📊 Comparing Decimals and Fractions
Complete Notes & Formulae for Grade 5 Math
🎯 What You'll Learn
In this unit, we'll master comparing decimals and fractions using different methods:
- Using number lines to compare values
- Converting between decimals and fractions
- Ordering mixed sets of decimals and fractions
- Working with mixed numbers
1️⃣ Understanding Decimals and Fractions
📝 Key Definitions
Decimal: A number that uses a decimal point to show parts of a whole. The digits after the decimal point represent tenths, hundredths, thousandths, etc.
Example: \(0.5\) means 5 tenths, \(0.75\) means 75 hundredths
Fraction: A number written as \(\frac{\text{numerator}}{\text{denominator}}\) showing parts of a whole.
Example: \(\frac{1}{2}\) means 1 out of 2 equal parts, \(\frac{3}{4}\) means 3 out of 4 equal parts
📍 Decimal Place Value Chart
| Ones | Decimal Point | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|
| 1 | . | 0.1 | 0.01 | 0.001 |
| \(1\) | . | \(\frac{1}{10}\) | \(\frac{1}{100}\) | \(\frac{1}{1000}\) |
2️⃣ Converting Fractions to Decimals
🔄 Main Formula
\[\text{Decimal} = \text{Numerator} \div \text{Denominator}\]
Or written as: \(\frac{a}{b} = a \div b\)
📋 Three Methods to Convert
Method 1: When denominator is 10, 100, or 1000
- \(\frac{3}{10} = 0.3\) (3 tenths)
- \(\frac{25}{100} = 0.25\) (25 hundredths)
- \(\frac{125}{1000} = 0.125\) (125 thousandths)
Method 2: Create equivalent fractions with 10, 100, or 1000
- \(\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10} = 0.5\)
- \(\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 0.75\)
- \(\frac{1}{5} = \frac{1 \times 2}{5 \times 2} = \frac{2}{10} = 0.2\)
Method 3: Divide numerator by denominator
- \(\frac{3}{8} = 3 \div 8 = 0.375\)
- \(\frac{5}{8} = 5 \div 8 = 0.625\)
- \(\frac{7}{20} = 7 \div 20 = 0.35\)
✏️ Example 1: Convert \(\frac{2}{5}\) to a decimal
1Method 1 - Create equivalent fraction:
\(\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10} = 0.4\)
2Method 2 - Division:
\(\frac{2}{5} = 2 \div 5 = 0.4\)
✓ Answer: \(\frac{2}{5} = 0.4\)
3️⃣ Converting Decimals to Fractions
🔄 Conversion Steps
Step 1: Count decimal places
Step 2: Write as fraction with denominator 10, 100, or 1000
Step 3: Simplify if possible
📋 Common Conversions
| Decimal | Fraction | Process |
|---|---|---|
| \(0.5\) | \(\frac{1}{2}\) | \(\frac{5}{10} = \frac{1}{2}\) |
| \(0.25\) | \(\frac{1}{4}\) | \(\frac{25}{100} = \frac{1}{4}\) |
| \(0.75\) | \(\frac{3}{4}\) | \(\frac{75}{100} = \frac{3}{4}\) |
| \(0.2\) | \(\frac{1}{5}\) | \(\frac{2}{10} = \frac{1}{5}\) |
| \(0.4\) | \(\frac{2}{5}\) | \(\frac{4}{10} = \frac{2}{5}\) |
| \(0.125\) | \(\frac{1}{8}\) | \(\frac{125}{1000} = \frac{1}{8}\) |
✏️ Example 2: Convert \(0.6\) to a fraction
1One decimal place means tenths:
\(0.6 = \frac{6}{10}\)
2Simplify by dividing by GCF (2):
\(\frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5}\)
✓ Answer: \(0.6 = \frac{3}{5}\)
4️⃣ Comparing Decimals and Fractions on Number Lines
📏 How Number Lines Work
A number line helps us see the size of numbers visually. Numbers increase in value from left to right.
- Smaller values are on the left
- Larger values are on the right
- Equal values occupy the same position
📍 Example Number Line: 0 to 1
0 ——— 0.25 ——— 0.5 ——— 0.75 ——— 1
\(0\) ——— \(\frac{1}{4}\) ——— \(\frac{1}{2}\) ——— \(\frac{3}{4}\) ——— \(1\)
↑ Notice: \(0.25 = \frac{1}{4}\), \(0.5 = \frac{1}{2}\), \(0.75 = \frac{3}{4}\)
✏️ Example 3: Compare \(\frac{3}{5}\) and \(0.7\) using a number line
1Convert fraction to decimal:
\(\frac{3}{5} = 3 \div 5 = 0.6\)
2Plot on number line:
0 ——— 0.5 ——— 0.6 ——— 0.7 ——— 1.0
↑ (\(\frac{3}{5}\)=0.6) ↑ (0.7)
3Compare positions:
0.6 is to the LEFT of 0.7, so 0.6 < 0.7
✓ Answer: \(\frac{3}{5} < 0.7\) or \(0.6 < 0.7\)
5️⃣ Comparing Decimals and Fractions Directly
🎯 Two Main Strategies
Strategy A: Convert fractions to decimals
Strategy B: Convert decimals to fractions
💡 Tip: Converting to decimals is usually easier!
📋 Steps to Compare
1Convert all numbers to the same form (all decimals OR all fractions)
2Compare the values using place value or common denominators
3Use comparison symbols:
- \(>\) means "greater than"
- \(<\) means "less than"
- \(=\) means "equal to"
✏️ Example 4: Compare \(\frac{7}{8}\) and \(0.8\)
1Convert fraction to decimal:
\(\frac{7}{8} = 7 \div 8 = 0.875\)
2Compare decimals:
\(0.875\) vs \(0.8\)
Look at tenths place: both have 8
Look at hundredths place: 7 > 0
So \(0.875 > 0.8\)
✓ Answer: \(\frac{7}{8} > 0.8\)
✏️ Example 5: Compare \(0.45\) and \(\frac{9}{20}\)
1Convert fraction to decimal:
\(\frac{9}{20} = 9 \div 20 = 0.45\)
2Compare:
\(0.45 = 0.45\)
✓ Answer: \(0.45 = \frac{9}{20}\) (They are equal!)
6️⃣ Ordering Decimals and Fractions
📋 Steps to Order Numbers
1Convert all numbers to the same form (decimals work best)
2Arrange from least to greatest (ascending order) or greatest to least (descending order)
3Write the answer using the original forms
💡 Pro Tips for Ordering:
- Ascending order: From smallest to largest (↑)
- Descending order: From largest to smallest (↓)
- Always convert to decimals first - it's easier!
- Line up decimal points when comparing
✏️ Example 6: Order from least to greatest: \(\frac{3}{4}\), \(0.8\), \(\frac{2}{5}\), \(0.65\)
1Convert all to decimals:
- \(\frac{3}{4} = 3 \div 4 = 0.75\)
- \(0.8 = 0.8\) (already decimal)
- \(\frac{2}{5} = 2 \div 5 = 0.4\)
- \(0.65 = 0.65\) (already decimal)
2Arrange in order:
\(0.4\), \(0.65\), \(0.75\), \(0.8\)
3Write using original forms:
\(\frac{2}{5}\), \(0.65\), \(\frac{3}{4}\), \(0.8\)
✓ Answer: \(\frac{2}{5} < 0.65 < \frac{3}{4} < 0.8\)
7️⃣ Working with Mixed Numbers
📝 What is a Mixed Number?
A mixed number has a whole number part AND a fraction part.
Example: \(2\frac{3}{4}\) means 2 whole units plus \(\frac{3}{4}\)
🔄 Converting Mixed Numbers to Decimals
\[a\frac{b}{c} = a + (b \div c)\]
Step 1: Convert the fraction part to a decimal
Step 2: Add the whole number
✏️ Example 7: Convert \(3\frac{1}{2}\) to a decimal
1Convert fraction part:
\(\frac{1}{2} = 1 \div 2 = 0.5\)
2Add whole number:
\(3 + 0.5 = 3.5\)
✓ Answer: \(3\frac{1}{2} = 3.5\)
✏️ Example 8: Order from greatest to least: \(2.5\), \(\frac{5}{2}\), \(2\frac{3}{4}\), \(2.25\)
1Convert all to decimals:
- \(2.5 = 2.5\)
- \(\frac{5}{2} = 5 \div 2 = 2.5\)
- \(2\frac{3}{4} = 2 + (3 \div 4) = 2 + 0.75 = 2.75\)
- \(2.25 = 2.25\)
2Arrange from greatest to least:
\(2.75\), \(2.5\), \(2.5\), \(2.25\)
3Write using original forms:
\(2\frac{3}{4}\), \(2.5\) and \(\frac{5}{2}\) (tied), \(2.25\)
✓ Answer: \(2\frac{3}{4} > 2.5 = \frac{5}{2} > 2.25\)
8️⃣ Quick Reference Guide
📌 Essential Formulas
| Formula | What It Does |
|---|---|
| \(\frac{a}{b} = a \div b\) | Convert fraction to decimal |
| \(0.a = \frac{a}{10}\) | One decimal place = tenths |
| \(0.ab = \frac{ab}{100}\) | Two decimal places = hundredths |
| \(a\frac{b}{c} = a + (b \div c)\) | Convert mixed number to decimal |
🌟 Common Equivalent Values (Memorize These!)
| Fraction | Decimal | Fraction | Decimal |
|---|---|---|---|
| \(\frac{1}{2}\) | 0.5 | \(\frac{1}{10}\) | 0.1 |
| \(\frac{1}{4}\) | 0.25 | \(\frac{3}{10}\) | 0.3 |
| \(\frac{3}{4}\) | 0.75 | \(\frac{7}{10}\) | 0.7 |
| \(\frac{1}{5}\) | 0.2 | \(\frac{9}{10}\) | 0.9 |
| \(\frac{2}{5}\) | 0.4 | \(\frac{1}{8}\) | 0.125 |
| \(\frac{3}{5}\) | 0.6 | \(\frac{3}{8}\) | 0.375 |
| \(\frac{4}{5}\) | 0.8 | \(\frac{5}{8}\) | 0.625 |
🎯 Comparison Symbols
| Symbol | Meaning | Example |
|---|---|---|
| \(>\) | Greater than | \(0.8 > 0.75\) |
| \(<\) | Less than | \(0.3 < 0.5\) |
| \(=\) | Equal to | \(0.5 = \frac{1}{2}\) |
9️⃣ Study Tips & Strategies
✨ Success Strategies
- Always convert to the same form before comparing - decimals are usually easier!
- Use number lines to visualize the relative size of numbers
- Memorize common equivalents like \(\frac{1}{2} = 0.5\) and \(\frac{1}{4} = 0.25\)
- Line up decimal points when comparing decimal numbers
- Check your work by converting back to the original form
- Remember: The alligator mouth \(>\) or \(<\) always "eats" the bigger number!
🔍 Common Mistakes to Avoid
- ❌ Comparing \(0.5\) and \(0.25\) and thinking \(0.25\) is bigger because 25 > 5
- ✅ Remember place value! \(0.5 = 0.50\), and \(50 > 25\), so \(0.5 > 0.25\)
- ❌ Forgetting to add the whole number in mixed numbers
- ✅ \(2\frac{1}{4} = 2.25\), not just \(0.25\)!
- ❌ Not simplifying fractions when converting decimals
- ✅ \(0.5 = \frac{5}{10} = \frac{1}{2}\) (simplest form)
📚 Practice Problems
Try these on your own!
1. Compare: \(\frac{2}{3}\) ___ \(0.7\)
2. Order from least to greatest: \(\frac{3}{8}\), \(0.5\), \(\frac{1}{4}\), \(0.45\)
3. Convert to decimal: \(4\frac{2}{5}\)
4. Compare: \(1.5\) ___ \(\frac{3}{2}\)
5. Order from greatest to least: \(2.8\), \(2\frac{3}{4}\), \(\frac{11}{4}\), \(2.65\)
💡 Answers (Check after trying!):
1. \(\frac{2}{3} < 0.7\) (since \(0.667 < 0.7\))
2. \(\frac{1}{4}\), \(\frac{3}{8}\), \(0.45\), \(0.5\)
3. \(4.4\)
4. \(1.5 = \frac{3}{2}\)
5. \(2.8\), \(2\frac{3}{4}\), \(\frac{11}{4}\), \(2.65\)
🎉 You've Got This!
Keep practicing and you'll master comparing decimals and fractions in no time!