Operations with Decimals
Operations with decimals work like whole-number math, with one extra step: placing the decimal point. To add or subtract, line up the decimal points. To multiply, multiply as usual, then count the decimal places in both numbers. To divide by a decimal, move both decimal points to the right until the divisor is a whole number.
Before this lesson: You should know how to add, subtract, multiply and divide whole numbers, and what tenths and hundredths are from fractions.
Decimal Place Value
Each place to the right of the decimal point is worth one tenth of the place before it. In 12.345, the 3 is in the tenths place, the 4 is in the hundredths place and the 5 is in the thousandths place.
| Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|
| 1 | 2 | . | 3 | 4 | 5 |
| 10 | 1 | \frac{1}{10} | \frac{1}{100} | \frac{1}{1000} |
Adding zeros at the end of a decimal doesn’t change its value. So 0.6, 0.60 and 0.600 are all the same number.
Rules for Operations with Decimals
| Operation | What to do | Example |
|---|---|---|
| Add or subtract | Line up the decimal points. Fill empty places with zeros. Then add or subtract like whole numbers. | 0.75 + 0.60 = 1.35 |
| Multiply | Multiply like whole numbers. Count the decimal places in both numbers. Put that many decimal places in the answer. | 0.4 \times 0.6 = 0.24 |
| Divide by a whole number | Divide as usual. Put the decimal point in the answer right above the one in the number you are dividing. | 9.6 \div 3 = 3.2 |
| Divide by a decimal | Move the decimal point in both numbers the same number of places to the right, until the divisor is a whole number. Then divide. | 4.2 \div 0.06 = 420 \div 6 = 70 |
The grid below shows why the multiplying rule works. Four tenths of six tenths is 24 hundredths.
Worked Examples
Example 1: Add decimals
Find 3.75 + 12.6.
Step 1. Line up the decimal points. Write 12.6 as 12.60 so both numbers have two decimal places.
\begin{array}{r} 3.75 \\ +\ 12.60 \\ \hline 16.35 \end{array}Step 2. Add like whole numbers and bring the decimal point straight down.
Answer: 16.35.
Example 2: Subtract from a whole number
Find 8 − 2.47.
Step 1. Write 8 as 8.00, so it has two decimal places like 2.47.
\begin{array}{r} 8.00 \\ -\ 2.47 \\ \hline 5.53 \end{array}Step 2. Subtract, borrowing as you would with whole numbers.
Answer: 5.53. Check: 5.53 + 2.47 = 8.
Example 3: Multiply decimals
Find 2.4 × 0.35.
Step 1. Ignore the decimal points and multiply: 24 × 35 = 840.
Step 2. Count decimal places. 2.4 has one and 0.35 has two, so the answer needs 3.
Step 3. Start at the right end of 840 and move the decimal point 3 places left: 0.840.
Answer: 0.84. The zero at the end can be dropped.
Example 4: Divide by a decimal
Find 7.56 ÷ 0.4.
Step 1. The divisor 0.4 has one decimal place. Move both decimal points one place right.
7.56 \div 0.4 = 75.6 \div 4
Step 2. Divide by the whole number 4: 75.6 ÷ 4 = 18.9.
Answer: 18.9. Check: 18.9 × 0.4 = 7.56.
Example 5: Word problem
You buy 3 notebooks that cost $2.85 each. You pay with a $10 bill. How much change do you get?
Step 1. Find the total cost: 3 × 2.85 = 8.55. The notebooks cost $8.55.
Step 2. Subtract from $10.00: 10.00 − 8.55 = 1.45.
Answer: You get $1.45 in change.
Common Mistakes
| Mistake | What it looks like | How to fix it |
|---|---|---|
| Lining up the last digits instead of the decimal points | 3.75 + 12.6 = 4.01 | Line up the decimal points: 3.75 + 12.60 = 16.35. |
| Lining up decimal points when multiplying | Placing the point in the answer under the others | Count the total decimal places instead: 2.4 \times 0.35 = 0.84. |
| Dropping a needed zero | 0.4 \times 0.2 = 0.8 | 4 × 2 = 8 needs two decimal places, so write 0.08. |
| Moving only one decimal point when dividing | 7.56 \div 0.4 = 7.56 \div 4 | Move the decimal point in both numbers by the same number of places. |
| Forgetting zeros when subtracting | 8 − 2.47 = 6.47 | Write 8 as 8.00 first, then subtract: 5.53. |
Multiplying and Dividing by 10, 100 and 1,000
When you multiply by 10, 100 or 1,000, move the decimal point right one place for each zero. When you divide, move it left.
- 4.6 × 10 = 46 and 4.6 × 100 = 460
- 4.6 ÷ 10 = 0.46 and 4.6 ÷ 100 = 0.046
This is why powers of 10 are so useful. If an answer doesn’t come out even, round it to the nearest hundredth unless the problem says otherwise.
Practice Problems
- 4.5 + 2.38
- 10.2 − 3.75
- 6 − 1.08
- 0.3 × 0.7
- 1.25 × 4
- 3.6 × 0.05
- 9.6 ÷ 3
- 4.2 ÷ 0.06
- Find 2.5 × 100 and 2.5 ÷ 100.
- Gas costs $3.40 per gallon. How much do 12.5 gallons cost?
Show answers
- 4.50 + 2.38 = 6.88
- 10.20 − 3.75 = 6.45
- 6.00 − 1.08 = 4.92
- 3 × 7 = 21 with two decimal places: 0.21
- 125 × 4 = 500 with two decimal places: 5.00, or 5
- 36 × 5 = 180 with three decimal places: 0.180, or 0.18
- 9.6 ÷ 3 = 3.2
- Move both decimal points two places: 420 ÷ 6 = 70
- 2.5 × 100 = 250 and 2.5 ÷ 100 = 0.025
- 12.5 × 3.40 = 42.5, so the gas costs $42.50.
Operations with Decimals FAQ
Do you line up the decimal points when you multiply?
No. Line up decimal points only to add or subtract. To multiply, ignore the points, multiply, and then count decimal places.
How do you divide by a decimal?
Move the decimal point in the divisor to the right until it is a whole number. Move the point in the other number the same number of places, then divide.
What is 0.1 times 0.1?
0.1 \times 0.1 = 0.01. One times one is 1, and the answer needs two decimal places.
Can there be decimals in order of operations?
Yes. Decimals follow the same order of operations as whole numbers. For example, 2 \times 1.5 + 0.5 = 3 + 0.5 = 3.5.
Summary
- To add or subtract decimals, line up the decimal points and fill in zeros.
- To multiply, multiply like whole numbers, then count the total decimal places.
- To divide by a decimal, move both decimal points right until the divisor is a whole number.
- Multiplying by 10, 100 or 1,000 moves the decimal point right. Dividing moves it left.
- Check answers with the opposite operation.
Previous lesson: Mixed Numbers and Improper Fractions
Next lesson: Converting Fractions and Decimals
Back to the Pre-Algebra Index