Casting Out Nines

Distribute the smaller factor across the place values of the larger number, then add the partial products.

1. 01

   9 × 3 = 27

   Multiply just the units part (9) by 3. We do one small piece at a time, then add them up at the end. 9 × 3 = 27 — running product 27.

2. 02

   9 ✓

   Apply this piece of the shortcut — running product 27.

3. →

   = Passes

Left-to-right multiplication (Benjamin): break the 2-digit number into tens + units, multiply each, add.

1. 01

   20 × 8 = 160

   Tens first — biggest piece always goes first to keep the running total anchored.

2. 02

   2 × 8 = 16

   Now the units piece.

3. 03

   160 + 16 = 176

   Add the two partial products — left-to-right addition.

4. →

   = 176

Benjamin, Secrets of Mental Math — distribute by place

Left-to-right multiplication (Benjamin): break the 2-digit number into tens + units, multiply each, add.

1. 01

   40 × 7 = 280

   Tens first — biggest piece always goes first to keep the running total anchored.

2. 02

   8 × 7 = 56

   Now the units piece.

3. 03

   280 + 56 = 336

   Add the two partial products — left-to-right addition.

4. →

   = 336

Benjamin, Secrets of Mental Math — distribute by place

Left-to-right multiplication (Benjamin): break the 2-digit number into tens + units, multiply each, add.

1. 01

   80 × 8 = 640

   Tens first — biggest piece always goes first to keep the running total anchored.

2. 02

   5 × 8 = 40

   Now the units piece.

3. 03

   640 + 40 = 680

   Add the two partial products — left-to-right addition.

4. →

   = 680

Benjamin, Secrets of Mental Math — distribute by place

Left-to-right multiplication (Benjamin): break the 2-digit number into tens + units, multiply each, add.

1. 01

   80 × 6 = 480

   Tens first — biggest piece always goes first to keep the running total anchored.

2. 02

   8 × 6 = 48

   Now the units piece.

3. 03

   480 + 48 = 528

   Add the two partial products — left-to-right addition.

4. →

   = 528

Benjamin, Secrets of Mental Math — distribute by place