$3897 \times 999 = x$

### Concept & Formula Similar to previous problems, calculating products involving 999 is highly efficient when utilizing the distributive property over subtraction. $$a \times (10^n - 1) = a \times 10^n - a$$ ### Step-by-Step Solution * Express 999 as a base of 10: $$999 = 1000 - 1$$ * Substitute into the expression: $$3897 \times 999 = 3897 \times (1000 - 1)$$ * Multiply and subtract: $$3897000 - 3897$$ * Complete the subtraction: $$3897000 - 3897 = 3893103$$ ### Exam Strategy & Shortcut Use the subtraction shortcut quickly: $3897000 - 3897$. To do this mentally, subtract 4000 (which is easier) and then add back 103 (since $4000 - 3897 = 103$). $$3897000 - 4000 = 3893000$$ $$3893000 + 103 = 3893103$$ ### Common Pitfall Trying to use the "subtract 1 and find the 9's complement" trick without adjusting for the fact that 3897 has 4 digits while 999 has only 3 digits. That specific trick only works when the number of 9s is greater than or equal to the digits in the other number. The zero-appending subtraction method is foolproof. ### Final Answer **Therefore, the correct answer is 3893103.**