Partial-quotients division replaces the guessing step of long division with something safer. Instead of finding the exact digit that fits, a student subtracts any easy multiple of the divisor, records how many groups that was, and repeats. Adding the recorded numbers gives the quotient.
The advantage is that there is no such thing as guessing too low. A student who only trusts multiples of ten will take more steps than one who reaches for a hundred at a time, and both will finish with exactly the same answer.
How the method works
- Write the dividend, and keep a column to the right for the partial quotients.
- Pick an easy multiple of the divisor that is not larger than what is left. Ten times, a hundred times, and doubles are the usual choices.
- Subtract it, and write the multiplier in the right-hand column.
- Repeat until what is left is smaller than the divisor.
- Add the right-hand column. That is the quotient. Whatever is left over is the remainder.
A worked example
1,032 divided by 24
1032 | 24 x 40 - 960 | 40 ------- | 72 | 24 x 3 - 72 | 3 ------- | 0 | --- | 43 so 1032 / 24 = 43
A slower route reaches the same answer. A student who only uses ten at a time subtracts 240 four times, leaving 72, then subtracts 72. The right-hand column reads 10, 10, 10, 10, 3, which still adds to 43. Nobody is penalised for taking small steps.
An example with a remainder
3,847 divided by 26
3847 | 26 x 100 - 2600 | 100 ------- | 1247 | 26 x 40 - 1040 | 40 ------- | 207 | 26 x 7 - 182 | 7 ------- | 25 | --- | 147 3847 / 26 = 147 remainder 25
The remainder is 25 because it is smaller than the divisor 26, so no further group can be taken out. Checking: 26 x 147 = 3822, and 3822 + 25 = 3847.
Why it works
Division asks how many groups of the divisor fit inside the dividend. Every subtraction removes a known number of groups and records it. Since the groups removed never overlap and the process stops only when no whole group is left, the recorded numbers add up to the total number of groups that fit. That total is the quotient by definition.
Common mistakes
Subtracting more than is left
If a subtraction gives a negative result, the multiple chosen was too big. Undo it and take a smaller one. Nothing else is affected.
Stopping too early
The process ends when what is left is smaller than the divisor, not when it looks small. With a divisor of 26, a leftover of 52 still holds two more groups.
Adding the wrong column
The quotient is the sum of the right-hand column, not the last entry in it. Writing 3 instead of 43 in the first example is the classic slip.
Check yourself
918 divided by 17
918 | 17 x 50 - 850 | 50 ------ | 68 | 17 x 4 - 68 | 4 ------ | 0 | 54 918 / 17 = 54
5,624 divided by 32
5624 | 32 x 100 - 3200 | 100 ------ | 2424 | 32 x 70 - 2240 | 70 ------ | 184 | 32 x 5 - 160 | 5 ------ | 24 | 175 5624 / 32 = 175 remainder 24
Check: 32 x 175 = 5600, and 5600 + 24 = 5624.
If two students use different steps, can they get different answers?
No. They can write different right-hand columns, but those columns must add to the same total, because the same number of whole groups fits inside the dividend either way. Different routes, same destination.
See also
Sources
- Partial quotients, sometimes called the friendly-numbers method, is the division algorithm introduced in Everyday Mathematics before the standard long-division algorithm. Examples on this page are our own.