Addition and Subtraction

What do addition and subtraction tell us?

Addition finds a total by combining quantities. Subtraction finds an amount left, a difference between quantities, or a missing part when the total is known. Both operations rely on place value: combine or subtract ones with ones, tens with tens, and so on.

For example, if a shelf has 18 red folders and 7 blue folders, \(18+7=25\) gives the total number of folders. If 7 of those 25 folders are removed, \(25-7=18\) gives the number left. The same subtraction also answers how many more folders 25 is than 7.

This lesson focuses on whole numbers. For subtraction examples here, the starting amount is at least the amount subtracted, so the result is a whole number. Subtractions such as \(7-18\) have negative results and are studied in the integer lessons.

Name the parts of each operation

In \(18+7=25\), 18 and 7 are addends, and 25 is the sum. In \(25-7=18\), 25 is the minuend (starting amount), 7 is the subtrahend (amount subtracted), and 18 is the difference.

An equals sign states that the expressions on its two sides have the same value. It does not simply mean “write the next step.” For a two-step problem, write separate equations or a single expression whose value remains consistent.

Place-value alignment and regrouping

For a written calculation, line up the rightmost digits first. In 3,246 + 58, the 8 belongs under the 6 in the ones column, and the 5 belongs under the 4 in the tens column. Do not line up the numbers by their first digit.

Regrouping changes the grouping of a quantity without changing its total value:

• Ten ones have the same value as one ten.
• Ten tens have the same value as one hundred.
• Ten hundreds have the same value as one thousand.

In addition, regrouping is often called carrying. In subtraction, it is often called borrowing, but nothing needs to be returned: you are exchanging one larger place unit for ten smaller ones.

Worked example: unequal lengths and several addends

Problem: Add 3,246 + 58 + 709.

Align each number by place value. An empty leading position contributes zero.

1. Ones: \(6+8+9=23\). Record 3 ones and carry 2 tens.
2. Tens: \(4+5+0+2=11\). Record 1 ten and carry 1 hundred.
3. Hundreds: \(2+0+7+1=10\). Record 0 hundreds and carry 1 thousand.
4. Thousands: \(3+0+0+1=4\).

The sum is 4,013. A carry is not always 1: adding several digits can produce 20 or more units in a column. Check by grouping differently: \(58+709=767\), and \(3{,}246+767=4{,}013\).

Worked example: subtraction with regrouping

Problem: A library has 527 books and lends 189. How many remain?

The amount remaining is \(527-189\). Seven ones are insufficient to remove nine ones, so exchange one of the two tens for ten ones. The starting amount is now represented as 5 hundreds, 1 ten, and 17 ones.

After \(17-9=8\) ones, the tens column has only 1 ten, not the original 2. To remove 8 tens, exchange one hundred for ten tens. That leaves 4 hundreds and 11 tens.

• Ones: \(17-9=8\).
• Tens: \(11-8=3\).
• Hundreds: \(4-1=3\).

There are 338 books remaining. Check: \(338+189=527\). Regrouping has not changed the initial count; it has only made the subtraction possible column by column.

Worked example: subtract across zeros

Problem: Calculate 4,002 − 1,786.

Two ones are insufficient to subtract six, and there are no tens or hundreds available to exchange directly. Start at the thousands place and regroup through the empty positions.

Every row represents the same quantity, 4,002. The last row gives \(3{,}000+900+90+12=4{,}002\). Now subtract:

\[12-6=6,\quad9-8=1,\quad9-7=2,\quad3-1=2.\]

Reading the result by place gives 2,216. Check: \(2{,}216+1{,}786=4{,}002\). Intermediate zero columns become 9 after passing a unit to the next column; they do not all stay 10.

Addition properties and subtraction order

For whole numbers \(a\), \(b\), and \(c\):

• Commutative property of addition: \(a+b=b+a\). Changing addend order does not change the sum.
• Associative property of addition: \((a+b)+c=a+(b+c)\). Changing the grouping does not change the sum.
• Additive identity: \(a+0=a\).

Subtraction does not obey the first two properties. For example, \(12-5=7\), but \(5-12=-7\). Also, \((20-8)-3=9\), whereas \(20-(8-3)=15\). Keep subtraction order and parentheses intact.

Subtracting zero leaves a number unchanged, \(a-0=a\), and subtracting a number from itself gives zero, \(a-a=0\).

Worked example: mental calculation and missing parts

Problem A: Find 298 + 157 mentally.

Move 2 from the second addend to the first: \(298+157=300+155=455\). This is exact compensation, not an estimate, because one addend increased by exactly the amount the other decreased.

Problem B: Find 703 − 298 mentally.

Subtract 300 and return the extra 2 removed: \(703-298=703-300+2=405\). Check: \(405+298=703\).

Problem C: Solve \(268+\square=700\).

The missing addend is the total minus the known part: \(700-268=432\). Check by substitution: \(268+432=700\). This connection between addition and subtraction is useful when equations become more complicated.

Worked example: translate a two-step situation

Problem: A club begins with 840 tickets, receives 175 more, and then sells 628. How many unsold tickets remain?

First identify what each action does: receiving tickets increases the amount; selling tickets decreases it.

\[840+175=1{,}015\text{ tickets},\] \[1{,}015-628=387\text{ tickets}.\]

The answer is 387 unsold tickets. Check the complete story: \(387+628=1{,}015\), the total available before sales. As a rough check, \(840+180-630\approx390\), which is near 387.

Do not choose an operation from a single keyword. “Mina has 35 cards, which is 8 more than Leo has” asks for Leo's count; the equation is \(35-8=27\), even though the phrase contains “more.” Identify the known total or larger amount and the unknown quantity.

Check your understanding

Try these independently, then read the reasoning. These self-checks belong to this lesson; the separate Quizzes and Practice tabs are unchanged.

1. Calculate 386 + 457. Explain both carries.
2. Calculate 2,708 + 96 + 415, aligning each place.
3. Calculate 812 − 367 and check by addition.
4. Calculate 3,000 − 1,478. Describe the regrouped representation of 3,000.
5. Find the missing addend in \(\square+485=920\).
6. Use compensation to calculate 596 + 248 exactly.
7. A store has 1,250 notebooks, receives 375, and sells 486. How many remain?
8. Explain why “subtract the smaller digit from the larger digit in each column” is not a valid way to calculate 52 − 38.

Solutions and reasoning

1. 843. Ones give 13, so carry 1 ten; tens give \(8+5+1=14\), so carry 1 hundred; hundreds give \(3+4+1=8\).
2. 3,219. Ones give 19; tens give \(0+9+1+1=11\); hundreds give \(7+0+4+1=12\); thousands give \(2+1=3\).
3. 445. Regroup 812 as 7 hundreds, 10 tens, and 12 ones. Subtract 3 hundreds, 6 tens, and 7 ones. Check: \(445+367=812\).
4. 1,522. Regroup 3,000 as 2 thousands, 9 hundreds, 9 tens, and 10 ones. Subtract column by column. Check: \(1{,}522+1{,}478=3{,}000\).
5. 435. \(920-485=435\), and \(435+485=920\).
6. 844. \(596+248=600+244=844\). Moving 4 between addends preserves the total.
7. 1,139 notebooks. The total after delivery is \(1{,}250+375=1{,}625\); then \(1{,}625-486=1{,}139\). Check: \(1{,}139+486=1{,}625\).
8. The starting number must remain 52. Regroup it as 4 tens and 12 ones, giving \(12-8=4\) ones and \(4-3=1\) ten, or 14. Reversing the ones subtraction would incorrectly produce 26.

Common mistakes

• Misaligning place values: Align ones first, not leftmost digits.
• Forgetting a carried unit: Include it when adding the next column, and remember which place unit it represents.
• Using the original digit after regrouping: In 527 − 189, the tens digit becomes 1 before the next exchange.
• Turning every crossed zero into 10: A column that passes one unit to its right has 9 units left.
• Reversing subtraction within a column: Regroup the starting amount instead; do not swap the digits.
• Treating subtraction as commutative: Changing the order can change both the value and the sign.
• Dropping units or mixing quantities: Add or subtract quantities measured in compatible units, and label the final answer.
• Mistaking an estimate for proof: Use an inverse operation for an exact check.

Continue learning

• Place Value and Whole Numbers: understand the units exchanged during regrouping.
• Rounding and Estimating: check whether an exact answer is reasonable.
• Order of Operations: interpret calculations with several operations and parentheses.
• Integer Addition: extend addition to positive and negative numbers.
• Integer Subtraction: understand differences that can be negative.

Sources and further reading

Public textbook materials were consulted for topic coverage. This lesson's explanations, numerical examples, regrouping tables, and self-checks are independently written; no textbook prose, exercises, or artwork are reproduced or closely paraphrased.

• OpenStax Prealgebra 2e: Add Whole Numbers — addition notation and whole-number calculations.
• OpenStax Prealgebra 2e: Subtract Whole Numbers — subtraction notation, calculations, and applications.
• OpenStax Prealgebra 2e: preface and reuse conditions — consult current licensing restrictions before reproducing textbook material.
• Siyavula: Working with Whole Numbers — related publicly accessible topic outline.
• Siyavula: open-textbook licensing — distinguishes adaptable unbranded editions from branded editions with no-derivatives restrictions.