Divide Decimals by Powers of Ten - Fifth Grade Math
Complete Notes & Formulas
1. Divide by Powers of Ten
What are Powers of Ten?
Powers of ten are numbers like 10, 100, 1000, 10000, etc. They start with 1 and end with zeros.
10 = 1 zero
100 = 2 zeros
1000 = 3 zeros
10000 = 4 zeros
The Golden Rule
When dividing by a power of 10, move the decimal point to the LEFT by the number of zeros in the divisor.
Steps to Divide by Powers of Ten
Step 1: Identify the decimal number (dividend)
Step 2: Count the zeros in the power of 10 (divisor)
Step 3: Move the decimal point LEFT by that many places
Step 4: Add zeros on the left if needed
Step 5: Write the final answer
Formula
Examples
Example 1: 45.32 ÷ 10
• Count zeros in 10: 1 zero
• Move decimal LEFT 1 place: 45.32 → 4.532
Answer: 4.532
Example 2: 2.7 ÷ 1000
• Count zeros in 1000: 3 zeros
• Move decimal LEFT 3 places: 2.7 → 0.0027
• Add zeros: Need 2 zeros before 2
Answer: 0.0027
Example 3: 674 ÷ 1000
• Decimal is after 4: 674.
• Move LEFT 3 places: 674. → 0.674
Answer: 0.674
Important Note
When dividing, the decimal point moves LEFT ←
When multiplying, the decimal point moves RIGHT →
2. Decimal Division Patterns Over Increasing Place Values
Understanding the Pattern
When we divide by increasing powers of ten (10, 100, 1000...), we see a clear pattern in how the decimal point moves.
Pattern Table
| Divide By | Zeros | Decimal Moves | Example |
|---|---|---|---|
| 1 | 0 zeros | No movement | 97.5 ÷ 1 = 97.5 |
| 10 | 1 zero | 1 place LEFT ← | 97.5 ÷ 10 = 9.75 |
| 100 | 2 zeros | 2 places LEFT ← | 97.5 ÷ 100 = 0.975 |
| 1000 | 3 zeros | 3 places LEFT ← | 97.5 ÷ 1000 = 0.0975 |
| 10000 | 4 zeros | 4 places LEFT ← | 97.5 ÷ 10000 = 0.00975 |
Pattern Rule
Visualizing the Pattern
Starting number: 456.8
÷ 1 → 456.8 (no change)
÷ 10 → 45.68 (1 place left)
÷ 100 → 4.568 (2 places left)
÷ 1000 → 0.4568 (3 places left)
÷ 10000 → 0.04568 (4 places left)
3. Divide by a Power of Ten: With Exponents
Understanding Exponents
An exponent shows how many times to multiply a number by itself. For powers of 10, the exponent tells us how many zeros follow the 1.
101 = 10 (1 zero)
102 = 100 (2 zeros)
103 = 1000 (3 zeros)
104 = 10000 (4 zeros)
105 = 100000 (5 zeros)
The Exponent Division Rule
When dividing by 10n, move the decimal point LEFT by n places.
Formula
(where n = exponent)
Steps to Divide Using Exponents
Step 1: Look at the exponent on 10
Step 2: The exponent = number of places to move decimal LEFT
Step 3: Move the decimal point
Step 4: Add zeros if necessary
Examples
Example 1: 234.5 ÷ 102
• Exponent is 2
• Move decimal LEFT 2 places: 234.5 → 2.345
Answer: 2.345
Example 2: 8.6 ÷ 103
• Exponent is 3
• Move decimal LEFT 3 places: 8.6 → 0.0086
• Need to add 2 zeros
Answer: 0.0086
Example 3: 5432.1 ÷ 104
• Exponent is 4
• Move decimal LEFT 4 places: 5432.1 → 0.54321
Answer: 0.54321
4. Multiply and Divide by a Power of Ten: With Exponents
Comparing Multiplication and Division
Understanding the difference between multiplying and dividing by powers of ten is crucial!
Key Rules Comparison
| Operation | Direction | Rule | Example |
|---|---|---|---|
| Multiply × 10n | RIGHT → | Move decimal n places RIGHT | 3.5 × 102 = 350 |
| Divide ÷ 10n | LEFT ← | Move decimal n places LEFT | 3.5 ÷ 102 = 0.035 |
Complete Formula Guide
Multiplication: Number × 10n → Move decimal RIGHT n places
Division: Number ÷ 10n → Move decimal LEFT n places
Practice Examples
Multiplication Examples:
① 4.56 × 101 = 45.6 (move 1 place right)
② 4.56 × 102 = 456 (move 2 places right)
③ 4.56 × 103 = 4560 (move 3 places right, add zero)
Division Examples:
① 4.56 ÷ 101 = 0.456 (move 1 place left)
② 4.56 ÷ 102 = 0.0456 (move 2 places left)
③ 4.56 ÷ 103 = 0.00456 (move 3 places left)
Memory Tip
🔢 Multiply = Make bigger → Move RIGHT →
➗ Divide = Make smaller → Move LEFT ←
5. Divide by a Power of Ten with Decimals: Find the Missing Number
Understanding Missing Number Problems
Sometimes we need to find the power of ten that was used in division. We work backwards by looking at how the decimal point moved!
Three Types of Missing Number Problems
Type 1: Dividend ÷ ? = Quotient
Example: 12.1 ÷ ? = 1.21
Type 2: ? ÷ Power of 10 = Quotient
Example: ? ÷ 10 = 4.56
Type 3: With exponents: Dividend ÷ 10? = Quotient
Example: 8.5 ÷ 10? = 0.085
Steps to Find Missing Power of Ten
Step 1: Compare the dividend and quotient
Step 2: Count how many places the decimal moved
Step 3: The number of places = number of zeros (or exponent)
Step 4: Write the power of ten
Formula
Detailed Examples
Example 1: 12.1 ÷ ? = 1.21
Compare: 12.1 → 1.21
Decimal moved: 12.1 → 1.21 (1 place LEFT)
1 place = divide by 10
Answer: Missing number is 10
Example 2: 345.6 ÷ ? = 3.456
Compare: 345.6 → 3.456
Decimal moved: 345.6 → 3.456 (2 places LEFT)
2 places = divide by 100 or 102
Answer: Missing number is 100
Example 3: ? ÷ 100 = 4.56
Work backwards!
If we divided by 100, we moved decimal 2 places LEFT
To reverse: move decimal 2 places RIGHT
4.56 → 456
Answer: Missing number is 456
Example 4: 8.5 ÷ 10? = 0.085
Compare: 8.5 → 0.085
Decimal moved: 8.5 → 0.085 (2 places LEFT)
2 places = exponent is 2
Answer: Missing exponent is 2 (8.5 ÷ 102 = 0.085)
6. Divide by 0.1 or 0.01
The Surprising Rule!
⚡ Dividing by 0.1 or 0.01 makes numbers BIGGER! ⚡
Understanding Why
• 0.1 means "one tenth" (1/10)
• Asking "how many tenths are in a number?" gives a bigger result!
• 0.01 means "one hundredth" (1/100)
• Asking "how many hundredths?" gives an even bigger result!
Key Rules
Rule 1: Dividing by 0.1 = Multiplying by 10
Number ÷ 0.1 = Number × 10
Rule 2: Dividing by 0.01 = Multiplying by 100
Number ÷ 0.01 = Number × 100
Formulas
÷ 0.1 → Move decimal 1 place RIGHT →
÷ 0.01 → Move decimal 2 places RIGHT →
Why the Decimal Moves Right
Because dividing by a decimal less than 1 is the same as multiplying by a whole number greater than 1!
Examples with 0.1
Example 1: 50 ÷ 0.1
Method 1: 50 ÷ 0.1 = 50 × 10 = 500
Method 2: Move decimal 1 place RIGHT → 50. → 500
Answer: 500
Example 2: 5 ÷ 0.1
5 ÷ 0.1 = 5 × 10
Move decimal RIGHT: 5. → 50
Answer: 50
Example 3: 0.5 ÷ 0.1
0.5 ÷ 0.1 = 0.5 × 10
Move decimal RIGHT: 0.5 → 5
Answer: 5
Examples with 0.01
Example 1: 19.21 ÷ 0.01
19.21 ÷ 0.01 = 19.21 × 100
Move decimal 2 places RIGHT: 19.21 → 1921
Answer: 1921
Example 2: 7190 ÷ 0.01
7190 ÷ 0.01 = 7190 × 100
Move decimal 2 places RIGHT: 7190. → 719000
Answer: 719,000
Example 3: 0.05 ÷ 0.01
0.05 ÷ 0.01 = 0.05 × 100
Move decimal 2 places RIGHT: 0.05 → 5
Answer: 5
Think About It!
Question: How many 0.1 (tenths) are in 5?
Answer: 5 ÷ 0.1 = 50 tenths!
Question: How many 0.01 (hundredths) are in 2?
Answer: 2 ÷ 0.01 = 200 hundredths!
📊 Quick Reference Summary
| Divide By | Decimal Movement | Same As | Example |
|---|---|---|---|
| 10 or 101 | 1 place LEFT ← | × 0.1 | 45.6 ÷ 10 = 4.56 |
| 100 or 102 | 2 places LEFT ← | × 0.01 | 45.6 ÷ 100 = 0.456 |
| 1000 or 103 | 3 places LEFT ← | × 0.001 | 45.6 ÷ 1000 = 0.0456 |
| 0.1 | 1 place RIGHT → | × 10 | 4.56 ÷ 0.1 = 45.6 |
| 0.01 | 2 places RIGHT → | × 100 | 4.56 ÷ 0.01 = 456 |
💡 Important Tips to Remember
✓ Dividing by whole powers of 10 (10, 100, 1000) → Decimal moves LEFT ←
✓ Dividing by decimal powers (0.1, 0.01) → Decimal moves RIGHT →
✓ Exponent tells you how many places to move the decimal point
✓ Number of zeros = Number of places to move
✓ Add zeros on the left if you run out of digits when moving left
✓ Add zeros on the right if you run out of digits when moving right
✓ Dividing by 0.1 is the same as multiplying by 10
✓ Dividing by 0.01 is the same as multiplying by 100
✓ Numbers get smaller when dividing by 10, 100, 1000
✓ Numbers get bigger when dividing by 0.1, 0.01
🎯 Practice Makes Perfect!
Master these concepts by practicing daily with different numbers.
Remember: The decimal point is your friend - just know which way to move it! 📍