gives the total weight of the cake. The reasoning is straightforward, but we must correctly interpret that only one half is cut into small pieces and count the number of small pieces accordingly.

Step-by-Step Solution:
Step 1: Let the weight of the original cake be W grams. Then each half weighs W/2 grams. Step 2: One half remains uncut as a single large piece of weight W/2 grams. Step 3: The other half is divided into equal small pieces, each weighing 15 grams. Step 4: Hemavathi has 9 pieces in total. One of these is the uncut half; therefore, the number of small pieces is 9 - 1 = 8. Step 5: The total weight of the small pieces is 8 × 15 grams = 120 grams. Step 6: This total 120 grams comes from the half that was cut, so W/2 = 120 grams. Step 7: Multiply both sides by 2 to obtain W = 240 grams.

Verification / Alternative check:
We can verify by reconstructing the situation with W = 240 grams. Each half weighs 240/2 = 120 grams. One half remains as a single piece of 120 grams. The other half is cut into 120/15 = 8 pieces, each of 15 grams. Thus there are 8 small pieces plus 1 large half, making 9 pieces in total. All details of the story are satisfied, so the solution is consistent.

Why Other Options Are Wrong:
Option 280 g or 180 g or 170 g: For any of these values, the weight of a half would not be a multiple of 15 that allows exactly 8 small pieces and a total of 9 pieces. For example, if the cake weighed 180 g, half would be 90 g and cutting that into 15 g pieces would yield 6 pieces, not 8, contradicting the total piece count.
Option None of these: Incorrect because 240 g is one of the options and matches the scenario precisely.

Common Pitfalls:
A common misunderstanding is to assume both halves are cut into small pieces, leading to the wrong number of pieces and an incorrect total weight. Another mistake is to divide the total number of pieces incorrectly between large and small pieces or to forget that the large half is one whole piece. Carefully distinguishing the uncut half from the cut half and counting pieces accurately leads to the correct answer.

Final Answer:
The original cake weighed 240 grams.