Question

Simplify: 9 X 99 X 999

• A. 890019
• B. 890190
• C. 890109
• D. 891009

Hint:

The \( (10^k - 1) \) trick is extremely powerful for multiplication problems involving numbers like 9, 99, 999, etc. It converts a complex multiplication into a much simpler operation of adding zeros and then subtracting.

Answer 1

The Correct Option is C

Step 1: Understanding the Problem
We need to calculate the product of 9, 99, and 999. We can use a shortcut based on the distributive property.

Step 2: Key Formula or Approach
We can write the numbers as differences from powers of 10. \[ 9 = (10 - 1) \] \[ 99 = (100 - 1) \] \[ 999 = (1000 - 1) \] So, the problem is \( (10 - 1) \times 99 \times 999 \). Let's group them differently for easier calculation.
\[ 9 \times (100 - 1) \times 999 \] \[ 9 \times (99 \times 999) \]
Step 3: Detailed Explanation
Let's calculate step-by-step.
Step 1: Calculate \( 9 \times 99 \)
\[ 9 \times 99 = 9 \times (100 - 1) = 900 - 9 = 891 \]
Step 2: Calculate \( 891 \times 999 \)
We use the same trick again. \[ 891 \times 999 = 891 \times (1000 - 1) \] \[ = (891 \times 1000) - (891 \times 1) \] \[ = 891000 - 891 \] Now, we perform the subtraction: \[ \begin{array}{@{}c@{\,}c@{}c@{}c@{}c@{}c@{}c} & 8 & 9 & 1 & 0 & 0 & 0
- & & & & 8 & 9 & 1
\hline & 8 & 9 & 0 & 1 & 0 & 9
\end{array} \] The calculation is:

10 - 1 = 9
9 - 9 = 0
9 - 8 = 1
0 (from the 1 that was borrowed from) remains 0.
The rest remains: 89.
The result is 890109.

Step 4: Final Answer
The result of the simplification is 890109. Therefore, option (C) is the correct answer.