# 2.7: Adding and subtracting decimal numbers

Notes

##### Adding a positive decimal to a positive decimal

To add a positive decimal to a positive decimal, first line up the decimals one above the other, equalizing the lengths of the decimals with zeros if necessary. Then use the standard addition algorithm to add the digits in the smallest place value (regrouping or carrying if the sum is more than 9), then add the digits in the next place value to the left (regrouping if necessary), and so on. For example,

• 0.752 + 0.28.
• First, equalize the lengths by writing 0.752 and 0.280.
• Next, add 2 thousandths and 0 thousandths to get 2 thousandths.
• Next, add 5 hundredths and 8 hundredths to get 13 hundredths.
• Regroup the 13 hundredths into 1 tenth and 3 hundredths (carry the 1 tenth into the tenths column).
• Next, add 7 tenths and 2 tenths and the carried 1 tenth to get 10 tenths.
• Regroup the 10 tenths into 1 one and 0 tenths (carry the 1 one into the ones column).
• Next, add 0 ones and the carried 1 one to get 1 one.
• The answer is 1 one, 0 tenths, 3 hundredths, and 2 thousandths, i.e., 1.032.
```   0.7 5 2
+  0.2 8 0
1 1
1.0 3 2```
##### Subtracting a negative decimal from a positive decimal

Write A – (–B) = A + B and use the standard addition algorithm. For example, 0.752 – (–0.28) = 0.752 + 0.28 = 1.032.

##### Subtracting a positive decimal from a larger positive decimal

To subtract a positive decimal from a larger positive decimal, first line up the decimals one above the other, equalizing the lengths of the decimals with zeros if necessary. Then use the standard subtraction algorithm to subtract the digits in the smallest place value (regrouping or borrowing if the digit being subtracted is greater than the digit being subtracted from), then subtract the digits in the next place value to the left (regrouping if necessary), and so on. Here we’re assuming the number we’re subtracting is less than the number we’re subtracting from. For example,

• 0.752 – 0.28.
• First, equalize the lengths by writing 0.752 and 0.280.
• Next, subtract 0 thousandths from 2 thousandths to get 2 thousandths.
• Next, note that 8 hundredths is greater than 5 hundredths so we need to regroup the 7 tenths in 0.752 into 6 tenths and 10 hundredths (borrow the 10 hundredths to make 15 hundredths).
• Now we can subtract 8 hundredths from 15 hundredths to get 7 hundredths.
• Next, subtract 2 tenths from 6 tenths to get 4 tenths.
• The answer is 4 tenths, 7 hundredths, and 2 thousandths, i.e., 0.472.
```  6 15
0.7  5  2
- 0.2  8  0
0.4  7  2```
##### Subtracting a larger integer from a smaller integer

Write A – B = – (B – A) and use the standard subtraction algorithm. For example, 0.28 – 0.752 = –(0.752 – 0.28) = –0.472.

##### Adding a negative integer to a positive integer

Write A + (–B) = A – B and use the standard subtraction algorithm. For example, 0.752 + (–0.28) = 0.752 – 0.28 = 0.472. This assumes A is larger than B. If B is larger than A, then calculate A + (–B) = –(B – A). For example, 0.28 + (–0.752) = –(0.752 – 0.28) = –0.472.

##### Adding a positive integer to a negative integer

Write –A + B = B – A and use the standard subtraction algorithm. For example, –0.28 + 0.752 = 0.752 – 0.28 = 0.472. This assumes B is larger than A. If A is larger than B, then calculate –A + B = –(A – B). For example, –0.752 + 0.28 = –(0.752 – 0.28) = –0.472.

The video below works through some examples of adding and subtracting with decimal numbers.

Practice Exercises

Do the following exercises to practice adding and subtracting decimal numbers. 