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Addition and Subtraction with Decimal Numbers
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Introduction

In this lesson, you will learn how to add and subtract with decimal numbers. Addition and subtraction with decimals are the same as addition and subtraction with whole numbers. You’ll stack the numbers and line them up according to place values; when dealing with decimals, this means you line up the decimals.

There may be instances when there aren’t the same number of digits after the decimal place in both numbers. You can fill in any places to the right of the decimal with a zero. In numbers, this could look like the following example:

\begin{align*}
3.15 = 3.150 = 3.150000000 \\
\end{align*}


These videos illustrate the lesson material below. Watching the videos is optional.


Steps for Addition: Combining Two Positive or Two Negative Numbers


  1. Stack the numbers according to place value. This will line up the decimals.
  2. Add the numbers in the columns starting with the column on the right.
  3. Carry numbers into the next place value to the left as needed.
  4. Repeat this pattern for each remaining column.
  5. Bring the decimal down.
  6. Check the sign. If both numbers were positive, the answer is positive. If both numbers were negative, the answer is negative.

Addition with Decimals

The steps for addition with decimals are exactly the same as the steps for addition with whole numbers, except that you have to keep track of the decimal point.

Example 1
\(1.3+2.8\)

Start with 1.3, which is shown on the number line below, and add 2.8.

A number line that shows 2.8 being added to 1.3 which ends at 4.1. 

Figure 1

The only difference between adding whole numbers and adding decimal numbers is bringing the decimal down. When you put the numbers into their columns according to place value, the decimals automatically line up. In this example, the answer is 4.1.

Bringing the decimal down just means that you put the decimal in the answer in the same column as the other decimals. In the figure below, the other decimals go between the ones place and the tenths place, so that is where the decimal in the answer will go.

1.3 plus 2.8 equals 4.1. This image shows that the decimal goes in the same place in the answer, which is between the ones place and the tenths place. 

Figure 2

Example 2
\(-12.91 + (-3.82)\)

The figure below shows an addition problem where both numbers are negative. When you add them together, you will put negative signs in front of both of them to remind you that the answer will be negative. Follow the steps above and the answer is -16.73.

This figure shows how to add two negative numbers. Negative 12.91 plus negative 3.82 equals negative 16.73. 

Figure 3

Steps for Subtraction: Combining Both a Positive and a Negative Number

These are the steps for subtraction with both a positive and a negative number:

  1. Place the biggest number on top.
  2. Stack in columns according to place value. This will line up the decimals.
  3. Regroup as needed.
  4. Subtract in columns by place value starting on the right and going left.
  5. Bring the decimal down.
  6. The strongest number wins, meaning if the bigger number was negative then the answer is negative. If the bigger number was positive, then the answer is positive.

Subtraction with Decimal Numbers

Example 3
\(3.4-7.8\)

Imagine that these numbers don’t have a decimal in them. If there were no decimal, you would simply subtract the smaller number from the larger number. You would figure out if the remaining number was negative or positive, and that would be the answer. Thankfully, the steps for subtraction with decimal numbers are exactly the same as the steps for subtraction with whole numbers. The only difference is that you have to keep track of the decimal point.

A number line showing how to subtract a negative number from a positive number and showing that the rules for subtraction are the same for decimal numbers and whole numbers. Starting at 3.4, subtract 7.8 which ends at negative 4.4.

Figure 4

There are two ways to go about solving this problem. The first is converting it into an addition problem, because subtraction is the same as adding a negative number. This method is shown in the figure above, and the answer is -4.4.

\begin{align*}3.4-7.8 &=-4.4\\\\\text{OR}\\\\3.4+(-7.8) &=-4.4\end{align*}

The figure below shows the subtraction algorithm. Put the biggest number on top and the smaller number on the bottom and line up the place values, then subtract. Since the bigger number had a negative sign in front of it, the answer is negative. This method gives the same answer: -4.4.

7.8 minus 3.4 with negative 4.4 below the equation line.

Figure 5

Steps for Subtraction

Example 4
\(3.4 + (-2.1)\) or \(3.4 - 2.1\)

By following the steps, you will determine the answer is 1.3.

\begin{align*} &3.4\\ - &2.1\\ \hline &1.3 \end{align*}


Things to Remember


  • In addition, if both numbers are positive, the answer is positive.
  • In addition, if both numbers are negative, the answer is negative.
  • In subtraction, if the bigger number is positive, the answer is positive.
  • In subtraction, if the bigger number is negative, the answer is negative.

Practice Problems

Evaluate the following expressions:
  1. \(2.4 + 8.8 = ?\) (
    Solution
    x
    Solution: 11.2
    Details:
    Using place values, add 2.4 and 8.8.
    Line up the numbers in their corresponding places.
    This is a box with two columns. The ones column and the left and the tenths column on the right. Number 2 is placed in the ones column and point 4 is placed in the tenths column. Below these two numbers there is an 8 in the ones column and a point 8 in the tenths column.


    Next, add the numbers in the tenth column. Since \(4 + 8 = 12\), place the 2 below the tenths place and the 1 in the ones place.
    This is the same as the previous image except there is now an addition symbol to the left of all the numbers and a blue solution box at the very bottom. In the tenths column of the solution box is the number point 2. The number 1 is also written at the very top of the numbers in the ones column.

    Finally, add the numbers in the ones place. Include the 1 carried over from the tenths place.
    \(1 + 2 + 8 = 11\)
    This is the same as the previous image except there is now the number 11 written below the numbers in the ones column in the solution box. The solution box now contains 11.2

    The final answer: 11.2
    Be sure to include the decimal point between the tenths place and the ones place.
    )
  2. \(8.12 + 9.81 = ?\) (
    Solution
    x
    Solution: 17.93
    Details:
    Using place values, add 8.12 and 9.81.
    Line up the numbers in their corresponding place values.
    This is the same as the previous image except there is now the number 11 written below the numbers in the ones column in the solution box. The solution box now contains 11.2

    Next, add the numbers in the hundredths column. Since \(2 + 1 = 3\), place the 3 below the hundredths place.
    This is a box that has three column: Ones, Tenth, and Hundredths. Number 8 is placed in Ones Column, Number point 1 is placed in Tenth column, and Number 2 is placed in Hundredths column. Below Number 8, Number 9 is placed. Number point 8 is placed right below Number point 1. Number 1 is placed below Number 2. Add the numbers in the Hundredths column, then place 3 below the hundredths place.

    Next, add the numbers in the tenths column. Since \(1 + 8 = 9\), place the 9 below the tenths place.
    This image is the same as the previous image except there is now a point 9 in the tenths column.

    Finally, add the numbers in the ones column. Since \(8 + 9 = 17\), put a 7 in the ones place and the 1 in the tens place.
    This is the same as the previous image except there is now a 7 in the ones column and a new column to the left of the ones labeled the tens column. In it there is a number 1. The solution under the horizontal line now reads 17.93

    The final answer: 17.93
    Be sure to include the decimal point between the tenths place and the ones place.
    )
  3. \(5.92 + 3.22 = ?\) ( Transcript )
  4. \(7.2 - 1.5 = ?\) (
    Solution
    x
    Solution:
    5.7
    )
  5. \(5.2 - 5.4 = ?\) (
    Solution
    x
    Solution:
    -0.2
    )
  6. \(39.6 - 88.88 = ?\) (
    Video Solution
    | Transcript)

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