A tin filled with cookies weighs 2 pounds in total. After three fourth of the cookies are eaten, the tin with the remaining cookies weighs 0.8 pounds. What is the weight of the empty tin in pounds?

Introduction / Context:
This question is a word problem involving total weight, parts of a whole, and simple linear equations. You are asked to distinguish between the weight of the container and the weight of its contents. After a known fraction of the cookies is eaten, the total weight changes, and from these two conditions you can solve for the weight of the empty tin and the original weight of the cookies.

Given Data / Assumptions:

• The total weight of the tin and all the cookies initially is 2 pounds.
• After three fourth of the cookies are eaten, only one fourth of the cookies remain.
• The weight of the tin plus the remaining cookies is 0.8 pounds.
• We must find the weight of the empty tin alone, measured in pounds.

Concept / Approach:
Introduce variables for the weight of the empty tin and the initial weight of all the cookies. Then translate the statements into two linear equations. One equation expresses the initial total weight, and the other expresses the weight after only one fourth of the cookies remain. Solving this system gives the weight of the tin. The essential idea is that the tin weight stays constant, while the cookie weight decreases proportionally to the fraction remaining.

Step-by-Step Solution:
Step 1: Let T be the weight of the empty tin in pounds. Step 2: Let C be the initial weight of all the cookies in pounds. Step 3: Initially, the tin plus cookies weigh 2 pounds, so T + C = 2. Step 4: After three fourth of the cookies are eaten, only one fourth of the cookies remain, so the remaining cookie weight is C / 4. Step 5: At that time, the total weight of the tin and remaining cookies is 0.8 pounds, so T + C / 4 = 0.8. Step 6: Now we have a system of two equations: T + C = 2 and T + C / 4 = 0.8. Step 7: Subtract the second equation from the first: (T + C) − (T + C / 4) = 2 − 0.8. Step 8: Simplify the left side: C − C / 4 = (3 / 4) * C, and the right side is 1.2. Step 9: Thus (3 / 4) * C = 1.2, so C = 1.2 * (4 / 3) = 1.6. Step 10: Substitute C = 1.6 into T + C = 2 to get T + 1.6 = 2, so T = 0.4. Step 11: Therefore, the empty tin weighs 0.4 pounds.

Verification / Alternative check:
Check the values we found. If the cookie weight is 1.6 pounds and the tin weight is 0.4 pounds, the initial total is 0.4 + 1.6 = 2 pounds, matching the first condition. After eating three fourth of the cookies, the remaining cookie weight is 1.6 / 4 = 0.4 pounds. Adding the tin weight again gives 0.4 + 0.4 = 0.8 pounds, matching the second condition. This confirms that the solution is self consistent and correct.

Why Other Options Are Wrong:
If the tin weighed 0.2 pounds, the initial cookie weight would need to be 1.8 pounds, and one fourth of that is 0.45 pounds, making the later total 0.65 pounds, not 0.8 pounds. Similar inconsistencies appear for 0.3, 0.5, and 0.6 pounds when you set up and solve the equations. Only 0.4 pounds produces values for cookie weight and remaining weight that satisfy both given conditions at the same time.

Common Pitfalls:
Students sometimes misinterpret three fourth as the remaining fraction instead of the eaten fraction, leading them to use 3C / 4 instead of C / 4 in the second equation. Another common error is to assume that 0.8 is the weight of the cookies alone and ignore the tin, which breaks the logic of the problem. Careful reading and consistent use of variables for the tin and cookie weights help avoid these mistakes. Always check that your solution satisfies both original statements before finalizing the answer.

Final Answer:
The weight of the empty tin is 0.4 pounds.