Decimal Operations

Master all four operations with decimals! In this topic, you'll learn to add, subtract, multiply, and divide decimal numbers with confidence and precision.

Review: Understanding Decimals

Decimals are another way to represent parts of a whole, just like fractions. The decimal point separates the whole number part from the fractional part.

Place Value Review:

• Ones . Tenths Hundredths Thousandths
• Example: 12.345
  • 1 is in the tens place (10)
  • 2 is in the ones place (2)
  • 3 is in the tenths place (3/10 or 0.3)
  • 4 is in the hundredths place (4/100 or 0.04)
  • 5 is in the thousandths place (5/1000 or 0.005)

Adding Decimals

When adding decimals, the key is to line up the decimal points.

Example 1: Basic Addition

Add: 12.5 + 3.84

Step 1: Line up the decimal points vertically

Step 2: Add zeros as placeholders if needed 12.50 (added a 0 in hundredths place)

• 3.84 = 16.34

Answer: 16.34

Example 2: Adding Multiple Decimals

Add: 5.2 + 13.456 + 0.89

Solution: Line up decimals vertically: 5.200 (added zeros) 13.456

• 0.890 (added a zero) = 19.546

Answer: 19.546

Key Rule: Line up decimal points vertically, add zeros as placeholders, then add normally.

Subtracting Decimals

Subtraction follows the same alignment rule as addition.

Example 1: Basic Subtraction

Subtract: 15.8 - 7.23

Step 1: Line up decimal points: 15.80 (add zero)

• 7.23 = 8.57

Answer: 8.57

Example 2: Subtracting with Regrouping

Subtract: 10 - 3.76

Solution: 10.00 (whole number = 10.00)

• 3.76 = 6.24

Regroup as needed: Borrow from the ones place to subtract in the hundredths and tenths.

Answer: 6.24

Multiplying Decimals

When multiplying decimals, ignore the decimal points at first, then count total decimal places.

Example 1: Decimal × Whole Number

Multiply: 3.4 × 5

Step 1: Multiply without the decimal 34 × 5 = 170

Step 2: Count decimal places in the problem 3.4 has 1 decimal place 5 has 0 decimal places Total: 1 decimal place

Step 3: Place the decimal 170 → 17.0 = 17

Answer: 17

Example 2: Decimal × Decimal

Multiply: 2.5 × 1.3

Step 1: Multiply without decimals 25 × 13 = 325

Step 2: Count decimal places 2.5 has 1 decimal place 1.3 has 1 decimal place Total: 1 + 1 = 2 decimal places

Step 3: Place the decimal 325 → 3.25 (2 places from right)

Answer: 3.25

Example 3: Multiplying Small Decimals

Multiply: 0.4 × 0.3

Step 1: Multiply 4 × 3 = 12

Step 2: Count decimal places 0.4 has 1 decimal place 0.3 has 1 decimal place Total: 2 decimal places

Step 3: Place decimal 12 → 0.12 (2 places from right)

Answer: 0.12

Important: Sometimes you need to add zeros in front!

Dividing Decimals

Division is the trickiest operation with decimals. The goal is to make the divisor (outside number) a whole number.

Example 1: Decimal ÷ Whole Number

Divide: 15.6 ÷ 4

Step 1: Set up the division and divide normally, keeping the decimal point aligned

15.6 ÷ 4 = 3.9

(4 goes into 15 three times with remainder 3, bring down the 6 to get 36, and 4 goes into 36 nine times)

Answer: 3.9

Example 2: Whole Number ÷ Decimal

Divide: 12 ÷ 0.4

Step 1: Move decimal in divisor to make it whole 0.4 → 4 (moved 1 place right)

Step 2: Move decimal in dividend the same amount 12 → 120 (moved 1 place right, added zero)

Step 3: Divide 120 ÷ 4 = 30

Answer: 30

Example 3: Decimal ÷ Decimal

Divide: 3.6 ÷ 0.12

Step 1: Move decimal in divisor (0.12) to make it whole 0.12 → 12 (moved 2 places right)

Step 2: Move decimal in dividend the same amount 3.6 → 360 (moved 2 places right, needed to add a zero)

Step 3: Divide 360 ÷ 12 = 30

Answer: 30

Estimating with Decimals

Always estimate to check if your answer is reasonable!

Rounding Strategy

Addition/Subtraction: Round to the nearest whole number

• 12.8 + 5.3 ≈ 13 + 5 = 18 (actual: 18.1)

Multiplication: Round to make mental math easy

• 4.8 × 3.1 ≈ 5 × 3 = 15 (actual: 14.88)

Division: Round both numbers

• 23.7 ÷ 4.9 ≈ 24 ÷ 5 ≈ 5 (actual: 4.8...)

Real-World Applications

Money Problems

Problem: You buy 3 notebooks at $2.45 each and 2 pens at$1.30 each. How much do you spend?

Solution: Notebooks: 3 × 2.45 = 7.35 Pens: 2 × 1.30 = 2.60 Total: 7.35 + 2.60 = 9.95

Answer: $9.95

Measurement Problems

Problem: A recipe calls for 2.5 cups of flour. You want to make 1/2 of the recipe. How much flour do you need?

Solution: 2.5 ÷ 2 = 1.25 cups

Answer: 1.25 cups (or 1 1/4 cups)

Rate Problems

Problem: Gas costs $3.89 per gallon. How much does 12.5 gallons cost?

Solution: 3.89 × 12.5 = 48.625 ≈ $48.63 (round to nearest cent)

Answer: $48.63

Converting Between Fractions and Decimals

Fraction to Decimal: Divide the numerator by denominator

• 3/4 = 3 ÷ 4 = 0.75
• 1/8 = 1 ÷ 8 = 0.125

Decimal to Fraction: Use place value

• 0.6 = 6/10 = 3/5
• 0.25 = 25/100 = 1/4
• 0.125 = 125/1000 = 1/8

Common Decimal Equivalents to Memorize

• 1/2 = 0.5
• 1/4 = 0.25
• 3/4 = 0.75
• 1/5 = 0.2
• 1/10 = 0.1
• 1/8 = 0.125
• 1/3 ≈ 0.333... (repeating)
• 2/3 ≈ 0.666... (repeating)

Common Mistakes to Avoid

Mistake 1: Not lining up decimal points in addition/subtraction Wrong: Not aligning decimals properly Right: Line up decimal points vertically (12.50 + 3.84)

Mistake 2: Counting decimal places incorrectly in multiplication 4.5 × 2.3 has 2 total decimal places (1 + 1), not 3!

Mistake 3: Moving decimals different amounts in division When dividing 3.6 ÷ 0.12, move BOTH decimals the same number of places.

Mistake 4: Forgetting zeros as placeholders 0.4 × 0.2 = 0.08, not 0.8 (need the zero placeholder)

Mistake 5: Not checking reasonableness If 5.2 × 3.1 gives you 161.2, something's wrong! (Should be about 15)

Mental Math Tricks

Multiplying by 10, 100, 1000: Move decimal point right

• 3.45 × 10 = 34.5
• 3.45 × 100 = 345

Dividing by 10, 100, 1000: Move decimal point left

• 67.8 ÷ 10 = 6.78
• 67.8 ÷ 100 = 0.678

Multiplying by 0.1, 0.01: Same as dividing by 10, 100

• 50 × 0.1 = 5
• 50 × 0.01 = 0.5

Practice Strategies

1. Master place value: Know what each digit represents
2. Always line up decimals: For addition and subtraction
3. Count carefully: Track decimal places in multiplication
4. Make divisor whole: In division, always move decimals in divisor first
5. Estimate first: Check if your answer makes sense
6. Use graph paper: Helps align decimal points correctly
7. Practice with money: Real-world context makes it meaningful

Decimal operations are essential for everyday life - shopping, cooking, measuring, and so much more. Master these skills and you'll use them forever!