Division

What is Division?

Division splits a number into equal groups. It is the inverse (opposite) of multiplication.

12 ÷ 4 = 3 means: 12 split into 4 equal groups gives 3 in each group.

Symbol: ÷ | Key terms: dividend ÷ divisor = quotient (with a possible remainder)

Division answers the question: "How many times does one number go into another?"

Key Terms

Term	Meaning	Example: 27 ÷ 4 = 6 r 3
Dividend	The number being divided	27
Divisor	The number you divide by	4
Quotient	The whole-number result	6
Remainder	What is left over	3

Division Facts from Times Tables

The fastest way to divide is to use your times tables in reverse. Division and multiplication are opposites.

• 7 × 8 = 56, so 56 ÷ 7 = 8 and 56 ÷ 8 = 7
• 9 × 6 = 54, so 54 ÷ 9 = 6 and 54 ÷ 6 = 9
• 12 × 4 = 48, so 48 ÷ 12 = 4

Knowing your tables up to 12 × 12 makes division much faster.

Short Division (Bus Stop Method)

Short division is used when the divisor is a single digit. Write the dividend under the "bus stop" and work left to right, carrying any remainder to the next digit.

Example: 348 ÷ 6

• 3 ÷ 6: 6 does not go into 3, so carry 3 to make 34
• 34 ÷ 6 = 5 remainder 4 → write 5, carry the 4
• 48 ÷ 6 = 8 → write 8

Answer: 348 ÷ 6 = 58

Long Division

Long division handles larger divisors (two or more digits). Use the D-M-S-B cycle: Divide, Multiply, Subtract, Bring down. Repeat until done.

Example: 728 ÷ 14

• How many times does 14 go into 72? → 5 times (5 × 14 = 70)
• Subtract: 72 - 70 = 2, bring down 8 → 28
• How many times does 14 go into 28? → 2 times (2 × 14 = 28)
• Subtract: 28 - 28 = 0

Answer: 728 ÷ 14 = 52

Remainders

When a number does not divide evenly, the leftover is the remainder.

• 29 ÷ 6 = 4 remainder 5 (because 6 × 4 = 24, and 29 - 24 = 5)

The remainder is always less than the divisor. If your remainder is bigger than the divisor, your quotient digit is too small — increase it by 1.

Remainders can also be written as fractions (29 ÷ 6 = 4 and 5/6) or decimals (by continuing the division past the decimal point).

Divisibility Rules

These shortcuts let you check if a number divides exactly — no calculation needed:

Divisible by	Rule	Example
2	Last digit is even	348: 8 is even ✓
3	Digit sum divisible by 3	129: 1+2+9=12 ✓
4	Last two digits divisible by 4	312: 12÷4=3 ✓
5	Last digit is 0 or 5	75: ends in 5 ✓
9	Digit sum divisible by 9	81: 8+1=9 ✓
10	Last digit is 0	240: ends in 0 ✓

Core Division Rules

• Any number ÷ 1 = itself: 15 ÷ 1 = 15
• Any number ÷ itself = 1: 7 ÷ 7 = 1
• 0 ÷ any number = 0: 0 ÷ 9 = 0
• Dividing by 0 is undefined — it has no valid answer

Word Problems

Division word problems usually contain clue words: shared equally, split between, each, per, groups of, distributed.

Example 1: A baker makes 252 rolls and fills bags of 9. How many full bags?

• 252 ÷ 9 = 28 bags

Example 2: 175 children are put into teams of 8. How many full teams, and how many children are left?

• 175 ÷ 8 = 21 remainder 7 → 21 full teams, 7 children left over

Always ask: does the remainder matter? Sometimes you round up (e.g., how many buses are needed to carry everyone), sometimes you ignore it (e.g., how many whole boxes can be filled).

Common Mistakes

• Not carrying the remainder — in short division, always carry the leftover digit to the next column.
• Remainder bigger than the divisor — this means your quotient digit is too small; increase it.
• Wrong reading of the question — note whether the remainder should be a fraction, decimal, or left as is.
• Working right to left — always start from the leftmost digit.

Tips and Tricks

• Check by multiplying back: quotient × divisor + remainder = dividend. For 348 ÷ 6 = 58: 58 × 6 = 348 ✓
• Dividing by 4: halve twice. 348 ÷ 4 → 348 ÷ 2 = 174 → 174 ÷ 2 = 87
• Dividing by 5: multiply by 2 then divide by 10. 345 ÷ 5 → 690 ÷ 10 = 69
• Dividing by 25: multiply by 4 then divide by 100. 375 ÷ 25 → 1500 ÷ 100 = 15
• Dividing by 9: use the digit-sum rule first to check if it divides evenly before calculating.