All Operations with Decimals

Working with decimals is just like working with whole numbers, but you need to pay special attention to the decimal point. To solve multi-step problems, you need to know how to add, subtract, multiply, and divide decimals, as well as how to follow the standard order of operations (PEMDAS).

Adding and Subtracting Decimals

The golden rule for adding and subtracting decimals is to line up the decimal points. This ensures you are adding tenths to tenths, hundredths to hundredths, and so on.

1. Line up the numbers vertically by their decimal points.
2. Add placeholder zeros at the end of the numbers if necessary so they have the same number of decimal places.
3. Add or subtract as you would with whole numbers, and bring the decimal point straight down.

Example: $3.75 + 2.8$

\begin{array}{r@{\quad}l} 3.75 \\ + 2.80 & \text{(added a zero)} \\ \hline 6.55 \end{array}

Multiplying Decimals

When multiplying, you do not need to line up the decimal points.

1. Multiply the numbers just like they are whole numbers, ignoring the decimal points for a moment.
2. Count the total number of decimal places in both of your original factors.
3. Place the decimal point in your answer so it has that same total number of decimal places.

Example: $4.56 \times 2.3$

• $4.56$ has $2$ decimal places.
• $2.3$ has $1$ decimal place.
• Total decimal places needed in the answer: $2 + 1 = 3$.
• Multiply $456 \times 23 = 10488$.
• Put the decimal point $3$ places from the right: $10.488$.

Dividing Decimals

When dividing by a decimal, the goal is to change the divisor (the number you are dividing by) into a whole number.

1. Move the decimal point in the divisor to the right until it is a whole number.
2. Move the decimal point in the dividend (the number inside the division bracket) the exact same number of places to the right. Add zeros if needed.
3. Bring the decimal point straight up into your quotient (answer) and divide normally.

Example: $12.6 \div 0.04$

• Move the decimal in $0.04$ two places to the right to make it $4$.
• Move the decimal in $12.6$ two places to the right to make it $1260$.
• Now divide: $1260 \div 4 = 315$.

Order of Operations with Decimals

When a problem has multiple operations, you must follow the order of operations, often remembered by the acronym PEMDAS:

1. Parentheses
2. Exponents
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

Example: Evaluate $3.75 + 2.8 \times 1.5$

• Step 1: Multiply first. $2.8 \times 1.5 = 4.20$ (or just $4.2$)
• Step 2: Add. $3.75 + 4.2 = 7.95$

Example: Evaluate $4.56 \times 2.3 - 1.88$

• Step 1: Multiply. $4.56 \times 2.3 = 10.488$
• Step 2: Subtract. $10.488 - 1.880 = 8.608$

Estimating to Check Your Answer

A great way to check if your decimal calculation is correct is to estimate the answer using whole numbers.

Let's look at the previous multiplication: $4.56 \times 2.3$.

• Round $4.56$ up to $5$.
• Round $2.3$ down to $2$.
• $5 \times 2 = 10$.

Our exact answer was $10.488$, which is very close to our estimate of $10$. This tells us our decimal placement is highly likely to be correct!