Stop with a remainder
1,456 ÷ 32 = 45 remainder 16. After the 6 has been used, 16 remains. It is smaller than 32, so the answer is complete; what a worded question does with the 16 is explained on interpreting a remainder.
Before this page: know multiplication facts, column subtraction and division by a one-digit number using short division.
Dividing by a two-digit number
With a two-digit divisor, keeping the subtractions in your head can be awkward. Long division writes them down. The remainder from each subtraction joins the next digit to form the amount for the next step.
At each stage, choose the greatest multiple of the divisor that does not exceed the current amount. With the first nine multiples listed, the choice is a matter of looking.
1 × 2323
2 × 2346
3 × 2369
4 × 2392
5 × 23115
6 × 23138
7 × 23161
8 × 23184
9 × 23207
For 179, 7 × 23 = 161 fits and 8 × 23 = 184 does not, so the quotient digit is 7.
The long-division cycle
1,794 ÷ 23
Start with 1,794 ÷ 23
1,794 is 1 thousand, 7 hundreds, 9 tens and 4 ones.
Choose, write, subtract, bring down
- 1
Choose the greatest listed multiple not exceeding the current amount.
- 2
Write its multiplier in the quotient and its value underneath.
- 3
Subtract to find the remainder. It must be smaller than the divisor.
- 4
Bring down the next digit and repeat until every digit of the dividend has been used.
Keeping a zero in the quotient
Once the quotient has begun, every place needs a digit. If the current amount is smaller than the divisor, write a zero in that place before bringing down the next digit. This also applies to the final place of a whole-number quotient.
203
15
3045
- Start with 30: 2 × 15 = 30, so write 2 above the 0.
- Bring down 4: 4 is smaller than 15, so write 0 above the 4.
- Bring down 5: 45 ÷ 15 = 3, so write 3 above the 5.
Remainder or decimal
Once every original digit has been used, stop if the answer is required as a whole-number quotient with a remainder; append a decimal point and zeros only if the answer should continue as a decimal.
Continue as a decimal
1,456 ÷ 32 = 45.5. Write 1,456 as 1,456.0, then bring down the appended zero: 160 ÷ 32 = 5.
Continuing into decimals
In 68.4 ÷ 24 the 4 tenths are brought down like any other digit, and the quotient's decimal point goes directly above the dividend's point:
Example: calculate 68.4 ÷ 24 exactly
68.4 ÷ 24
Start with 68.4 ÷ 24
68.4 is 6 tens, 8 ones and 4 tenths.
Explore long division
Start with 2,231 ÷ 46
2,231 is 2 thousands, 2 hundreds, 3 tens and 1 one.
Checking by multiplication
Whole-number result
78 × 23 = 1,794, so 1,794 ÷ 23 = 78.
Decimal result
2.85 × 24 = 68.4, so 68.4 ÷ 24 = 2.85.
With a final remainder, multiply the quotient by the divisor and add the remainder: 45 × 32 + 16 = 1,456, the dividend back again.
Equal groups and equal lengths
Sharing 1,794 bolts among 23 crates and cutting 68.4 m into 24 equal lengths have the same structure: the divisor is the number of equal groups or parts, and the quotient is the size of each.
Equal packing
A supplier places 1,794 bolts equally into 23 crates. Each crate contains 78 bolts.
Equal lengths
A 68.4 m cable is cut into 24 equal lengths. Each length is 2.85 m.
Key points
- Use the largest multiple of the divisor no greater than the current amount.
- Subtract it, then bring down the next digit. Keep zero quotient digits in their places.
- For a decimal answer, continue past the point until the remainder is zero or the decimal continues indefinitely.
- Check by multiplication, adding any remainder back.