Long Multiplication and Division

Multiplying in Columns #

Multiply each digit, and carry the spill.

Multiply each digit in turn and carry whatever spills past nine

Start at the units: 7 × 3 = 21. Write the 1, and carry the 2 tens.

Tens: 4 × 3 = 12, and the carried 2 makes 14. Write 4, carry 1 hundred.

Hundreds: 2 × 3 = 6, and the carried 1 makes 7. So 247 × 3 = 741.

Two Digits by Two Digits #

Two rows, one shifting zero, then add.

Split the bottom number into tens and units, multiply by each, then add the two rows

To multiply two digits: start with the ones place: 24 × 6 = 144.

The 3 in 36 is not 3 units — it is 3 tens, and 24 × 3 = 72.

So the tens row is ten times 72: write a 0, then the 72 — 24 × 30 = 720.

Add the two rows: 144 + 720 = 864.

Long Division #

Divide, multiply, subtract, bring down, repeat.

Divide, multiply, subtract, bring the next digit down, and go round again

4 steps: Divide, Multiply, Subtract, Bring down. 5 is less than 12, so use 52.

12 × 4 = 48, which fits inside 52. Divide: write 4 above. Multiply: write 48 below.

Step 3: Subtract: 52 − 48 = 4. Step 4: Bring down the 8 to make 48.

Repeat until it is solved!

Remainders #

The bit too small to share out again.

When the sharing will not come out even, whatever is left over is the remainder

Thirty-seven sweets, five to a bag. Fill the bags until you run out of sweets.

Now as division: 7 bags of 5 make 7 × 5 = 35, and 37 − 35 = 2 are left.

Those 2 are the remainder: fewer than 5, so they cannot fill another bag.

Interpreting the Remainder #

Round it up, round it down, or keep it — the question decides.

The question decides whether the remainder rounds up, rounds down, or is the answer

26 children, 4 to a car: $26/4=6\text{r}2$26 / 4 = 6 r 2. The 2 still travel — so 7 cars.

Same division, $26 and $4 tickets: 6 tickets — the $2 cannot buy another.

26 eggs, boxes of 4: how many left unboxed? Now the remainder 2 IS the answer.