You know long division by a one-digit number: divide, multiply, subtract, bring down. With a two-digit divisor the steps are the same. The new part is finding each digit of the quotient, and an estimate does that.
Key idea
Round the divisor to the nearest ten to estimate each quotient digit. Multiply and subtract. If the product is too big, try one less. If what is left is as big as the divisor or bigger, try one more.
Why it works
Division asks for a missing factor: $672 \div 21$ asks $21 \times ? = 672$. Each digit you write is a piece of that factor: $21 \times 30 = 630$, and $21 \times 2 = 42$, so $21 \times 32 = 672$. A remainder must be smaller than the divisor; if it is not, another whole group still fits.
Worked examples
Worked example: An estimate that works
Divide $1{,}976 \div 38$.
- 38 is about 40.
- How many 38s in 197? Think $200 \div 40 = 5$. $5 \times 38 = 190$, and $197 - 190 = 7$. Write 5.
- Bring down the 6: 76. Think $80 \div 40 = 2$. $2 \times 38 = 76$, and $76 - 76 = 0$. Write 2.
- Check: $52 \times 38 = 1{,}976$.
Answer: $52$
Worked example: An estimate to adjust
Divide $1{,}640 \div 24$.
- 24 is about 20. How many 24s in 164? Think $160 \div 20 = 8$. But $8 \times 24 = 192$ is too big. Try 7: $168$, still too big. Try 6: $144$, and $164 - 144 = 20$. Write 6.
- Bring down the 0: 200. Think $200 \div 20 = 10$, but a digit can be at most 9. Try 8: $8 \times 24 = 192$, and $200 - 192 = 8$.
- The remainder 8 is less than 24, so it is finished. Check: $68 \times 24 + 8 = 1{,}640$.
Answer: $68$ R $8$
Common mistake
Keeping a quotient digit that is too small. If you wrote 5 for $164 \div 24$, you would subtract $120$ and have 44 left, which is more than 24: another 24 still fits. The rule to check: after each subtraction, what is left must be less than the divisor.