How to Multiply by 11 Mentally in Seconds
How to Multiply by 11 Mentally in Seconds
Multiplying by 11 is one of the most satisfying mental math tricks you can learn. With the Trachtenberg system's rule for ×11, you can multiply any number by 11 in your head — no paper required.
The Rule
The rule for multiplying by 11 is beautifully simple:
Add each digit to its neighbor (the digit on the right).
You work from right to left. Pad the number with a leading zero (and imagine a trailing zero), so every digit has a neighbor. Then for each digit, add it to the digit to its right.
Step-by-Step Examples
Example 1: 63 × 11
Let's work through this step by step:
| Position | Digit | Neighbor | Sum | Write |
|---|---|---|---|---|
| Rightmost | 3 | (none) | 3 | 3 |
| Middle | 6 | 3 | 6 + 3 = 9 | 9 |
| Leftmost | (none) | 6 | 6 | 6 |
Answer: 693
Example 2: 345 × 11
| Position | Digit | Neighbor | Sum | Write |
|---|---|---|---|---|
| Rightmost | 5 | (none) | 5 | 5 |
| Middle | 4 | 5 | 4 + 5 = 9 | 9 |
| Middle | 3 | 4 | 3 + 4 = 7 | 7 |
| Leftmost | (none) | 3 | 3 | 3 |
Answer: 3795
Example 3: 728 × 11 (with carrying)
What happens when the sum exceeds 9? You carry just like in normal addition.
| Position | Digit | Neighbor | Sum | Write | Carry |
|---|---|---|---|---|---|
| Rightmost | 8 | (none) | 8 | 8 | 0 |
| Middle | 2 | 8 | 2 + 8 = 10 | 0 | 1 |
| Middle | 7 | 2 | 7 + 2 + 1 = 10 | 0 | 1 |
| Leftmost | (none) | 7 | 0 + 7 + 1 = 8 | 8 | 0 |
Answer: 8008
The Pattern in Action
Here's the beautiful thing about this rule: it works for any size number. Whether you're multiplying a 2-digit number or a 10-digit number by 11, the same simple rule applies.
Try these in your head:
- 24 × 11 = 264 (2, then 2+4=6, then 4)
- 52 × 11 = 572 (5, then 5+2=7, then 2)
- 81 × 11 = 891 (8, then 8+1=9, then 1)
- 45 × 11 = 495 (4, then 4+5=9, then 5)
A Quick Mental Shortcut for Two-Digit Numbers
For two-digit numbers, there's an even faster shortcut: split the number and put the sum of the digits in the middle.
- 35 × 11: Split 3 and 5. Put 3+5=8 in the middle → 385
- 72 × 11: Split 7 and 2. Put 7+2=9 in the middle → 792
- 47 × 11: Split 4 and 7. Put 4+7=11 in the middle (carry) → 517
This shortcut is a special case of the general rule and works great for quick calculations.
Practice Makes Perfect
The key to mastering any mental math technique is consistent practice. Start with 2-digit numbers and work your way up to 3-digit and 4-digit numbers.
Here are some practice problems to get you started:
- 36 × 11 = ?
- 89 × 11 = ?
- 234 × 11 = ?
- 567 × 11 = ?
- 1234 × 11 = ?
- 36 × 11 = 396
- 89 × 11 = 979
- 234 × 11 = 2574
- 567 × 11 = 6237
- 1234 × 11 = 13574
Take It Further
The ×11 rule is just the beginning. The Trachtenberg system has similar rules for every multiplier from 2 to 12. Each rule follows a simple pattern that becomes automatic with practice.