If each child is given 10 sweets, there are 3 sweets left over. But if each is given 11, then the number of sweets is 4 less. Find the number of sweets.

### Concept & Logic This requires setting up a basic system of linear equations comparing total sweets based on different distribution scenarios to find the unknown number of children and sweets. ### Step-by-Step Solution * **Given:** Let the number of children be $c$ and the total number of sweets be $s$. * **Scenario 1:** If each gets 10 sweets, 3 are leftover. $$s = 10c + 3$$ * **Scenario 2:** If each gets 11 sweets, they are 4 sweets short (4 less). $$s = 11c - 4$$ * **Equate the two expressions for total sweets ($s$):** $$10c + 3 = 11c - 4$$ * **Solve for $c$ (number of children):** $$11c - 10c = 3 + 4$$ $$c = 7 \text{ children}$$ * **Find total sweets ($s$):** Substitute $c = 7$ into the first equation: $$s = 10(7) + 3$$ $$s = 70 + 3 = 73$$ ### Exam Strategy & Shortcut The difference in the number of sweets given per child ($11 - 10 = 1$ sweet) accounts for the total difference in the pile of sweets (from having $+3$ extra to being $-4$ short). Total difference = $3 - (-4) = 7$ sweets. Therefore, Number of children = $\frac{\text{Total Difference}}{\text{Difference per child}} = \frac{7}{1} = 7$. Total sweets = $(7 \times 10) + 3 = 73$. This logic is lightning fast. ### Common Pitfall Setting up the second equation as $s = 11c + 4$ instead of $11c - 4$. If you are "4 short", it means you need 4 more to fulfill the $11c$ requirement, hence $11c$ is 4 greater than the actual sweets $s$. ### Final Answer Therefore, the correct answer is **73**.