While out on picnic, a group of boys came upon an apple tree. One of the boys climbed the tree and picked enough apples for each boy to have three, with none left over. Then along came three boys, making it impossible to divide the picked apples evenly. However, after picking one more apple and adding it to the total, every boy had two apples, with none left over. How many apples were finally divided?

### Concept & Logic Use basic linear equations to balance the total number of apples before and after the new boys arrive. ### Step-by-Step Solution * Let the initial number of boys be $x$. * Initial total apples = $3x$. * After 3 more boys join, total boys = $x + 3$. * After picking 1 more apple, total apples = $3x + 1$. * The new distribution gives 2 apples per boy, setting up the equation: $$2(x + 3) = 3x + 1$$ * Expand the equation: $$2x + 6 = 3x + 1$$ * Solving for $x$, we get $x = 5$. * Initial boys = 5. Total apples finally divided = $3(5) + 1 = 16$. ### Exam Strategy & Shortcut Test the options to find a valid integer for the boys. If 16 apples were finally divided, initially there were 15 apples. 15 is divisible by 3, meaning there were 5 boys initially. $5 + 3 = 8$ boys finally, and $16 / 8 = 2$ apples each. The conditions perfectly match without needing algebra. ### Common Pitfall A frequent error is calculating the initial number of apples (15) and forgetting to add the 1 extra apple picked at the end, leading to a frustrating search for an option that isn't there. ### Final Answer Therefore, the correct answer is 16.