Adding and Subtracting Decimals

Line up the points

Everything about adding decimals is in one instruction: the decimal points go in a vertical line, and the answer’s point sits directly below them.

$$\begin{array}{r} 12.40 \\[-1pt] +\;\; 3.75 \\ \hline 16.15 \end{array}$$

Pad, add each column, bring the point down. No other rule is needed.

Why alignment is the whole method

Column arithmetic works because each column adds matching places: ones to ones, tenths to tenths, hundredths to hundredths.

Lining up the decimal points is what puts matching places in the same column. It is not a formatting preference — it is the thing that makes the columns mean anything.

Line up the right-hand ends instead and you get this:

$$\begin{array}{r} 12.4 \\[-1pt] +\;3.75 \end{array} \quad\text{wrongly stacked as}\quad \begin{array}{r} 12.4\phantom{0} \\[-1pt] +\;3.75 \end{array}$$

Now the $4$ tenths sits above the $7$ hundredths, and the sum adds four tenths to seven hundredths as though they were the same size. They are not — one is ten times the other.

Padding with zeros

When one number has fewer decimal places, add trailing zeros until they match. This changes nothing, because $3.5 = 3.50$.

Given	Padded
$12.4$ and $3.75$	$12.40$ and $3.75$
$7$ and $0.28$	$7.00$ and $0.28$
$9.6$ and $11.375$	$9.600$ and $11.375$

A whole number gets a point at its right-hand end: $7$ is $7.0$.

Padding is not required by the arithmetic — an empty column counts as zero anyway — but writing the zeros in makes it much harder to misalign, and that is the error worth designing out.

Subtracting works identically

Same alignment, same padding, and borrowing behaves exactly as it does with whole numbers.

$$\begin{array}{r} 8.30 \\[-1pt] -\;2.45 \\ \hline 5.85 \end{array}$$

Here the padding earns its keep: without the zero, there is nothing in the hundredths column to borrow from and the subtraction stalls.

Worked examples

Common mistakes

Practice problems