Compare Decimals and Fractions
Fifth Grade Math - Complete Guide
đź”— Why Compare Decimals and Fractions?
The Connection
Decimals and fractions are two different ways to represent the same value. Sometimes we need to compare them to find which is larger or smaller, or to arrange them in order.
đź“Ś Key Insight:
Before comparing decimals and fractions, we need to convert them to the same form - either all decimals OR all fractions!
⚖️ Methods to Compare Decimals and Fractions
Method 1: Convert Fractions to Decimals
Best Method! This is usually easier because comparing decimals is straightforward.
\[\frac{\text{Numerator}}{\text{Denominator}} = \text{Numerator} \div \text{Denominator}\]
- Convert all fractions to decimals by dividing
- Compare the decimals place by place (left to right)
- Determine which is greater or arrange in order
Example: Compare \(\frac{3}{4}\) and \(0.8\)
Step 1: Convert fraction to decimal → \(\frac{3}{4} = 3 \div 4 = 0.75\)
Step 2: Now compare → \(0.75\) and \(0.8\)
Step 3: Add trailing zeros → \(0.75\) and \(0.80\)
Step 4: Compare tenths (7 < 8)
âś“ Answer: \(0.75 < 0.8\) or \(\frac{3}{4} < 0.8\)
Method 2: Convert Decimals to Fractions
Use this method when working with simple fractions or when the problem requires fraction form.
- Convert all decimals to fractions using place value
- Find common denominators for all fractions
- Compare numerators
- The fraction with larger numerator is greater
Example: Compare \(0.6\) and \(\frac{3}{5}\)
Step 1: Convert decimal to fraction → \(0.6 = \frac{6}{10}\)
Step 2: Simplify → \(\frac{6}{10} = \frac{3}{5}\)
Step 3: Compare → \(\frac{3}{5}\) and \(\frac{3}{5}\)
âś“ Answer: \(0.6 = \frac{3}{5}\) (They are equal!)
đź“Ź Compare Decimals and Fractions on Number Lines
Number Line Rules
đź“Ś Important Rules:
• Numbers to the RIGHT are GREATER
• Numbers to the LEFT are SMALLER
• Numbers at the SAME POSITION are EQUAL
đź“ť Steps to Compare on Number Line
- Convert fractions to decimals (or vice versa)
- Identify the range (between which whole numbers)
- Mark both numbers on the number line
- The number on the right is greater
đź’ˇ Example
Compare \(\frac{1}{2}\) and \(0.6\) on a number line
Step 1: Convert fraction to decimal → \(\frac{1}{2} = 0.5\)
Step 2: Both numbers are between 0 and 1
Step 3: Mark on number line:
0.0────0.1────0.2────0.3────0.4────0.5────0.6────0.7────0.8────0.9────1.0
↑ \(\frac{1}{2}\) = 0.5 0.6 is here ↑
✓ Since 0.6 is to the RIGHT of 0.5 → \(0.6 > \frac{1}{2}\)
đź“Š Put a Mix of Decimals and Fractions in Order
Two Types of Ordering
Ascending Order (Least to Greatest):
Arrange from smallest to largest
Example: \(0.2 < \frac{1}{2} < 0.8 < \frac{9}{10}\)
Descending Order (Greatest to Least):
Arrange from largest to smallest
Example: \(\frac{9}{10} > 0.8 > \frac{1}{2} > 0.2\)
đź“ť Steps to Order
- Convert all to decimals (easiest method)
- Compare the decimal values
- Arrange in ascending or descending order
- Write the answer using original forms
đź’ˇ Example
Order from least to greatest: \(\frac{3}{4}\), \(0.5\), \(\frac{7}{10}\), \(0.82\)
Step 1: Convert all fractions to decimals
• \(\frac{3}{4} = 3 \div 4 = 0.75\)
• \(0.5 = 0.50\) (already decimal)
• \(\frac{7}{10} = 7 \div 10 = 0.7 = 0.70\)
• \(0.82\) (already decimal)
Step 2: Now we have: 0.75, 0.50, 0.70, 0.82
Step 3: Compare and order
• 0.50 is smallest (5 tenths)
• 0.70 is next (7 tenths)
• 0.75 is next (7 tenths, 5 hundredths)
• 0.82 is largest (8 tenths)
Step 4: Write using original forms
âś“ Answer: \(0.5 < \frac{7}{10} < \frac{3}{4} < 0.82\)
🔢 Ordering Decimals, Fractions, and Mixed Numbers
What are Mixed Numbers?
Mixed numbers have a whole number and a fraction together. Examples: \(2\frac{1}{2}\), \(3\frac{3}{4}\), \(1\frac{1}{5}\)
đź“Ś Conversion Formula:
\[a\frac{b}{c} = a + \frac{b}{c} = a + (b \div c)\]
đź“ť Steps to Order Mixed Numbers with Decimals and Fractions
- Convert mixed numbers to decimals: Keep whole number, convert fraction part
- Convert all fractions to decimals
- Compare all decimals starting from the whole number part
- Arrange in order and write using original forms
đź’ˇ Examples
Example 1: Order from least to greatest: \(3\frac{1}{2}\), \(\frac{6}{4}\), \(1.8\), \(1.3\)
Step 1: Convert all to decimals
• \(3\frac{1}{2} = 3 + \frac{1}{2} = 3 + 0.5 = 3.5\)
• \(\frac{6}{4} = 6 \div 4 = 1.5\)
• \(1.8\) (already decimal)
• \(1.3\) (already decimal)
Step 2: Now we have: 3.5, 1.5, 1.8, 1.3
Step 3: Compare and order
• Compare whole numbers: 1, 1, 1, 3
• 3.5 is largest (whole number is 3)
• For the three 1's, compare decimals: 0.3, 0.5, 0.8
• Order: 1.3 < 1.5 < 1.8 < 3.5
âś“ Answer: \(1.3 < \frac{6}{4} < 1.8 < 3\frac{1}{2}\)
Example 2: Order from greatest to least: \(2.75\), \(2\frac{1}{2}\), \(\frac{11}{4}\), \(2.6\)
Step 1: Convert all to decimals
• \(2.75\) (already decimal)
• \(2\frac{1}{2} = 2 + 0.5 = 2.5\)
• \(\frac{11}{4} = 11 \div 4 = 2.75\)
• \(2.6\) (already decimal)
Step 2: Now we have: 2.75, 2.5, 2.75, 2.6
Step 3: Compare and order (greatest to least)
• 2.75 appears twice (largest)
• 2.6 is next
• 2.5 is smallest
âś“ Answer: \(2.75 = \frac{11}{4} > 2.6 > 2\frac{1}{2}\)
đź“Š Common Fractions and Decimal Equivalents
Memorize these to make comparisons faster!
| Fraction | Decimal | Fraction | Decimal |
|---|---|---|---|
| \(\frac{1}{2}\) | 0.5 | \(\frac{1}{8}\) | 0.125 |
| \(\frac{1}{4}\) | 0.25 | \(\frac{3}{8}\) | 0.375 |
| \(\frac{3}{4}\) | 0.75 | \(\frac{5}{8}\) | 0.625 |
| \(\frac{1}{5}\) | 0.2 | \(\frac{7}{8}\) | 0.875 |
| \(\frac{2}{5}\) | 0.4 | \(\frac{1}{10}\) | 0.1 |
| \(\frac{3}{5}\) | 0.6 | \(\frac{3}{10}\) | 0.3 |
| \(\frac{4}{5}\) | 0.8 | \(\frac{7}{10}\) | 0.7 |
✏️ Practice Problems
Problem 1: Compare
Which is greater: \(\frac{5}{8}\) or \(0.7\)?
Solution:
Convert: \(\frac{5}{8} = 5 \div 8 = 0.625\)
Compare: \(0.625\) and \(0.700\)
Tenths place: 6 < 7
âś“ Answer: \(0.7 > \frac{5}{8}\)
Problem 2: Order
Order from least to greatest: \(\frac{2}{3}\), \(0.5\), \(\frac{3}{5}\), \(0.72\)
Solution:
Convert all to decimals:
• \(\frac{2}{3} = 0.667\)
• \(0.5 = 0.500\)
• \(\frac{3}{5} = 0.600\)
• \(0.72 = 0.720\)
Order: 0.500 < 0.600 < 0.667 < 0.720
âś“ Answer: \(0.5 < \frac{3}{5} < \frac{2}{3} < 0.72\)
Problem 3: Mixed Numbers
Order from greatest to least: \(1\frac{3}{4}\), \(1.6\), \(\frac{8}{5}\), \(1.85\)
Solution:
Convert all to decimals:
• \(1\frac{3}{4} = 1.75\)
• \(1.6\) (already decimal)
• \(\frac{8}{5} = 1.6\)
• \(1.85\) (already decimal)
Order: 1.85 > 1.75 > 1.6 = 1.6
âś“ Answer: \(1.85 > 1\frac{3}{4} > 1.6 = \frac{8}{5}\)
đź’ˇ Quick Tips & Strategies
âś… Easiest Method
Convert everything to decimals first - it's faster!
âś… Number Lines
Remember: Right = Greater, Left = Smaller
âś… Mixed Numbers
Convert to improper fraction first, then to decimal
âś… Memorize Common Ones
Know \(\frac{1}{2}=0.5\), \(\frac{1}{4}=0.25\), \(\frac{3}{4}=0.75\)
🎯 Step-by-Step Summary
1. Convert ALL to decimals → 2. Compare digit by digit → 3. Arrange in order
Don't forget: Write your final answer using the ORIGINAL forms!