1.2.1 Multiplying by 11 Trick
The simplest multiplication trick is the 11’s trick. It is a mundane version of “moving down the line,” where you add consecutive digits and record the answer.
Basic Example: 523 × 11
| Step | Calculation | Result |
|---|---|---|
| Ones | 1 × 3 | 3 |
| Tens | 1 × 2 + 1 × 3 | 5 |
| Hundreds | 1 × 5 + 1 × 2 | 7 |
| Thousands | 1 × 5 | 5 |
Answer: 5753
As one can see, the result can be obtained by subsequently adding the digits along the number you’re multiplying. Be sure to keep track of the carries as well:
Example with Carries: 6798 × 11
| Step | Calculation | Result | Carry |
|---|---|---|---|
| Ones | 8 | 8 | 0 |
| Tens | 9 + 8 | 17 | 1 |
| Hundreds | 7 + 9 + 1 | 17 | 1 |
| Thousands | 6 + 7 + 1 | 14 | 1 |
| Ten Thousands | 6 + 1 | 7 | — |
Answer: 74778
Extending to 111 or 1111
The trick can also be extended to 111 or 1111 (and so on). Where as in the 11’s trick you are adding pairs of digits “down the line,” for 111 you will be adding triples:
Example: 6543 × 111
| Step | Calculation | Result | Carry |
|---|---|---|---|
| Ones | 3 | 3 | 0 |
| Tens | 4 + 3 | 7 | 0 |
| Hundreds | 5 + 4 + 3 | 12 | 1 |
| Thousands | 6 + 5 + 4 + 1 | 16 | 1 |
| Ten Thousands | 6 + 5 + 1 | 12 | 1 |
| Hun. Thousands | 6 + 1 | 7 | — |
Answer: 726273
Division by 11 (Reverse Trick)
Another common form of the 11’s trick is used in reverse. For example:
1353 ÷ 11 = ? (or 11 × x = 1353)
- Ones Digit of x = Ones Digit of 1353: 3
- Tens Digit: 5 = 3 + x_tens → x_tens = 2
- Hundreds Digit: 3 = 2 + x_hund → x_hund = 1
Answer: 123
Division by 111 Example: 46731 ÷ 111
- Ones Digit of x = 1
- Tens Digit: 3 = 1 + x_tens → x_tens = 2
- Hundreds Digit: 7 = 2 + 1 + x_hund → x_hund = 4
Answer: 421
Tip: The hardest part is knowing when to stop. Think about how many digits the answer should have. For example, dividing a 5-digit number by ~100 leaves a 3-digit answer.
Problem Set 1.2.1
Practice these problems. Type your answer and press Enter to check: