Division with Remainders

4.D.1Divide by 1-digit01/04

Divide up to 4-digit numbers by a 1-digit number.

Step01of 04

Long division — the standard algorithm for dividing big numbers. Repeat: divide, multiply, subtract, bring down. Repeat until you've used every digit.

4.D.1 — Divide up to 4-digit numbers by a 1-digit number.

Step 01 of 04

Long division — the standard algorithm for dividing big numbers. Repeat: divide, multiply, subtract, bring down. Repeat until you've used every digit.

Step 02 of 04

Worked example. 728 ÷ 4.

First, can 4 go into 7? Yes — 1 time. Write 1 above the 7. $4 \times 1 = 4$ ; $7 - 4 = 3$ . Bring down the 2 to make 32.

Can 4 go into 32? Yes — 8 times. Write 8 above the 2. $4 \times 8 = 32$ ; $32 - 32 = 0$ . Bring down the 8 to make 8.

Can 4 go into 8? Yes — 2 times. $4 \times 2 = 8$ ; $8 - 8 = 0$ . No more digits to bring down. Done.

728 \div 4 = 182

Step 03 of 04

When the divisor is bigger than the leading digit.

231 ÷ 6. Can 6 go into 2? No (2 < 6) — write nothing. Try 23. 6 goes into 23 → 3 times (since 6 × 3 = 18, 6 × 4 = 24 too big). Continue normally.

Step 04 of 04

Estimating. Before dividing, mentally check: 728 ÷ 4 ≈ 700 ÷ 4 = 175. Our answer 182 is close — sanity check passed.

Key insight

Divide-multiply-subtract-bring down, repeat. Skip a position if the divisor is bigger than the working number. Estimate first to catch big mistakes.

4.D.2 — Division with remainders.

Step 01 of 04

When a number doesn't divide evenly, what's left over is the remainder. Standard notation: write "r" followed by the remainder.

23 \div 4 = 5 r 3

23 contains five 4's (= 20), with 3 left over.

Step 02 of 04

Worked example with long division. 379 ÷ 6.

6 into 37 → 6 times (36); 37 − 36 = 1. Bring down 9 → 19. 6 into 19 → 3 times (18); 19 − 18 = 1. No more digits. Quotient 63, remainder 1.

379 \div 6 = 63 r 1

Verify: $63 \times 6 + 1 = 378 + 1 = 379$ $✓$ .

Step 03 of 04

What does the remainder mean in context? Three common interpretations:

"How many full groups?"	→use the QUOTIENT, ignore remainder
"How many groups needed (round up)?"	→add 1 to quotient if remainder > 0
"How much left over?"	→use the REMAINDER
"Express as fraction"	→remainder/divisor (e.g. 23/4 = 5 3/4)

Step 04 of 04

Worked context example. A van seats 9. 41 students need a ride. How many vans?

41 \div 9 = 4 r 5

4 vans full, 5 students left over needing transport too. Need ONE MORE van (since you can't leave 5 students behind). Answer: 5 vans.

Key insight

Verify division by multiplying back and adding the remainder: result should equal the original. ALWAYS read the question to interpret the remainder correctly — sometimes use it, sometimes ignore it, sometimes round up.

4.D.3 — Division word problems.

Step 01 of 04

Two main kinds of division.

Equal SHARING	→how many in each group?
Equal GROUPING	→how many groups?

"Share 24 cookies among 4 friends" → SHARING (each gets 24/4 = 6). "Pack 24 cookies into bags of 4" → GROUPING (need 24/4 = 6 bags). Same arithmetic, different setup.

Step 02 of 04

Sharing example. A teacher splits 132 pencils equally among 6 groups. How many pencils per group?

132 \div 6 = 22 pencils per group

Step 03 of 04

Grouping example. A factory packages 240 candies into bags of 8. How many bags?

240 \div 8 = 30 bags

Step 04 of 04

With remainders. "Pack 60 books into boxes that hold 7 each. How many boxes do you need?"

60 \div 7 = 8 r 4

Need 9 boxes — 8 full and 1 partial for the leftover 4 books.

Key insight

Both sharing and grouping use division. Read the question to decide if the remainder gets ignored, used directly, or causes a "round up." Word problems are mostly about reading carefully; the arithmetic is just long division.

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