Converting Between Decimals and Fractions

For Elementary Students

What Are We Learning?

Decimals and fractions are two different ways to show the same number!

Think about it like this: It's like saying "half" and "0.5" — they mean the SAME thing, just written differently!

1/2 = 0.5  (Same amount!)

Why Learn Both Ways?

Sometimes fractions are easier:

• "I ate 3/4 of the pizza"

Sometimes decimals are easier:

• "My height is 1.5 meters"

Knowing how to switch between them is super useful!

Decimal to Fraction: The Steps

Step 1: Say the decimal out loud — this tells you the bottom number!

Step 2: Write what you said as a fraction

Step 3: Simplify if you can!

Example 1: Convert 0.7

Step 1: Say it out loud

"0.7" = "seven tenths"

Step 2: Write as fraction

7/10

Step 3: Can we simplify? Nope! 7 and 10 don't share any common divisors.

Answer: 7/10 ✓

Example 2: Convert 0.25

Step 1: Say it out loud

"0.25" = "twenty-five hundredths"

Step 2: Write as fraction

25/100

Step 3: Simplify!

Both divide by 5: 25/100 = 5/20
Both divide by 5 again: 5/20 = 1/4

Answer: 1/4 ✓

Example 3: Convert 0.125

Step 1: Say it

"0.125" = "one hundred twenty-five thousandths"

Step 2: Write it

125/1000

Step 3: Simplify (divide by 125)

125/1000 = 1/8

Answer: 1/8 ✓

Place Value Trick

The number of decimal places tells you the bottom!

0.7    → 1 place  → tenths      → /10
0.25   → 2 places → hundredths  → /100
0.125  → 3 places → thousandths → /1000

Pattern:

• 1 decimal place → put over 10
• 2 decimal places → put over 100
• 3 decimal places → put over 1000

Fraction to Decimal: Just Divide!

To change a fraction to a decimal, divide the top by the bottom!

Remember: The fraction bar means "divided by"

3/4 = 3 ÷ 4

Example 1: Convert 1/2

1 ÷ 2 = 0.5

Answer: 0.5 ✓

Example 2: Convert 3/4

3 ÷ 4 = 0.75

Answer: 0.75 ✓

Example 3: Convert 1/8

1 ÷ 8 = 0.125

Answer: 0.125 ✓

Two Kinds of Decimals

Terminating decimals: They STOP!

1/4 = 0.25 (stops)
3/8 = 0.375 (stops)

Repeating decimals: They go on forever, repeating!

1/3 = 0.333333... (3 repeats forever)
2/3 = 0.666666... (6 repeats forever)
1/6 = 0.166666... (6 repeats)

We write repeating decimals with a line over the repeating part:

1/3 = 0.3̄  (bar over the 3)

Fractions You Should Memorize

These come up ALL the time — knowing them saves time!

Try to remember these! They're super common!

Mixed Numbers to Decimals

Step 1: Convert the fraction part

Step 2: Add it to the whole number

Example: 2 1/4

Step 1: 1/4 = 0.25
Step 2: 2 + 0.25 = 2.25

Answer: 2.25 ✓

Quick Practice

Let's try some together!

Convert 0.5 to a fraction:

"Five tenths" = 5/10 = 1/2 ✓

Convert 1/5 to a decimal:

1 ÷ 5 = 0.2 ✓

For Junior High Students

Understanding the Relationship

Decimals and fractions are two different representations of the same rational number.

Key insight: Both represent parts of a whole using base-10 (decimal) or ratio notation (fraction).

Why convert?

• Certain operations are easier with fractions (multiplication, division)
• Certain contexts prefer decimals (measurements, money, calculators)
• Some problems require converting between forms

Converting Decimal to Fraction

General algorithm:

1. Identify the place value of the rightmost digit
2. Write the decimal as a fraction with that place value as denominator
3. Simplify to lowest terms

Formal approach:

For a decimal with n digits after the decimal point:

decimal × 10ⁿ / 10ⁿ

Place Value System

Decimal to Fraction: Examples

Example 1: Convert 0.6

Step 1: Identify place value
Last digit (6) is in tenths place

Step 2: Write as fraction
0.6 = 6/10

Step 3: Simplify
GCD`(6, 10)` = 2
6/10 = 3/5

Answer: 3/5

Example 2: Convert 0.45

Step 1: Two decimal places → hundredths
0.45 = 45/100

Step 2: Simplify
GCD`(45, 100)` = 5
45/100 = 9/20

Answer: 9/20

Example 3: Convert 0.875

Step 1: Three decimal places → thousandths
0.875 = 875/1000

Step 2: Simplify
GCD`(875, 1000)` = 125
875 ÷ 125 = 7
1000 ÷ 125 = 8

Answer: 7/8

Example 4: Convert 2.4

For mixed decimal:
2.4 = 2 + 0.4
    = 2 + 4/10
    = 2 + 2/5
    = 2 2/5

Or as improper fraction:
2.4 = 24/10 = 12/5

Both forms are correct

Converting Fraction to Decimal

Method: Divide the numerator by the denominator.

Formal definition:

a/b = a ÷ b

Process:

1. Set up division: numerator ÷ denominator
2. Perform long division or use calculator
3. Result is the decimal equivalent

Fraction to Decimal: Examples

Example 1: Convert 3/8

3 ÷ 8 = 0.375

Verification: 0.375 = 375/1000 = 3/8 ✓

Example 2: Convert 7/20

7 ÷ 20 = 0.35

Check: 0.35 = 35/100 = 7/20 ✓

Example 3: Convert 5/6

5 ÷ 6 = 0.8333...

This is a repeating decimal: 0.83̄

Example 4: Convert 11/4

11 ÷ 4 = 2.75

This is an improper fraction, giving a decimal > 1

Terminating vs. Repeating Decimals

Terminating decimal: The division ends with remainder 0.

Condition: A fraction in simplest form gives a terminating decimal if and only if the denominator has only factors of 2 and/or 5.

Why? Because 10 = 2 × 5, and our decimal system is base 10.

Examples of terminating:

1/2 = 0.5       (denominator: 2 = 2¹)
1/4 = 0.25      (denominator: 4 = 2²)
1/5 = 0.2       (denominator: 5 = 5¹)
1/8 = 0.125     (denominator: 8 = 2³)
3/20 = 0.15     (denominator: 20 = 2² × 5¹)

Repeating decimal: The division never terminates; digits repeat in a pattern.

Condition: Denominator (in simplest form) has prime factors other than 2 and 5.

Examples of repeating:

1/3 = 0.333...   (denominator: 3)
1/6 = 0.1666...  (denominator: 6 = 2 × 3, has factor 3)
2/7 = 0.285714285714... (denominator: 7)
5/11 = 0.4545... (denominator: 11)

Notation for repeating decimals:

We use a bar over the repeating digits:

1/3 = 0.3̄
1/6 = 0.16̄ (only 6 repeats)
2/11 = 0.1̄8̄ (both 1 and 8 repeat)

Mixed Numbers and Decimals

Converting mixed number to decimal:

Method 1: Convert fraction part, add whole part

5 3/4 = 5 + 3/4
      = 5 + 0.75
      = 5.75

Method 2: Convert to improper fraction, then divide

5 3/4 = 23/4
      = 23 ÷ 4
      = 5.75

Converting decimal to mixed number:

The whole number part stays as is; convert the decimal part:

3.625 = 3 + 0.625
      = 3 + 625/1000
      = 3 + 5/8
      = 3 5/8

Special Fractions and Decimals

Common benchmarks:

Memorizing these accelerates problem-solving.

Applications

Money: Decimals are standard

$3.75 = 3 75/100 dollars = 3 3/4 dollars

Measurements: Often use decimals

1.5 meters = 1 1/2 meters

Cooking: Often use fractions

3/4 cup = 0.75 cup

Science: Typically uses decimals

0.625 grams rather than 5/8 grams

Real-Life Applications

Shopping: "$25.99 is approximately $26 = 26/1"

Baking: "Recipe calls for 0.75 cups = 3/4 cup"

Sports: "Runner completed 3.25 laps = 3 1/4 laps"

Construction: "Board is 2.5 inches thick = 2 1/2 inches"

Common Mistakes

Mistake 1: Incorrect denominator

❌ 0.25 = 25/10 ✓ 0.25 = 25/100 = 1/4 (two decimal places → hundredths)

Mistake 2: Not simplifying

❌ 0.5 = 5/10 (leaving answer unsimplified) ✓ 0.5 = 5/10 = 1/2

Mistake 3: Division error in fraction to decimal

❌ 3/4 = 4 ÷ 3 = 1.33... (divided wrong way) ✓ 3/4 = 3 ÷ 4 = 0.75

Mistake 4: Misplacing decimal point

❌ 1/8 = 1.25 (incorrect division) ✓ 1/8 = 0.125

Mistake 5: Confusing repeating and terminating

Not recognizing that 1/3 = 0.333... repeats infinitely

Tips for Success

Tip 1: Count decimal places to determine denominator (1 place = 10, 2 places = 100, etc.)

Tip 2: Always simplify fractions to lowest terms

Tip 3: Memorize common fraction-decimal equivalents

Tip 4: For fraction to decimal: numerator ÷ denominator (top ÷ bottom)

Tip 5: Check if denominator has only factors of 2 and 5 to predict terminating/repeating

Tip 6: Use calculator for complex divisions, but understand the process

Checking Your Work

Decimal to fraction:

• Convert back: divide numerator by denominator to verify

Fraction to decimal:

• Convert back: write decimal as fraction and simplify to verify

Example verification:

Claim: 0.6 = 3/5

Check: 3 ÷ 5 = 0.6 ✓ Correct!

Advanced: Converting Repeating Decimals to Fractions

Example: Convert 0.6̄ (0.666...) to a fraction

Let x = 0.666...
Multiply by 10: 10x = 6.666...
Subtract: 10x − x = 6.666... − 0.666...
         9x = 6
         x = 6/9 = 2/3

Therefore: 0.6̄ = 2/3

This technique works for any repeating decimal.

Practice