Convert Between Decimals and Fractions

Fifth Grade Math - Complete Guide

📚 Understanding Decimals and Fractions

What's the Connection?

Decimals and fractions are two different ways to represent the same value. They both show parts of a whole number.

Key Relationship:

• Fraction: Represents parts divided by a whole → \(\frac{1}{2}\)
• Decimal: Represents parts using place value → \(0.5\)
• Both mean the same thing: one-half!

Common Examples:

\(\frac{1}{2} = 0.5\) | \(\frac{1}{4} = 0.25\) | \(\frac{3}{4} = 0.75\) | \(\frac{1}{10} = 0.1\)

🎨 Model Decimals and Fractions

Visual Models

We can use grids, number lines, and shapes to represent both decimals and fractions visually.

Using a 10×10 Grid (100 squares):

• Each square = \(\frac{1}{100} = 0.01\) (one hundredth)
• 10 squares (1 column) = \(\frac{10}{100} = \frac{1}{10} = 0.1\) (one tenth)
• 25 squares = \(\frac{25}{100} = \frac{1}{4} = 0.25\) (one quarter)
• 50 squares = \(\frac{50}{100} = \frac{1}{2} = 0.5\) (one half)

💡 Key Insight:

When you shade 35 squares out of 100, you're showing both \(\frac{35}{100}\) as a fraction and \(0.35\) as a decimal!

➡️ Convert Fractions to Decimals

Basic Formula

\[\frac{\text{Numerator}}{\text{Denominator}} = \text{Numerator} \div \text{Denominator}\]

Simply divide the top number by the bottom number!

Method 1: Long Division Method

Use this method for any fraction. Simply divide the numerator by the denominator.

Set up division: numerator ÷ denominator
If numerator is smaller, add a decimal point and zeros
Perform long division
Stop when remainder is 0 or after 3 decimal places

Example: Convert \(\frac{3}{4}\) to a decimal

Step 1: Divide 3 by 4
Step 2: 3 is smaller than 4, so add decimal point and zero → 3.0
Step 3: \(30 \div 4 = 7\) remainder 2
Step 4: Bring down another 0 → \(20 \div 4 = 5\)
Step 5: No remainder, division complete

✓ Answer: \(\frac{3}{4} = 0.75\)

Method 2: Convert Denominator to Power of 10

Use this method when the denominator can be multiplied to get 10, 100, or 1000.

Find what number multiplies the denominator to make 10, 100, or 1000
Multiply both numerator and denominator by that number
Write the numerator with decimal point based on zeros
Count zeros in denominator = decimal places

Example: Convert \(\frac{7}{20}\) to a decimal

Step 1: Think: What × 20 = 100? → Answer: 5
Step 2: Multiply numerator and denominator by 5
\(\frac{7 \times 5}{20 \times 5} = \frac{35}{100}\)
Step 3: 100 has 2 zeros, so move decimal 2 places from right
\(\frac{35}{100} = 0.35\)

✓ Answer: \(\frac{7}{20} = 0.35\)

🔢 Convert Mixed Numbers to Decimals

What is a Mixed Number?

A mixed number has a whole number and a fraction together. Example: \(3\frac{1}{4}\) (three and one-fourth)

Method 1: Convert to Improper Fraction First

\[\text{Mixed Number} \rightarrow \text{Improper Fraction} \rightarrow \text{Decimal}\]

Convert mixed number to improper fraction: \(\frac{(\text{Whole} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}}\)
Divide the numerator by denominator
Write the result as a decimal

Example: Convert \(2\frac{3}{5}\) to a decimal

Step 1: Convert to improper fraction
\(2\frac{3}{5} = \frac{(2 \times 5) + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5}\)
Step 2: Divide numerator by denominator
\(13 \div 5 = 2.6\)

✓ Answer: \(2\frac{3}{5} = 2.6\)

Method 2: Keep Whole Number, Convert Fraction

Keep the whole number part aside
Convert only the fraction part to decimal
Add the whole number to the decimal

Example: Convert \(5\frac{1}{4}\) to a decimal

Step 1: Keep whole number → 5
Step 2: Convert fraction → \(\frac{1}{4} = 1 \div 4 = 0.25\)
Step 3: Add them → \(5 + 0.25 = 5.25\)

✓ Answer: \(5\frac{1}{4} = 5.25\)

⬅️ Convert Decimals to Fractions

Basic Steps

Identify the place value of the last digit
Determine the denominator based on place value
Remove the decimal point (numerator)
Simplify the fraction to lowest terms

Place Value to Denominator Guide

Decimal Places	Place Value	Denominator	Example
1 place	Tenths	10	\(0.3 = \frac{3}{10}\)
2 places	Hundredths	100	\(0.25 = \frac{25}{100}\)
3 places	Thousandths	1000	\(0.125 = \frac{125}{1000}\)

💡 Examples

Example 1: Convert \(0.6\) to a fraction

Step 1: Last digit is in tenths place → Denominator = 10
Step 2: Remove decimal point → Numerator = 6
Step 3: Write as fraction → \(\frac{6}{10}\)
Step 4: Simplify → \(\frac{6 \div 2}{10 \div 2} = \frac{3}{5}\)

✓ Answer: \(0.6 = \frac{3}{5}\)

Example 2: Convert \(0.75\) to a fraction

Step 1: Last digit is in hundredths place → Denominator = 100
Step 2: Remove decimal point → Numerator = 75
Step 3: Write as fraction → \(\frac{75}{100}\)
Step 4: Simplify → \(\frac{75 \div 25}{100 \div 25} = \frac{3}{4}\)

✓ Answer: \(0.75 = \frac{3}{4}\)

Example 3: Convert \(0.125\) to a fraction

Step 1: Last digit is in thousandths place → Denominator = 1000
Step 2: Remove decimal point → Numerator = 125
Step 3: Write as fraction → \(\frac{125}{1000}\)
Step 4: Simplify → \(\frac{125 \div 125}{1000 \div 125} = \frac{1}{8}\)

✓ Answer: \(0.125 = \frac{1}{8}\)

🔄 Convert Decimals to Mixed Numbers

When to Use Mixed Numbers

Use mixed numbers when the decimal is greater than 1. Example: \(2.5\), \(3.75\), \(5.2\)

📝 Steps to Convert

Separate the whole number part from the decimal part
Convert the decimal part to a fraction
Simplify the fraction
Combine the whole number with the fraction

💡 Examples

Example 1: Convert \(3.5\) to a mixed number

Step 1: Separate → Whole number = 3, Decimal part = 0.5
Step 2: Convert 0.5 to fraction → \(\frac{5}{10}\)
Step 3: Simplify → \(\frac{5}{10} = \frac{1}{2}\)
Step 4: Combine → \(3\frac{1}{2}\)

✓ Answer: \(3.5 = 3\frac{1}{2}\)

Example 2: Convert \(2.25\) to a mixed number

Step 1: Separate → Whole number = 2, Decimal part = 0.25
Step 2: Convert 0.25 to fraction → \(\frac{25}{100}\)
Step 3: Simplify → \(\frac{25}{100} = \frac{1}{4}\)
Step 4: Combine → \(2\frac{1}{4}\)

✓ Answer: \(2.25 = 2\frac{1}{4}\)

Example 3: Convert \(4.8\) to a mixed number

Step 1: Separate → Whole number = 4, Decimal part = 0.8
Step 2: Convert 0.8 to fraction → \(\frac{8}{10}\)
Step 3: Simplify → \(\frac{8}{10} = \frac{4}{5}\)
Step 4: Combine → \(4\frac{4}{5}\)

✓ Answer: \(4.8 = 4\frac{4}{5}\)

📊 Quick Conversion Chart

Fraction	Decimal	Fraction	Decimal
\(\frac{1}{2}\)	0.5	\(\frac{1}{5}\)	0.2
\(\frac{1}{4}\)	0.25	\(\frac{2}{5}\)	0.4
\(\frac{3}{4}\)	0.75	\(\frac{3}{5}\)	0.6
\(\frac{1}{8}\)	0.125	\(\frac{4}{5}\)	0.8
\(\frac{1}{10}\)	0.1	\(\frac{1}{3}\)	0.333...