The words used in division
| Term | Meaning | In 4,587 ÷ 13 = 352 r 11 |
|---|---|---|
| Dividend | The number being divided | 4,587 |
| Divisor | The number you divide by | 13 |
| Quotient | How many whole times the divisor fits | 352 |
| Remainder | What is left over, always smaller than the divisor | 11 |
Every division answer fits this check:
How long division works
Long division breaks a big division into a chain of small ones, working from left to right through the dividend. Each round has the same four steps, which many teachers remember as “Does McDonald’s Sell Burgers?”:
- Divide. How many times does the divisor fit into the current number? Write that digit in the answer line.
- Multiply. Multiply that digit by the divisor and write the product underneath.
- Subtract. Take the product away. The result must be smaller than the divisor, or the digit was too small.
- Bring down. Bring the next digit of the dividend down next to the result, and start again.
When there are no digits left to bring down, what remains is the remainder. If you want a decimal answer instead, put a decimal point after the dividend, write zeros after it, and keep bringing them down.
Worked examples
Example 1: 4,587 ÷ 13.
- 13 does not fit into 4, so look at 45. 13 fits 3 times (3 × 13 = 39). 45 − 39 = 6. Bring down 8 to make 68.
- 13 fits into 68 five times (65). 68 − 65 = 3. Bring down 7 to make 37.
- 13 fits into 37 twice (26). 37 − 26 = 11. No digits left.
- Answer: 352 remainder 11. Check: 352 × 13 = 4,576, and 4,576 + 11 = 4,587.
Carry on past the decimal point and the remainders start to cycle, giving 352.846153846153…, written 352.(846153).
Example 2: 3 ÷ 4 as a decimal.
- 4 does not fit into 3, so write 0 and a decimal point, then bring down a zero to make 30.
- 4 fits into 30 seven times (28), leaving 2. Bring down another zero to make 20.
- 4 fits into 20 exactly five times, leaving 0.
- Answer: 0.75, a terminating decimal.
Example 3: decimals in the divisor, 12.5 ÷ 0.25.
Long division needs a whole-number divisor, so move the decimal point in both numbers the same number of places. Multiplying both by 100 gives 1,250 ÷ 25, which has exactly the same answer: 50. The calculator does this scaling for you and tells you which power of ten it used. If there is a remainder, it shows it in the scaled units in the layout and converts it back for the answer.
Terminating or repeating?
When you keep dividing past the decimal point, one of two things happens. Either a remainder of 0 appears and the decimal stops, or a remainder you have already seen comes back. From then on the same digits repeat forever. The calculator tracks every remainder to find the repeating block exactly, and shows it in brackets and with a bar over it.
A fraction in its lowest terms gives a terminating decimal only when the denominator has no prime factors other than 2 and 5. That is why eighths and fifths stop, but thirds, sixths and sevenths repeat.
| Fraction | Decimal | Type |
|---|---|---|
| 1/2 | 0.5 | Terminating |
| 1/3 | 0.(3) = 0.333… | Repeating |
| 1/4 | 0.25 | Terminating |
| 1/6 | 0.1(6) = 0.1666… | Repeating |
| 1/7 | 0.(142857) | Repeating, 6 digits |
| 1/8 | 0.125 | Terminating |
| 1/9 | 0.(1) = 0.111… | Repeating |
| 1/11 | 0.(09) = 0.0909… | Repeating |
| 1/12 | 0.08(3) = 0.08333… | Repeating |
Writing the remainder three ways
The same answer can be written as a remainder, a fraction or a decimal. For 4,587 ÷ 13:
- Remainder: 352 r 11
- Mixed number: 352 11/13 (the remainder over the divisor)
- Decimal: 352.8462 to four decimal places
Which one is right depends on the question. If 4,587 sweets are shared between 13 children, each gets 352 and 11 are left over. If 4,587 metres of rope are cut into 13 equal pieces, each piece is about 352.85 m.
Tips and common mistakes
- Line up the digits. Each digit of the answer sits above the last digit you used. Squared paper helps.
- Write the zeros. If the divisor does not fit after bringing down a digit, write 0 in the answer before bringing down the next one. Forgetting this is the most common error: 1,015 ÷ 5 is 203, not 23.
- Check each subtraction. If the result is as big as the divisor or bigger, your digit was too small.
- Estimate first. 4,587 ÷ 13 is a bit more than 4,500 ÷ 15 = 300, so an answer of 352 looks sensible.
- Use the check. Quotient × divisor + remainder should give back the dividend every time.
Related calculators
For quick answers without the steps, use the calculator. The fraction calculator simplifies remainders written as fractions, and the percentage calculator helps when the division is part of a percentage question. For square roots and powers, try the scientific calculator.