Adding and Subtracting Large Numbers

Line the places up first

$$\begin{array}{r} 346 \\ +\; 271 \\ \hline 617 \end{array}$$

Ones under ones, tens under tens, hundreds under hundreds. Only matching places can be added together, because three hundreds and three tens are not the same amount.

Work from the right, one column at a time.

Why carrying works

$$\begin{array}{r} _{1}\phantom{0}\phantom{0} \\ 168 \\ +\; 57 \\ \hline 225 \end{array}$$

The ones column gives $8 + 7 = 15$. That is more than one column can hold, since each column has room for a single digit.

So ten of the ones are traded for one ten. The $5$ stays and the $1$ moves into the tens column.

$$15 \text{ ones} = 1 \text{ ten} + 5 \text{ ones}$$

That is all carrying is — an exchange between neighboring places, made possible because each place is ten times the next.

What borrowing does

The same trade, run backwards.

$$\begin{array}{r} 52 \\ -\; 27 \\ \hline 25 \end{array}$$

The ones column asks for $2 - 7$, and there is not enough. So one ten is broken open:

$$5 \text{ tens} + 2 \text{ ones} \;=\; 4 \text{ tens} + 12 \text{ ones}$$

Now the ones column has $12 - 7 = 5$, and the tens column has $4 - 2 = 2$.

Nothing was added or lost. The number was rewritten, with one ten swapped for ten ones.

Borrowing across a zero

$$\begin{array}{r} 403 \\ -\; 158 \\ \hline 245 \end{array}$$

The ones need to borrow, but the tens column holds a zero — there is nothing there to break.

So go one further left. Borrow from the hundreds: one hundred becomes ten tens, which fills the empty tens column. Now the tens can give one to the ones.

$$403 = 3 \text{ hundreds} + 9 \text{ tens} + 13 \text{ ones}$$

The zero becomes a nine on the way through, because it received ten and gave away one.

Checking

Two checks, and both are quick.

Estimate. Round and add:

$$346 + 271 \approx 350 + 270 = 620$$

The answer $617$ is close, so it is plausible.

Work backwards. Subtraction is undone by addition:

$$52 - 27 = 25 \quad\Rightarrow\quad 25 + 27 = 52 \; ✓$$

The second check is the stronger one. It uses a different operation, so a mistake would have to happen twice in exactly matching ways to slip through.

Worked examples

Common mistakes

Practice problems