Partial-differences subtraction breaks a subtraction problem into one small subtraction per place value. Hundreds are subtracted from hundreds, tens from tens, ones from ones. Each of those answers is called a partial difference. Some partial differences come out negative, and that is allowed. Adding all the partial differences gives the answer.
The method is taught in elementary classrooms as an alternative to the traditional borrowing algorithm. Nothing is regrouped and nothing is crossed out, so a child who loses track of borrowing can still work accurately. It also gives young students an early, concrete reason to add a negative number to a positive one.
How the method works
- Write both numbers in expanded form, or line them up by place value.
- Starting from the largest place value, subtract each place on its own.
- Write each answer, including a minus sign when the bottom digit is larger.
- Add the partial differences together. That total is the difference.
Working from left to right is deliberate. The first partial difference is the largest piece of the answer, so a student knows the rough size of the result before finishing, and a wrong answer usually stands out immediately.
A worked example
Take 471 - 293. In expanded form the numbers are 400 + 70 + 1 and 200 + 90 + 3.
471 minus 293
400 + 70 + 1- 200 + 90 + 3-------------------hundreds 400 - 200 = 200tens 70 - 90 = -20ones 1 - 3 = -2------------------- 200 - 20 - 2 = 178
Two of the three partial differences are negative, and the answer is still exact: 471 - 293 = 178.
An example with no negative parts
When every top digit is larger than the digit below it, no partial difference is negative and the method looks very plain.
958 minus 342
hundreds 900 - 300 = 600tens 50 - 40 = 10ones 8 - 2 = 6------------------- 600 + 10 + 6 = 616
A four-digit example
6,204 minus 2,876
thousands 6000 - 2000 = 4000hundreds 200 - 800 = -600tens 0 - 70 = -70ones 4 - 6 = -2------------------- 4000 - 600 - 70 - 2 = 3328
Three negative parts in a row is normal and is not a sign that something went wrong. Zeros in the top number, which are the hardest case for borrowing, need no special handling here.
Why the negative parts are allowed
Subtraction distributes across the expanded form. Writing the numbers as sums of place values and subtracting term by term is the same calculation, only reordered:
471 - 293= (400 + 70 + 1) - (200 + 90 + 3)= (400 - 200) + (70 - 90) + (1 - 3)= 200 + (-20) + (-2)= 178
Nothing is lost when a middle term turns negative, because that term is still added back into the total with its sign. The borrowing algorithm avoids negatives by moving value between columns first. Partial differences accepts the negatives instead and settles up at the end. Both reach the same number, which is worth showing a class side by side once they know both.
How it compares with other subtraction methods
| Method | What the student does | Where it usually breaks down |
|---|---|---|
| Partial differences | Subtracts each place value, keeps negative parts, adds the parts | Dropping a minus sign when adding the parts at the end |
| Trade-first | Does all the regrouping first, then subtracts every column | Forgetting to reduce the column that was traded from |
| Counting up | Starts at 293 and counts up to 471 in friendly jumps | Losing one of the jumps before adding them together |
Which one should a student use? Whichever one they can explain. A child who can say why the tens column came out as negative twenty understands place value better than a child who borrows correctly but silently. Speed matters later than understanding does.
Common mistakes
Dropping the minus sign
By far the most common error. A student writes the tens partial difference as 20 instead of -20 and adds it, which overshoots the answer by exactly twice the missing amount. In the first example that produces 218 instead of 178.
Subtracting the digits in the wrong direction
In the tens column of 471 - 293, the calculation is 70 - 90, not 90 - 70. The top number always stays on top. Reversing it to avoid a negative answer is the single fastest way to get a wrong result.
Using bare digits instead of place values
Writing 7 - 9 = -2 for the tens column, rather than 70 - 90 = -20, gives a total that is eighteen too large, 196 instead of 178. Writing the zeros out is what keeps the method honest, at least until a student is fluent.
Check yourself
Work each one with partial differences before opening the answer.
624 - 358
600 - 300 = 300 20 - 50 = -20 4 - 8 = -4300 - 20 - 4 = 266
805 - 267
800 - 200 = 600 0 - 60 = -60 5 - 7 = -2600 - 60 - 2 = 538
7,431 - 2,659
7000 - 2000 = 5000 400 - 600 = -200 30 - 50 = -20 1 - 9 = -85000 - 200 - 20 - 8 = 4772
500 - 138
500 - 100 = 400 0 - 30 = -30 0 - 8 = -8400 - 30 - 8 = 362
Rows of zeros are where this method is at its most useful. Nothing has to be traded across three columns.
Why does 71 - 38 give a negative ones part, and what is the answer?
The ones column is 1 - 8, which is -7, because the digit on the bottom is larger. The tens column gives 70 - 30 = 40. Adding the parts, 40 - 7 = 33.
See also
Sources
- Partial-differences subtraction is one of the subtraction algorithms in the Everyday Mathematics curriculum (University of Chicago School Mathematics Project), where it is worked left to right by place value and negative partial differences are expected.
- This page was first published on this site in 2005 and rewritten in August 2026. The examples and diagrams are our own.