Mental Math Shortcuts: ×9, ×11

×11 shortcut (Trachtenberg / Benjamin): split the digits and put their sum in the middle.

1. 01

   2 _ 3

   Pull the digits of 23 apart, leaving a gap in the middle.

2. 02

   2 + 3 = 5

   Add the two digits — that goes in the gap.

3. 03

   253 = 253

   Drop the sum in the middle. Done in one beat.

4. →

   = 253

×11 shortcut with carry (Trachtenberg / Benjamin).

1. 01

   8 _ 5

   Split the digits of 85.

2. 02

   8 + 5 = 13 → carry 1

   Sum is 10 or more — write the units digit (3) in the middle and carry the 1 to the leftmost digit.

3. 03

   (8+1)35 = 935

   Front digit becomes 9; result is 935.

4. →

   = 935

×11 shortcut (Benjamin): multiply by 10, then add the number.

1. 01

   314 × 10 = 3140

   Shift one place left.

2. 02

   3140 + 314 = 3454

   Add the original 314.

3. →

   = 3454

×9 shortcut (Benjamin): use (n × 10) − n.

1. 01

   47 × 10 = 470

   Multiplying by 10 just shifts a zero on the end.

2. 02

   470 − 47 = 423

   Subtract one copy of 47, because 9 = 10 − 1.

3. →

   = 423

×9 shortcut (Benjamin): use (n × 10) − n.

1. 01

   52 × 10 = 520

   Multiplying by 10 just shifts a zero on the end.

2. 02

   520 − 52 = 468

   Subtract one copy of 52, because 9 = 10 − 1.

3. →

   = 468

Benjamin, Secrets of Mental Math — multiply by 10 and subtract