Working with Decimals
Decimals are how we handle money, measurements, and precision. Master them and everyday math becomes effortless.
What Are Decimals?
A decimal is another way to write a fraction with a denominator of 10, 100, 1000, etc. The decimal point separates whole numbers from fractional parts.
In the number 45.728:
- 4 = tens (40)
- 5 = ones (5)
- . = decimal point
- 7 = tenths (7/10)
- 2 = hundredths (2/100)
- 8 = thousandths (8/1000)
Adding and Subtracting Decimals
Rule: Line up the decimal points, then add/subtract as with whole numbers.
Example 1: 14.6 + 3.85
Line up: 14.60 + 3.85 (add a trailing zero for clarity)
Add: 60 + 85 = 145 in hundredths. 14 + 3 + 1 (carry) = 18. Answer: 18.45
Example 2: 20.00 - 7.43
Borrow as needed: 20.00 - 7.43 = 12.57
Multiplying Decimals
Step 1: Ignore the decimals and multiply as whole numbers.
Step 2: Count total decimal places in both factors. Place that many in the answer.
Example 3: 2.5 × 1.3
25 × 13 = 325. Total decimal places: 1 + 1 = 2.
Place decimal: 3.25
Example 4: 0.04 × 0.7
4 × 7 = 28. Total decimal places: 2 + 1 = 3.
Place decimal: 0.028
Example 5: 12.5 × 4.2
125 × 42 = 5,250. Total decimal places: 1 + 1 = 2.
Place decimal: 52.50 (or 52.5)
Dividing Decimals
If dividing by a decimal, move the decimal point in the divisor to make it a whole number, then move the dividend's decimal the same number of places.
Example 6: 8.4 ÷ 0.2
Move both decimals 1 place right: 84 ÷ 2 = 42
Example 7: 15.6 ÷ 0.04
Move both decimals 2 places right: 1560 ÷ 4 = 390
Converting Fractions to Decimals
Divide the numerator by the denominator.
- 1/2 = 1 ÷ 2 = 0.5
- 1/4 = 1 ÷ 4 = 0.25
- 3/4 = 3 ÷ 4 = 0.75
- 1/3 = 1 ÷ 3 = 0.333... (repeating)
- 1/8 = 1 ÷ 8 = 0.125
- 2/5 = 2 ÷ 5 = 0.4
Converting Decimals to Fractions
Read the decimal as a fraction, then simplify.
- 0.75 = 75/100 = 3/4
- 0.6 = 6/10 = 3/5
- 0.125 = 125/1000 = 1/8
Rounding Decimals
Look at the digit ONE place to the right of where you're rounding:
- If it's 0-4: round down (keep the digit)
- If it's 5-9: round up (increase the digit by 1)
Round 3.746 to nearest tenth:
Tenths digit: 7. Look at hundredths: 4. Since 4 < 5, round down → 3.7
Round 3.746 to nearest hundredth:
Hundredths digit: 4. Look at thousandths: 6. Since 6 ≥ 5, round up → 3.75
Comparing Decimals
Compare digit by digit from left to right:
Which is larger: 0.35 or 0.305?
0.350 vs 0.305 — tenths are equal (3), hundredths: 5 > 0, so 0.35 > 0.305
🎯 Try It Yourself
Test your understanding with these practice problems.
1. What is 3.47 + 12.8?
💡 Hint: Line up the decimal points
2. What is 7.5 × 0.4?
💡 Hint: 75 × 4 = 300, then place 2 decimal places
3. Convert 3/8 to a decimal
💡 Hint: Divide 3 by 8
4. Round 14.685 to the nearest tenth
💡 Hint: Look at the hundredths digit (8)
5. What is 24.6 ÷ 0.3?
💡 Hint: Move decimal: 246 ÷ 3
⚠️ Common Mistakes
- ✗Not lining up decimal points when adding or subtracting — this leads to wildly wrong answers.
- ✗Forgetting to count total decimal places when multiplying decimals.
- ✗When dividing by a decimal, forgetting to move the decimal in BOTH the divisor AND dividend.
- ✗Thinking 0.5 is less than 0.25 because 5 < 25 — compare place values, not digit counts!
- ✗Rounding 2.45 down to 2.4 — when the digit is exactly 5, round UP to 2.5.
🌍 Real Life Example
Money Math at the Store
You buy 3 items: $12.99, $7.50, and $3.49. Total: $12.99 + $7.50 + $3.49 = $23.98. You hand the cashier $25.00. Change: $25.00 - $23.98 = $1.02. With 8.5% tax: $23.98 × 0.085 = $2.04 tax, new total $26.02. Every time you handle money, you're doing decimal math.
💡 Key Takeaway
Decimals are fractions written in base-10 notation. The key rules: line up decimal points for addition/subtraction, count decimal places for multiplication, and move decimals when dividing. Decimals, fractions, and percentages are three ways of expressing the same thing.