The value of $999\frac{995}{999} \times 999$ is

### Concept & Strategy When a mixed fraction is multiplied by its exact denominator, separating the integer part from the fractional part makes the calculation straightforward. You can apply the distributive property $(a + b)c = ac + bc$. $$A\frac{B}{C} \times C = \left(A + \frac{B}{C}\right) \times C = (A \times C) + B$$ ### Step-by-Step Solution Write the mixed fraction as an addition: $$\left(999 + \frac{995}{999}\right) \times 999$$ Distribute the multiplication by $999$: $$= (999 \times 999) + \left(\frac{995}{999} \times 999\right)$$ $$= 999^2 + 995$$ To evaluate $999^2$ easily, rewrite $999$ as $(1000 - 1)$: $$999^2 = (1000 - 1)^2 = 1000^2 - 2(1000)(1) + 1^2$$ $$= 1000000 - 2000 + 1 = 998001$$ Add the remaining $995$ to the squared value: $$998001 + 995 = 998996$$ ### Exam Strategy & Shortcut An alternate trick for calculating $999^2$ is using the base $1000$. The number is $1$ less than $1000$. Subtract $1$ from the number itself: $999 - 1 = 998$. Write the square of the difference ($1^2 = 1$) in a 3-digit format: $001$. Concatenate: $998001$. Then simply add $995$ to get $998996$. ### Common Pitfall A very common mistake is assuming the $999$ in the integer part and the $999$ multiplier cancel out entirely, wrongly leading to simple addition. The multiplier must be distributed to BOTH the integer and the fractional component. ### Final Answer **Therefore, the correct answer is 998996.**