Lessons 35: Adding, Subtracting, Multiplying, and Dividing Decimal Numbers

New Concept

Performing arithmetic with decimals requires specific rules for placing the decimal point to maintain correct place value for each operation.

• Adding and Subtracting Decimal Numbers: We align the decimal points to ensure that we are adding or subtracting digits that have the same place value.
• Multiplying Decimal Numbers: When we multiply decimal numbers, the product has as many decimal places as there are in all the factors combined.
• Dividing Decimal Numbers: When we use long division, the decimal point in the quotient is lined up with the decimal point in the dividend.

What’s next

Next, we'll walk through worked examples for each operation, tackling problems involving perimeter, area, and unit conversions.

Property

We align the decimal points to ensure that we are adding or subtracting digits that have the same place value.

Examples

$4.2 + 0.42 + 42 \rightarrow \begin{array}{r} 4.2\phantom{0} \ 0.42 \ + 42.\phantom{00} \\ \hline 46.62 \end{array}$
$15.4 - 3.289 \rightarrow \begin{array}{r} 15.\stackrel{3}{\cancel{4}}\stackrel{9}{\cancel{0}}\stackrel{10}{\cancel{0}} \\ - \quad 3.289 \\ \hline 12.111 \end{array}$
$8 - 5.43 \rightarrow \begin{array}{r} \stackrel{7}{\cancel{8}}.\stackrel{9}{\cancel{0}}\stackrel{10}{\cancel{0}} \\ - \quad 5.43 \\ \hline 2.57 \end{array}$

Explanation

Think of it like stacking money. You must line up the decimal points to make sure you're adding dollars to dollars and cents to cents. This keeps all the place values in their correct columns so you don't accidentally mix them up. It's the secret to getting the right answer!

Property

When we multiply decimal numbers, the product has as many decimal places as there are in all the factors combined.

Examples

$0.35 \times 0.2 \rightarrow \begin{array}{rl} 0.35 & \text{(2 places)} \\ \times \quad 0.2 & \text{(1 place)} \\ \hline 0.070 & \text{(3 places) simplifies to 0.07} \end{array}$
$(0.04)^2 = 0.04 \times 0.04 = 0.0016$
$15 \times 2.54 \rightarrow \begin{array}{r} 2.54 \\ \times \quad 15 \\ \hline 1270 \\ 254\phantom{0} \\ \hline 38.10 \end{array}$

Explanation

Don't worry about lining up the decimals here. Just multiply the numbers as if they were whole numbers. Afterward, count the total decimal places in your original factors. Your final answer must have that same total number of decimal places. It's a simple counting trick!

Property

When we use long division, the decimal point in the quotient is lined up with the decimal point in the dividend.

Examples

$9.2 \div 4 \rightarrow \begin{array}{r} 2.3 \\ 4 \overline{) 9.2} \\ -8\phantom{.} \\ \hline 1\phantom{.}2 \\ -1\phantom{.}2 \\ \hline 0 \end{array}$
$1.4 \div 5 \rightarrow \begin{array}{r} 0.28 \\ 5 \overline{) 1.40} \\ -1\phantom{.}0\phantom{0} \\ \hline 40 \\ -40 \\ \hline 0 \end{array}$
$0.0245 \div 5 \rightarrow \begin{array}{r} 0.0049 \\ 5 \overline{) 0.0245} \\ -20\phantom{5} \\ \hline 45 \\ -45 \\ \hline 0 \end{array}$

Explanation

It’s the 'bring it straight up' rule! In long division, just move the decimal point from the number inside the box straight up into your answer line. Then, divide like normal. If you run out of digits, add a zero to the end and keep dividing until you are finished.