Subtraction: Methods and Written Subtraction

Subtraction finds what remains after a quantity is removed, or the difference between two quantities. In brief: 63 − 28 = 35 because removing 20 leaves 43 and removing another 8 leaves 35. You can also count from 28 up to 63 to find the same difference.

Written subtraction is useful for large numbers, while mental methods can make nearby or rounded numbers easier to handle. This guide explains both approaches using whole-number examples. You will also learn what regrouping really means, how to subtract across zeros and how addition provides a reliable check.

What does subtraction mean and what is a difference?

In a − b, a is the starting quantity, traditionally called the minuend, and b is the quantity subtracted, called the subtrahend. The result is the difference. These names are useful for identifying the roles, but the key point is that order matters: the first number is the value from which you subtract.

A take-away problem asks what remains. If 53 cards are available and 18 are handed out, 53 − 18 gives the remaining count. A comparison problem asks how much larger one quantity is than another. If two collections contain 53 and 18 cards, their difference is also 35.

Subtraction can describe a missing part as well. In 18 + ? = 53, the missing addend equals 53 − 18. This connection to addition explains why counting upward is a legitimate subtraction strategy, even though it looks different from taking away. Choose the interpretation that makes the quantities in the problem easiest to understand.

Which mental subtraction methods are useful?

Partition the number being subtracted into convenient parts. For 84 − 37, subtract 30 to get 54, then subtract 7 to get 47. You can split that final 7 into 4 and 3 if reaching 50 first is easier. The parts must total the original amount being removed.

Count up when the two values are close or cross friendly landmarks. To find 102 − 87, count 3 from 87 to 90, 10 to 100 and 2 to 102. The combined jumps are 3 + 10 + 2 = 15. Record the jumps so that none are lost or counted twice.

Compensation uses a nearby convenient number. For 352 − 198, subtract 200 to obtain 152, then add back 2 because you removed two too many. The result is 154. Unlike addition compensation in a similar-looking problem, the correction here increases the provisional answer because the adjusted subtrahend was larger.

Why do you regroup in written subtraction?

Align the numbers by place value, with ones under ones and tens under tens. Work from right to left. If a top digit contains too few units to subtract the bottom digit, exchange one unit from the next available larger place for ten units in the current place.

For example, 52 can be represented as 5 tens and 2 ones or as 4 tens and 12 ones. Both forms equal 52. To subtract 8 ones, the second representation is more convenient: 12 − 8 = 4. The exchange must reduce the tens count, otherwise you accidentally increase the starting number.

The word borrowing is common, but nothing is returned later. Regrouping better describes the permanent exchange in the written representation. You are not changing the value of the minuend; you are expressing it in units suitable for column subtraction. This distinction makes it easier to understand why every changed digit must be recorded carefully.

How do you subtract using the column method?

Calculate 734 − 286. An estimate of 700 − 300 = 400 suggests a difference of a few hundred, while the exact column method will determine the units. Set out the digits by place value and begin with the ones.

The 4 ones cannot supply 6 ones, so exchange one of the 3 tens. You now have 2 tens and 14 ones; subtract 6 to leave 8 ones. The remaining 2 tens cannot supply 8 tens, so exchange one of the 7 hundreds, giving 6 hundreds and 12 tens.

Subtract 8 tens from 12 tens to leave 4 tens, then subtract 2 hundreds from 6 hundreds to leave 4 hundreds. The result is 448. Check with 448 + 286 = 734. This inverse calculation confirms that the difference and the removed amount reconstruct the original quantity.

1. Align 734 and 286 by hundreds, tens and ones.
2. Regroup to obtain 14 ones and subtract 6, leaving 8.
3. Regroup to obtain 12 tens and subtract 8, leaving 4 tens.
4. Subtract hundreds: 6 − 2 = 4; combine the digits to get 448.

How do you subtract across zeros?

Zeros require a chain of exchanges when no unit is available in the immediately adjacent place. For 1002 − 578, the ones need regrouping, but both the tens and hundreds digits are zero. Begin with the thousands place, where one unit is available.

Exchange 1 thousand for 10 hundreds. Exchange one of those hundreds for 10 tens, leaving 9 hundreds. Exchange one of those tens for 10 ones, leaving 9 tens and increasing the original 2 ones to 12. The representation is now 0 thousands, 9 hundreds, 9 tens and 12 ones.

Subtract by columns: 12 − 8 = 4 ones, 9 − 7 = 2 tens and 9 − 5 = 4 hundreds. The difference is 424. Check that 424 + 578 = 1002. The intermediate 9s are not arbitrary marks; each records what remains after passing one exchanged unit toward the right.

What subtraction rules differ from addition?

Subtraction is not commutative: 12 − 5 = 7, but 5 − 12 = −7. Swapping the numbers changes the result. If a problem requires a nonnegative difference between two counts, subtract the smaller count from the larger one; if it specifies a signed change, preserve the given order.

Subtraction is not associative either. The expression (20 − 6) − 3 equals 11, while 20 − (6 − 3) equals 17. In a string such as 20 − 6 − 3 with no parentheses, perform operations from left to right. Do not regroup the subtractions as though they were additions.

Subtracting zero leaves a number unchanged, and subtracting a number from itself gives zero. A further useful rule is equal adjustment: adding the same amount to both numbers preserves their difference. Thus 503 − 298 equals 505 − 300 = 205. The distance between the adjusted values stays the same.

Where is subtraction used and which mistakes are common?

Subtraction supports inventory counts, comparisons of scores, elapsed quantities and remaining distances. Place-value notation makes the written algorithm practical by keeping the same-sized units in columns. Alternative classroom layouts may track exchanges differently, but each valid method must preserve the value of the starting quantity.

A common error is subtracting the smaller digit from the larger digit in every column regardless of which is on top. For 52 − 38, that approach would incorrectly give 26. Regrouping instead gives 4 tens and 12 ones, producing 14. Column position represents the full subtraction, not separate unordered digit differences.

Other mistakes include forgetting to reduce the place that supplied an exchanged unit, crossing zeros without recording intermediate changes and reversing a compensation correction. Always check the answer by adding it to the subtrahend. In a whole-number take-away context, also ask whether the remaining quantity should lie between zero and the original amount.

Frequently asked questions

What is the result of subtraction called?

The result is called the difference. In 32 − 9 = 23, the difference is 23.

Can I subtract by counting up?

Counting up gives the difference between the two values. Add the jumps from the smaller value to the larger one to obtain a nonnegative difference.

Why does 1000 need several regrouping steps?

Several steps are needed because the intermediate hundreds and tens positions contain zero. One thousand must be exchanged through those positions before ones become available.

Can subtraction give a negative answer?

Subtraction gives a negative answer when the second number exceeds the first. For example, 4 − 9 = −5, which is an integer outside the whole numbers.

How do I check a subtraction?

Add the difference to the number subtracted. If the subtraction is correct, that sum equals the original starting number.

Practice exercises

1. Calculate 72 − 0.
2. Complete 28 + ___ = 61.
3. Correct the answer in ’52 − 38 = 26′.
4. Calculate 903 − 457.
5. A store begins with 1200 notebooks and sells 685. How many remain?

Answers to the exercises

1. 72. Subtracting zero leaves the starting value unchanged.
2. 33. The missing part is 61 − 28; count 2 to 30, 30 to 60 and 1 to 61.
3. 14. Regroup 52 as 4 tens and 12 ones, then subtract 3 tens and 8 ones.
4. 446. Regroup through the zero tens to get 8 hundreds, 9 tens and 13 ones, then subtract.
5. 515 notebooks. Calculate 1200 − 685 = 515 and check that 515 + 685 = 1200.