Start Free
Level 5Grade 5skill: division_remainder· 7 min read

Division With Remainders

In short: To divide with a remainder, find the largest multiple of the divisor that fits under your number — the number you multiplied by is the quotient, and what's left after subtracting is the remainder. Always check the remainder is smaller than the divisor; if it isn't, you stopped too early.

How to divide when it doesn't come out even

Most divisions in real life don't split cleanly. Share 47 stickers among 6 friends and someone is left holding the leftovers. Division with remainders is just an honest way to say: here's how many go in each group, and here's what's still in your hand.

The whole method rests on one move you already know — multiplication. You're hunting for the largest multiple of the divisor that still fits inside your number without going over. Everything else is subtraction and a quick check.

  1. Find the largest multiple of the divisor that is less than or equal to your number (this is where your times tables do the work).
  2. The number you multiplied the divisor by is your quotient — the whole number of groups.
  3. Subtract that multiple from your number. What's left is the remainder.
  4. Check: the remainder must be smaller than the divisor. If it isn't, take one more group.
  5. Write the answer as 'quotient r remainder', for example 7 r 5.

Why the remainder is always smaller than the divisor Why it works

÷ 4
remainder can only be 0, 1, 2 or 3

27 ÷ 4 = 6 r 3 — you can never be left with 4, that's another whole group

÷ 6
remainder can only be 0, 1, 2, 3, 4 or 5

47 ÷ 6 = 7 r 5 — a remainder of 6 would mean one more six fits

÷ 9
remainder can only be 0 through 8

100 ÷ 9 = 11 r 1 — the biggest possible leftover is 8

Here's the one idea worth carrying: the remainder rule isn't something to memorise — it's a completeness check. If your leftover were as big as the divisor (or bigger), you could fit at least one more group, which means you stopped too early. So 'remainder smaller than divisor' is really the question 'am I actually finished?' in disguise.

Worked examples

47 ÷ 6 = 7 r 5
  1. Count up in sixes: 6, 12, 18, 24, 30, 36, 42, 48. The last one that fits under 47 is 42, which is 6 × 7.
  2. So the quotient is 7.
  3. Subtract: 47 − 42 = 5.
  4. Check: 5 is smaller than 6, so we're done.
85 ÷ 7 = 12 r 1
  1. Find the biggest 7-times fact under 85: 7 × 12 = 84 (and 7 × 13 = 91 is too big).
  2. The quotient is 12.
  3. Subtract: 85 − 84 = 1.
  4. Since 1 is well under 7, that's the final answer.
96 ÷ 8 (watch for a zero remainder) = 12 r 0 (or simply 12)
  1. 8 × 12 = 96 exactly.
  2. The quotient is 12.
  3. Subtract: 96 − 96 = 0.
  4. A remainder of 0 is fine — it just means the division came out even.
55 ÷ 7 (a big remainder is still legal) = 7 r 6
  1. 7 × 7 = 49 fits; 7 × 8 = 56 is over 55.
  2. The quotient is 7.
  3. Subtract: 55 − 49 = 6.
  4. Check: 6 is close to 7 but still smaller than 7, so 6 is a perfectly good remainder — don't be tempted to round the quotient up.
131 ÷ 8 (two-digit answer) = 16 r 3
  1. Work in stages: 8 × 16 = 128, and 8 × 17 = 136 overshoots 131.
  2. The quotient is 16.
  3. Subtract: 131 − 128 = 3.
  4. With 3 left over and 3 smaller than 8, the working is complete.

Common mistakes

Throwing away the leftover

✗ 7

For 47 ÷ 6 you correctly find that 6 × 7 = 42 fits, but then you write just 7 and quietly drop the 5 that was left over. An answer of 7 alone means 6 × 7 = 47, which isn't true.

Fix: A division that doesn't come out even has two parts. After the quotient, always subtract and report the remainder: 47 ÷ 6 = 7 r 5.

forgot_remainder

Grabbing a neighbouring times-table fact

✗ 7 r 2

For 50 ÷ 7 you want 7 × 7 = 49, but you reach for the nearby fact 6 × 8 = 48 instead. That leaves 50 − 48 = 2, so the remainder comes out 2 rather than the correct 1.

Fix: You're dividing by 7, so use a 7-times fact: 7 × 7 = 49, and 50 − 49 = 1. Answer: 7 r 1.

table_neighbour

Stopping before the division is finished

✗ 1 r 24

For 84 ÷ 6 you divide the 8 tens by 6, get 1, and stop — never bringing the 4 down. That leaves 24 sitting there as a 'remainder', but 24 is much bigger than 6, and a remainder can never be as large as the divisor.

Fix: Keep going until every digit is used and the leftover is smaller than the divisor: 6 × 14 = 84, so 84 ÷ 6 = 14 (remainder 0).

partial_computation

Practice: division with remainders

Work each division to a whole-number quotient and a remainder, then check that your remainder is smaller than the divisor before you write it down.

Related skills

First, make sure of

Frequently asked questions

What does the 'r' mean in an answer like 7 r 5?

The 'r' stands for remainder — the amount left over after you've made as many equal groups as possible. In 47 ÷ 6 = 7 r 5, you can make 7 groups of 6 (that's 42), and 5 counters are left that can't form another full group of 6.

Can the remainder ever be bigger than the number you're dividing by?

No. The remainder is always smaller than the divisor. If your leftover is equal to or bigger than the divisor, you could fit at least one more group, which means your quotient is too small. A remainder of 6 in a division by 6, for example, is impossible — that's another whole six.

Is a remainder of 0 a real answer?

Yes. A remainder of 0 simply means the division came out even, like 96 ÷ 8 = 12 r 0. You can write it as 12 r 0 or just 12 — both mean the same thing.

How do I check my division with remainders is correct?

Multiply the quotient by the divisor and add the remainder — you should get back the original number. For 85 ÷ 7 = 12 r 1, check 7 × 12 = 84, then 84 + 1 = 85. It matches, so the answer is right.

How is this different from just knowing my times tables?

Times tables give you the exact multiples (like 6 × 7 = 42). Division with remainders uses those facts but adds a step: when the number isn't an exact multiple, you find the closest multiple that fits underneath and record what's left over. Strong times tables make this fast, which is why they're the prerequisite.