Decimal
Operations
Master how to add, subtract, multiply, and divide decimals with confidence: align, count, and conquer.
Pre-Algebra
What Is a Decimal?
A decimal is a way of writing a number that is not whole. The digits to the right of the decimal point represent parts of one: tenths, hundredths, thousandths, and so on.
Why Decimals Matter
Decimals appear everywhere in daily life: money, measurements, sports statistics, and science. Mastering decimal operations lets you solve real-world problems with precision.
Money
\( \$4.99 + \$2.50 \): every receipt is a decimal addition problem.
Measurement
\( 2.54 \) cm per inch: unit conversion uses decimal multiplication.
Science
\( 9.81 \text{ m/s}^2 \): gravity itself is expressed as a decimal.
Key Vocabulary
1
Decimal Point: The dot that separates the whole number part from the fractional part.
2
Place Value: The value of a digit based on its position (tenths, hundredths, etc.).
3
Equivalent Decimals: Decimals that name the same value: \( 0.5 = 0.50 = 0.500 \).
4
Leading Zero: The zero before the decimal point in a number less than one, e.g. \( 0.4 \).
Rule 1: Adding & Subtracting
Rule 2: Multiplying Decimals
Rule 3: Dividing Decimals
Align vs Count: Know the Difference
| Add / Subtract | Multiply | Divide | |
|---|---|---|---|
| Decimal Rule | Align the points | Count total places | Shift to whole divisor |
| Pad Zeros? | Yes, trailing | No | Only if needed |
| When to Place Point | Before you compute | After you compute | After shifting, before dividing |
Step-by-Step Checklist
1
Identify the operation. Is this addition, subtraction, multiplication, or division?
2
Handle the decimal point. Align (add/sub), count places (multiply), or shift (divide).
3
Operate as with whole numbers. Carry, borrow, multiply, or long-divide normally.
4
Place or verify the decimal point in your answer using the rule from Step 2.
5
Estimate to check. Round the originals and see if your answer is reasonable.
Example 1: Addition
Compute \( 5.7 + 13.482 \).
Example 2: Subtraction
Compute \( 20.1 - 8.347 \).
Example 3: Multiplication
Compute \( 0.45 \times 1.2 \).
1
Count decimal places. \( 0.45 \) has 2 places; \( 1.2 \) has 1 place. Total: 3 places.
2
Multiply as whole numbers: \( 45 \times 12 = 540 \).
3
Place the decimal point 3 places from the right in \( 540 \): that gives us \( 0.540 = \) \( 0.54 \).
Example 4: Division
Compute \( 4.68 \div 0.12 \).
1
Make the divisor whole. \( 0.12 \) has 2 decimal places, so multiply both by \( 100 \).
2
Rewrite: \( 4.68 \times 100 = 468 \) and \( 0.12 \times 100 = 12 \). New problem: \( 468 \div 12 \).
3
Divide: \( 468 \div 12 = \) \( 39 \).
The Power of Estimation
After every decimal computation, round each number and perform a quick mental-math check. If your exact answer is far from the estimate, look for a decimal-placement error.
Problem
\( 3.87 \times 4.2 \)
Estimate
\( 4 \times 4 = 16 \)
Exact
\( 16.254 \) ✓ Close!
Common Mistakes
✕ Misaligned Decimals
Adding \( 3.5 + 0.42 \) by lining up the last digits instead of the decimal points gives a wrong answer.
✕ Forgetting to Count Places
In \( 0.3 \times 0.7 \), the answer is \( 0.21 \) (2 places), not \( 2.1 \) (1 place).
✕ Not Shifting Both Numbers
In division, if you move the decimal in the divisor but not the dividend, the quotient will be off by a power of 10.
Operations Summary
| Operation | Decimal Rule | Example |
|---|---|---|
| Add | Align points, pad zeros | \( 3.2 + 1.85 = 5.05 \) |
| Subtract | Align points, pad zeros | \( 7.4 - 2.16 = 5.24 \) |
| Multiply | Count total decimal places | \( 0.6 \times 0.4 = 0.24 \) |
| Divide | Shift decimal to make divisor whole | \( 3.6 \div 0.4 = 9 \) |
Question 1 of 5
What is \( 6.35 + 2.8 \)?
Align the decimal points and pad: \( 6.35 + 2.80 = 9.15 \). Choice (d), \( 8.15 \), results from forgetting to carry the 1 from the tenths column. Choice (c), \( 9.05 \), can result from an addition error like \( 3+8=10 \).
Question 2 of 5
What is \( 10.03 - 4.7 \)?
Pad to \( 10.03 - 4.70 \). Subtract column by column: \( 3 - 0 = 3 \) hundredths, \( 0 - 7 \) requires borrowing → \( 10 - 7 = 3 \) tenths (ones drops from \( 0 \) to \( 9 \), tens drops from \( 1 \) to \( 0 \)), then \( 9 - 4 = 5 \). Answer: \( 5.33 \).
Question 3 of 5
What is \( 0.6 \times 0.7 \)?
\( 6 \times 7 = 42 \). Total decimal places: \( 1 + 1 = 2 \). Place the decimal 2 places from the right: \( 0.42 \). Choice (b) only counted 1 decimal place, and (c) counted 3.
Question 4 of 5
What is \( 8.4 \div 0.06 \)?
Multiply both by \( 100 \): \( 840 \div 6 = 140 \). If you only multiplied by \( 10 \), you'd get \( 14 \) (choice b). Always match the shift to the divisor's decimal places.
Question 5 of 5
A shirt costs \( \$12.75 \). You pay with a \( \$20 \) bill. How much change do you receive?
\( 20.00 - 12.75 \): from the hundredths: \( 0 - 5 \) → borrow to get \( 10 - 5 = 5 \). Tenths: \( 9 - 7 = 2 \). Ones: \( 9 - 2 = 7 \). Tens: \( 1 - 1 = 0 \). Answer: \( \$7.25 \).