Addition and subtraction are the two everyday actions behind almost all the arithmetic you'll ever do: putting amounts together, and taking them apart. Once you see how they lean on place value, even big numbers become a calm, step-by-step routine.
Two simple ideas
Addition means combining. If you have 3 apples and someone hands you 4 more, you put the groups together and count: \( 3 + 4 = 7 \). The two numbers you add are called addends, and the result is the sum.
Subtraction has two faces that turn out to be the same thing. One is taking away: you have 7 apples, you eat 2, and \( 7 - 2 = 5 \) remain. The other is finding the difference: how much taller is a 7-step ladder than a 2-step stool? The gap between them is also \( 5 \). Whether you remove or compare, the answer is the same.
In words Addition pushes two groups together into one bigger group. Subtraction either pulls a group apart or measures the distance between two amounts. They are the same number line, walked in opposite directions.
Line up by place value
In the place value lesson you saw that every digit has a home: ones, tens, hundreds, and so on. The whole trick of written addition and subtraction is this: only combine digits that live in the same place. So we stack the numbers with ones over ones, tens over tens, hundreds over hundreds, and work one column at a time, always starting from the right.
Most of the time a column behaves. But sometimes a column overflows or comes up short — and that's where regrouping steps in.
Carrying: when a column overflows
A single place can only hold the digits \( 0 \) through \( 9 \). If a column of ones adds up to \( 13 \), that's too big to fit — but \( 13 \) is just \( 1 \) ten and \( 3 \) ones. So we leave the \( 3 \) in the ones place and carry the \( 1 \) ten over to the tens column. That carried ten then joins the next addition. This is called carrying (or regrouping up).
- Ones column. \( 7 + 6 = 13 \). Write the \( 3 \), carry the \( 1 \) to the tens.
- Tens column. \( 4 + 8 = 12 \), plus the carried \( 1 \) makes \( 13 \). Write the \( 3 \), carry the \( 1 \) to the hundreds.
- Hundreds column. \( 3 + 5 = 8 \), plus the carried \( 1 \) makes \( 9 \). Write the \( 9 \).
- Read it off: \[ 347 + 586 = 933 \]
Notice how each overflow simply nudged one extra unit into the next column to the left — that's all carrying ever does.
Borrowing: when a column comes up short
Subtraction has the mirror-image situation. If the top digit in a column is smaller than the bottom one, you can't take away enough — so you borrow from the column to the left. One unit borrowed from the tens is worth \( 10 \) ones; one borrowed from the hundreds is worth \( 10 \) tens. You shrink the neighbour by \( 1 \) and grow the current column by \( 10 \). This is borrowing (or regrouping down).
- Ones column. We need \( 3 - 8 \), but \( 3 \) is too small. We want to borrow from the tens — but the tens digit is \( 0 \), so first borrow from the hundreds.
- Set up the borrow. Take \( 1 \) hundred from the \( 5 \), leaving \( 4 \) hundreds. That hundred becomes \( 10 \) tens, so the tens column (was \( 0 \)) becomes \( 10 \).
- Now borrow \( 1 \) ten from those \( 10 \) tens, leaving \( 9 \) tens. That ten becomes \( 10 \) ones, so the ones column (was \( 3 \)) becomes \( 13 \).
- Ones. \( 13 - 8 = 5 \). Write the \( 5 \).
- Tens. \( 9 - 6 = 3 \). Write the \( 3 \).
- Hundreds. \( 4 - 2 = 2 \). Write the \( 2 \). The result: \[ 503 - 268 = 235 \]
Borrowing across a zero feels fiddly the first time, but it's just two borrows in a row — hundreds to tens, then tens to ones.
They undo each other
Here is the idea that ties the whole lesson together: addition and subtraction are inverse operations. One undoes the other. If you start with a number, add something, then subtract that same something, you land right back where you began:
\[ 235 + 268 = 503 \qquad \text{and} \qquad 503 - 268 = 235 \]
Because of this, every subtraction has a matching addition fact hiding inside it. That gives you a free, foolproof way to check your work.
Tip To check any subtraction, add your answer back to the number you took away — you should recover the number you started with. Since \( 235 + 268 = 503 \), our subtraction above is confirmed correct. If it doesn't match, you know to look again.
Two mental-math shortcuts
You won't always reach for pencil and paper. Two friendly strategies handle a lot of everyday sums in your head.
- Make ten. Tens are easy to add, so reshape a number to reach the nearest ten first. For \( 8 + 5 \), borrow \( 2 \) from the \( 5 \) to turn the \( 8 \) into \( 10 \): \( 8 + 2 = 10 \), and the \( 3 \) that's left gives \( 10 + 3 = 13 \).
- Count up for subtraction. Instead of taking away, walk forward from the smaller number to the bigger one. For \( 53 - 48 \), step from \( 48 \) up to \( 50 \) (that's \( 2 \)), then \( 50 \) up to \( 53 \) (that's \( 3 \)). Total distance: \( 2 + 3 = 5 \). This is the "difference" view of subtraction in action.
Practice
Try each one yourself, then reveal the full solution.
1. Add, carrying where needed: \( 478 + 365 \).
- Ones: \( 8 + 5 = 13 \). Write \( 3 \), carry \( 1 \).
- Tens: \( 7 + 6 = 13 \), plus the carried \( 1 \) is \( 14 \). Write \( 4 \), carry \( 1 \).
- Hundreds: \( 4 + 3 = 7 \), plus the carried \( 1 \) is \( 8 \). Write \( 8 \).
2. Subtract, borrowing where needed: \( 700 - 426 \).
- Borrow \( 1 \) hundred from \( 7 \), leaving \( 6 \) hundreds; the tens become \( 10 \).
- Borrow \( 1 \) ten from those \( 10 \) tens, leaving \( 9 \) tens; the ones become \( 10 \).
- Ones: \( 10 - 6 = 4 \). Tens: \( 9 - 2 = 7 \). Hundreds: \( 6 - 4 = 2 \).
- Check by adding back: \( 274 + 426 = 700 \). ✓
3. A bakery had \( 156 \) muffins in the morning and baked \( 89 \) more at noon. By closing time they had sold \( 178 \). How many muffins were left?
- First combine the muffins made: \( 156 + 89 \). Ones: \( 6 + 9 = 15 \), write \( 5 \), carry \( 1 \). Tens: \( 5 + 8 = 13 \), plus \( 1 \) is \( 14 \), write \( 4 \), carry \( 1 \). Hundreds: \( 1 + 1 = 2 \). So they had \( 156 + 89 = 245 \).
- Now take away what was sold: \( 245 - 178 \). Borrowing as needed gives \( 245 - 178 = 67 \).
- Check by adding back: \( 67 + 178 = 245 \). ✓