A total of 405 sweets were distributed equally among some children such that the number of sweets each child received was equal to 20% of the total number of children. How many sweets did each child receive?

Introduction / Context:
This problem combines equal distribution and percentages in a slightly indirect way. The total number of sweets and the condition relating sweets per child to the total number of children are given. Specifically, each child receives sweets equal to 20% of the number of children. We are asked to find how many sweets each child gets. This question requires forming an equation where both the number of children and sweets per child are unknown but linked through the given condition and the total number of sweets.

Given Data / Assumptions:
- Total sweets = 405.- Let the number of children be n.- Sweets are distributed equally among the n children.- Each child receives a number of sweets equal to 20% of the total number of children.- We must find the number of sweets each child receives.

Concept / Approach:
If there are n children, and each child receives k sweets, then the total sweets are n * k. The condition states that each child receives 20% of n sweets, that is 0.20 * n sweets. Therefore, k = 0.20 * n. Substituting this into n * k = 405, we get an equation involving n alone: n * 0.20n = 405. This simplifies to a quadratic equation in n, which we can solve to find n, and then compute k. Because the numbers are well chosen, n becomes a perfect square root and computation is straightforward.

Step-by-Step Solution:
Step 1: Let the number of children be n.Step 2: Let the number of sweets each child receives be k.Step 3: Total sweets = n * k = 405.Step 4: According to the problem, the sweets each child receives are equal to 20% of the number of children.Step 5: So k = 20% of n = (20 / 100) * n = 0.20n.Step 6: Substitute k into the total sweets equation: n * (0.20n) = 405.Step 7: This gives 0.20n^2 = 405.Step 8: Multiply both sides by 5 to eliminate the decimal: n^2 = 405 * 5.Step 9: 405 * 5 = 2,025.Step 10: So n^2 = 2,025.Step 11: Taking the positive square root (since number of children is positive), n = sqrt(2,025) = 45.Step 12: Now find k = 0.20 * n = 0.20 * 45 = 9.Therefore, each child receives 9 sweets.

Verification / Alternative check:
Check by recomputing the total sweets using n = 45 and k = 9.Total sweets = n * k = 45 * 9 = 405, which matches the given total.Also, k as 9 equals 20% of n = 20% of 45 = (20 / 100) * 45 = 9.Both conditions are satisfied, confirming that the solution is correct.

Why Other Options Are Wrong:
- 10: If each child got 10 sweets, then 405 / 10 = 40.5 children, which is impossible and does not give an integer number of children.- 11: 405 / 11 is not an integer, and 11 would not be equal to 20% of any whole number of children in this context.- 12: 405 / 12 is not 45, and 12 also does not correspond to 20% of a suitable integer number of children that yields exactly 405 sweets.

Common Pitfalls:
- Forgetting that both the number of children and sweets per child must be whole numbers.- Misinterpreting "20% of the total number of children" as 20% of sweets instead.- Errors in solving n^2 = 2,025, such as taking an incorrect square root.- Confusing multiplication steps when substituting k = 0.20n into the total equation.

Final Answer:
Each child received 9 sweets.