🌱 Starter Village · Arithmetic
Division
Divide whole numbers, with and without remainders, seeing division as sharing equally and as repeated subtraction.
In short
- Division splits a total into equal parts (sharing) or counts equal groups (grouping).
- Every division is a multiplication read backwards, which gives you a free check.
- A remainder is always smaller than the divisor: divisor x quotient + remainder = dividend.
- The context, not the arithmetic, decides whether a remainder is ignored, rounded up, or reported.
Two ways to read a division
24 / 4 can be read in two different ways, and both are useful:
- Sharing. Split 24 coins fairly between 4 adventurers. How many does each get? (6)
- Grouping. How many groups of 4 can be made from 24? (6)
The answer is the same, but the picture is different. Sharing fixes the *number of groups*; grouping fixes the *size of each group*. Word problems use both, so it is worth asking which one a question means.
Division undoes multiplication
Because 4 x 6 = 24, we know 24 / 4 = 6 and 24 / 6 = 4. Every division fact is a multiplication fact read backwards.
That gives two practical tools:
- To divide, ask "what do I multiply the divisor by to reach the dividend?"
- To check a division, multiply your answer by the divisor. You should get back to the number you started with.
Note the asymmetry: 24 / 4 is not the same as 4 / 24. Unlike multiplication, order matters.
Remainders
Not every division comes out exactly. 23 / 4 gives 5 whole groups with 3 left over, written 5 r 3.
Two rules keep remainders honest:
- The remainder must always be smaller than the divisor. If it is not, another whole group still fits.
- The check becomes: divisor x quotient + remainder = dividend. Here 4 x 5 + 3 = 23.
What to do with the remainder
The mathematics stops at 5 r 3; the story decides what that means. The same calculation gives three different answers depending on the question.
- *How many full crates can be packed?* Ignore the remainder: 5.
- *How many crates are needed so nothing is left behind?* Round up: 6.
- *How many items are left over?* The remainder itself: 3.
Always re-read the question after dividing and decide which of these is being asked.
Dividing larger numbers
For bigger dividends, chunk the work instead of guessing. To compute 372 / 6:
- 6 x 10 = 60, and 6 x 50 = 300. That is not more than 372, so the answer is at least 50.
- 372 - 300 = 72 left.
- 6 x 12 = 72 exactly.
- So 372 / 6 = 50 + 12 = 62.
This is what long division does; chunking simply lets you take whichever bites you are confident about.
Worked examples
Example 1
187 / 8 = ? Give the answer as "quotient r remainder".
- Find the largest multiple of 8 that does not pass 187.
- 8 x 20 = 160, and 187 - 160 = 27 remains.
- 8 x 3 = 24, which fits into 27, leaving 3.
- So the quotient is 20 + 3 = 23 and the remainder is 3.
- 187 / 8 = 23 r 3. Check: 8 x 23 + 3 = 184 + 3 = 187, and 3 < 8.
Example 2
124 lanterns must be carried out of the mine. Each cart holds 9 lanterns. How many carts are needed?
- Divide first: 9 x 13 = 117, and 124 - 117 = 7, so 124 / 9 = 13 r 7.
- 13 carts are completely full.
- The 7 remaining lanterns still have to be carried out.
- A partly full cart is still a cart, so round up: 13 + 1 = 14 carts.
Practice problems, with solutions
Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.
Problem 1
Difficulty 1 of 512 / 4 = ?
Answer: 3
- 4 x 3 = 12
- So 12 / 4 = 3.
Problem 2
Difficulty 3 of 589 / 10 = ? Write your answer as "quotient r remainder", for example 7 r 2.
Answer: 8 r 9
- Largest multiple of 10 below 89: 10 x 8 = 80
- Remainder: 89 - 80 = 9
- 89 / 10 = 8 r 9
- Check: 10 x 8 + 9 = 89
Problem 3
Difficulty 4 of 51,008 apples are packed into baskets that hold 13 each. How many baskets can be filled completely?
Answer: 77 baskets
- 1,008 / 13 = 77 r 7
- The 7 left over cannot fill a whole basket.
- So 77 baskets can be filled completely.
Common mistakes
- Leaving a remainder that is as large as (or larger than) the divisor.
- Rounding down when the story requires that nothing be left behind.
- Reversing the dividend and divisor, computing 4 / 24 instead of 24 / 4.
- Reporting the quotient only when the question asked for the leftover amount.
What you should be able to do
- Recall division facts as the inverse of multiplication facts.
- Divide with a remainder and state the result as "quotient r remainder".
- Interpret a remainder sensibly in a word problem.
- Check a division result by multiplying back.
Where this fits in the curriculum
Common Core
- 3.OA.A.2
Grade 3 — Interpret whole-number quotients as sharing equally or as repeated subtraction.
- 3.OA.C.7
Grade 3 — Fluently multiply and divide within 100, knowing division as the inverse of multiplication.
- 4.NBT.B.6
Grade 4 — Find whole-number quotients and remainders with up to four-digit dividends.
- 4.OA.A.3
Grade 4 — Solve multi-step word problems, including interpreting a remainder sensibly.