Five bags are arranged in order. The first bag contains some fruits. Each subsequent bag contains one quarter as many fruits as the previous bag, and the fifth (last) bag contains 4 fruits. How many fruits are in the first bag?

Introduction / Context: This is a geometric progression type problem disguised in a word puzzle about bags of fruits. We are told that each bag has one quarter as many fruits as the previous bag, and the last bag in a sequence of five bags contains 4 fruits. Because each bag is obtained from the previous one by multiplying by a fixed ratio, the numbers of fruits form a geometric sequence. The question asks us to determine how many fruits were initially in the first bag.

Given Data / Assumptions: - There are 5 bags, labeled from first to fifth in order. - Let the number of fruits in the first bag be F. - Each subsequent bag has one quarter as many fruits as the previous one, so the common ratio is 1/4. - The fifth bag contains 4 fruits. - All fruit counts are assumed to be whole numbers.

Concept / Approach: Since each bag contains 1/4 of the fruits of the previous bag, the sequence of fruit counts is a geometric progression: F, F/4, F/16, F/64, F/256. The fifth term is given as 4, so we can set F/256 = 4 and solve for F. This is a straightforward algebraic manipulation using the idea that dividing by 1/4 repeatedly is equivalent to dividing by 4 each step, or multiplying by 4 when moving backward in the sequence.

Step-by-Step Solution: Step 1: Write the fruit counts for the bags as a geometric sequence. First bag: F, Second bag: F/4, Third bag: F/4^2 = F/16, Fourth bag: F/4^3 = F/64, Fifth bag: F/4^4 = F/256. Step 2: We are told that the fifth bag has 4 fruits, so F/256 = 4. Step 3: Multiply both sides of the equation by 256: F = 4 × 256. Step 4: Compute 4 × 256 = 1024. Step 5: Hence the first bag contains F = 1024 fruits.

Verification / Alternative check: We can check by reconstructing the entire sequence: First bag: 1024 fruits, Second bag: 1024/4 = 256 fruits, Third bag: 256/4 = 64 fruits, Fourth bag: 64/4 = 16 fruits, Fifth bag: 16/4 = 4 fruits. This exactly matches the condition that each bag has one quarter of the previous and that the last bag contains 4 fruits.

Why Other Options Are Wrong: Option 2596, 256, 64: Substituting any of these as the first bag's count and successively dividing by 4 will not yield 4 fruits in the fifth bag. Option None of these: Incorrect because 1024 is one of the options and fits the requirements perfectly.

Common Pitfalls: A frequent misunderstanding is to interpret the phrase reduced by one quarter as meaning subtracting a fixed amount instead of multiplying by 1/4. In this problem, however, the context of each bag having one quarter as many fruits clearly indicates a multiplicative relationship. Recognizing the difference between additive and multiplicative change is crucial for solving such problems correctly.

Final Answer: The first bag contains 1024 fruits.