Converting Fractions, Decimals, and Percentages

For Elementary Students

The Big Three: Different Ways to Say the Same Thing!

Fractions, decimals, and percentages are just THREE different ways to write the SAME number!

Think about it like this: It's like saying "half" in three languages!

Fraction:   1/2
Decimal:    0.5
Percentage: 50%

All mean HALF!

The Super Important Table

Memorize these! You'll use them ALL the time:

Fraction   Decimal   Percentage
  1/2       0.5         50%      (half)
  1/4       0.25        25%      (quarter)
  3/4       0.75        75%      (three quarters)
  1/5       0.2         20%      (one fifth)
  1/10      0.1         10%      (one tenth)
  1/3       0.333...    33.3%    (one third - repeating!)
  2/3       0.666...    66.7%    (two thirds - repeating!)

Fraction to Decimal: DIVIDE!

Rule: Divide the top by the bottom!

numerator ÷ denominator = decimal

Example 1: Convert 3/4 to a decimal

3/4 means 3 ÷ 4

  0.75
4)3.00
  2.8
  ---
   20
   20
   --
    0

Answer: 0.75!

Example 2: Convert 1/8 to a decimal

1/8 means 1 ÷ 8

1 ÷ 8 = 0.125

Answer: 0.125!

Example 3: Convert 2/3 to a decimal

2/3 means 2 ÷ 3

2 ÷ 3 = 0.666666...  (it keeps going!)

Answer: 0.667 (rounded) or 0.6̄ (bar means repeating)

Decimal to Percentage: Move the Decimal!

Rule: Multiply by 100 (or move the decimal point 2 places RIGHT!)

Decimal × 100 = Percentage

Example 1: Convert 0.75 to a percentage

0.75 × 100 = 75%

Or move decimal 2 places right:
0.75 → 7.5. → 75.%

Answer: 75%!

Example 2: Convert 0.4 to a percentage

0.4 × 100 = 40%

Or: 0.4 → 4.0. → 40.%

Answer: 40%!

Example 3: Convert 1.5 to a percentage

1.5 × 100 = 150%

Yes! Percentages can be bigger than 100%!

Answer: 150%!

Percentage to Decimal: Move the Other Way!

Rule: Divide by 100 (or move the decimal point 2 places LEFT!)

Percentage ÷ 100 = Decimal

Example 1: Convert 35% to a decimal

35% ÷ 100 = 0.35

Or move decimal 2 places left:
35.% → 3.5% → .35 = 0.35

Answer: 0.35!

Example 2: Convert 6% to a decimal

6% ÷ 100 = 0.06

Or: 06.% → 0.6% → .06 = 0.06

Answer: 0.06!

Example 3: Convert 200% to a decimal

200% ÷ 100 = 2.0

Answer: 2.0!

Fraction to Percentage: Two Steps!

Method 1: Convert to decimal first, THEN to percentage

Fraction → Decimal → Percentage
(divide)  (multiply by 100)

Example: Convert 3/5 to a percentage

Step 1: Fraction to decimal
3/5 = 3 ÷ 5 = 0.6

Step 2: Decimal to percentage
0.6 × 100 = 60%

Answer: 60%!

Method 2 (Shortcut): Multiply the fraction by 100!

Fraction × 100 = Percentage

Example: Convert 3/5 to a percentage

3/5 × 100 = 300/5 = 60%

Same answer, one step!

Percentage to Fraction: Write Over 100!

Rule: Write the percentage over 100, then simplify!

Percentage
---------- then simplify
   100

Example 1: Convert 40% to a fraction

Step 1: Write over 100
40% = 40/100

Step 2: Simplify
40/100 = 4/10 = 2/5  (divide by 20)

Answer: 2/5!

Example 2: Convert 15% to a fraction

Step 1: 15% = 15/100

Step 2: Simplify
15/100 = 3/20  (divide by 5)

Answer: 3/20!

Example 3: Convert 75% to a fraction

Step 1: 75% = 75/100

Step 2: Simplify
75/100 = 3/4  (divide by 25)

Answer: 3/4!

Decimal to Fraction: Use Place Value!

Rule: Write what you see using place value, then simplify!

0.5 = 5 tenths = 5/10 = 1/2
0.25 = 25 hundredths = 25/100 = 1/4
0.125 = 125 thousandths = 125/1000 = 1/8

Example 1: Convert 0.65 to a fraction

Step 1: Read it "65 hundredths"
0.65 = 65/100

Step 2: Simplify
65/100 = 13/20  (divide by 5)

Answer: 13/20!

Example 2: Convert 0.2 to a fraction

Step 1: "2 tenths"
0.2 = 2/10

Step 2: Simplify
2/10 = 1/5  (divide by 2)

Answer: 1/5!

The Conversion Circle

        Fraction
        /      \
   divide     write over 100
      /          \
 Decimal  ←--→  Percentage
      ×100    ÷100

You can go any direction!

Visual Example: One Half

Fraction:   1/2    (one out of two parts)
            ■□

Decimal:    0.5    (five tenths)

Percentage: 50%    (50 out of 100)
            ■■■■■■■■■■
            ■■■■■■■■■■
            ■■■■■■■■■■
            ■■■■■■■■■■
            ■■■■■■■■■■
            ■■■■■■■■■■ (50 shaded)

Quick Tips

Tip 1: Memorize the common conversions (1/2, 1/4, 3/4, 1/10)

Tip 2: Fraction to decimal: DIVIDE (top ÷ bottom)

Tip 3: Decimal to percentage: Move decimal RIGHT 2 places

Tip 4: Percentage to decimal: Move decimal LEFT 2 places

Tip 5: Percentage to fraction: Write over 100, simplify!

For Junior High Students

Understanding Equivalent Representations

Fractions, decimals, and percentages are three notation systems representing rational numbers (and in extended contexts, real numbers).

Definition: These forms are equivalent if they represent the same numerical value.

Fundamental relationships:

a/b ↔ (a ÷ b) ↔ ((a ÷ b) × 100)%

Purpose of multiple representations:

• Different contexts favor different forms
• Percentages intuitive for proportions and comparisons
• Decimals efficient for computation
• Fractions exact for certain values

Common Equivalences

Essential conversions to memorize:

Note: Some fractions (like 1/3) produce repeating decimals (0.3̄).

Fraction to Decimal Conversion

Method: Perform division of numerator by denominator.

Mathematical operation: a/b = a ÷ b

Example 1: Convert 3/8 to decimal

3 ÷ 8 = 0.375

Long division verification:

     0.375
   ________
8 | 3.000
    2.4     (8 × 0.3)
    ---
     60
     56     (8 × 0.07)
     --
      40
      40    (8 × 0.005)
      --
       0

Result: 3/8 = 0.375

Example 2: Convert 5/6 to decimal

5 ÷ 6 = 0.8333...

Notation: 0.83̄ (bar indicates repeating digit)

Classification:

• Terminating decimal: Division eventually yields remainder 0 (e.g., 3/8 = 0.375)
• Repeating decimal: Digits repeat indefinitely (e.g., 1/3 = 0.3̄)

Theorem: A fraction a/b in lowest terms has a terminating decimal if and only if the denominator b has only prime factors 2 and/or 5.

Decimal to Percentage Conversion

Method: Multiply by 100.

Rationale: "Percent" means "per hundred," so conversion scales to denominator 100.

Formula: d × 100 = p%

Practical shortcut: Move decimal point two places right.

Example 1: Convert 0.75 to percentage

0.75 × 100 = 75%

Or: 0.75 → 75.

Example 2: Convert 0.4 to percentage

0.4 × 100 = 40%

Or: 0.4 → 0.40 → 40.

Example 3: Convert 1.5 to percentage

1.5 × 100 = 150%

Note: Percentages can exceed 100% (representing values greater than 1).

Example 4: Convert 0.06 to percentage

0.06 × 100 = 6%

Precision: Trailing zeros after decimal may be significant depending on context.

Percentage to Decimal Conversion

Method: Divide by 100.

Formula: p% ÷ 100 = d

Practical shortcut: Move decimal point two places left.

Example 1: Convert 35% to decimal

35 ÷ 100 = 0.35

Or: 35. → 0.35

Example 2: Convert 6% to decimal

6 ÷ 100 = 0.06

Or: 06. → 0.06

Example 3: Convert 200% to decimal

200 ÷ 100 = 2.0

Example 4: Convert 0.5% to decimal

0.5 ÷ 100 = 0.005

Or: 0.5. → 0.005 (move two places left)

Fraction to Percentage Conversion

Method 1: Via decimal (two-step)

1. Convert fraction to decimal (divide)
2. Convert decimal to percentage (multiply by 100)

Example: Convert 3/5 to percentage

Step 1: 3 ÷ 5 = 0.6
Step 2: 0.6 × 100 = 60%

Method 2: Direct (one-step)

Multiply fraction by 100.

Formula: (a/b) × 100 = (100a)/b %

Example: Convert 7/20 to percentage

(7/20) × 100 = 700/20 = 35%

Example with simplification: Convert 9/25 to percentage

(9/25) × 100 = 900/25 = 36%

Advantage of Method 2: Avoids intermediate decimal step for exact computation.

Percentage to Fraction Conversion

Method: Write percentage as fraction with denominator 100, then simplify.

Formula: p% = p/100 (then reduce to lowest terms)

Example 1: Convert 40% to fraction

Step 1: 40% = 40/100
Step 2: Simplify by dividing numerator and denominator by GCD`(40, 100)` = 20
        40/100 = 2/5

Example 2: Convert 15% to fraction

15% = 15/100 = 3/20  (÷ 5)

Example 3: Convert 75% to fraction

75% = 75/100 = 3/4  (÷ 25)

Example 4: Convert 8% to fraction

8% = 8/100 = 2/25  (÷ 4)

Verification: Convert back to check:

2/25 = 0.08 = 8% ✓

Decimal to Fraction Conversion

Method: Express decimal as fraction based on place value, then simplify.

Procedure:

1. Identify place value (tenths, hundredths, thousandths, etc.)
2. Write as fraction with appropriate power of 10 in denominator
3. Simplify to lowest terms

Example 1: Convert 0.5 to fraction

0.5 = 5 tenths = 5/10 = 1/2  (÷ 5)

Example 2: Convert 0.25 to fraction

0.25 = 25 hundredths = 25/100 = 1/4  (÷ 25)

Example 3: Convert 0.65 to fraction

0.65 = 65 hundredths = 65/100 = 13/20  (÷ 5)

Example 4: Convert 0.125 to fraction

0.125 = 125 thousandths = 125/1000 = 1/8  (÷ 125)

Repeating decimals: Require algebraic method.

Example: Convert 0.3̄ to fraction

Let x = 0.3̄ = 0.333...

10x = 3.333...
10x - x = 3.333... - 0.333...
9x = 3
x = 3/9 = 1/3

Conversion Summary Table

Applications

Example 1: Test score

Score: 18 out of 24 questions correct

As fraction: 18/24 = 3/4 (simplified)

As decimal: 18 ÷ 24 = 0.75

As percentage: 0.75 × 100 = 75%

Example 2: Sale discount

"Save 1/4 off original price"

As decimal: 1/4 = 0.25

As percentage: 0.25 × 100 = 25% off

Example 3: Survey results

"0.68 of respondents agree"

As fraction: 0.68 = 68/100 = 17/25

As percentage: 0.68 × 100 = 68%

Common Errors

Error 1: Incorrect decimal placement

❌ 35% = 3.5 (moved only one place) ✓ 35% = 0.35 (move two places left)

Error 2: Forgetting to simplify fractions

❌ 50% = 50/100 (not fully simplified) ✓ 50% = 1/2

Error 3: Multiplying instead of dividing (or vice versa)

❌ To convert 0.6 to percentage: 0.6 ÷ 100 = 0.006 ✓ To convert 0.6 to percentage: 0.6 × 100 = 60%

Error 4: Place value confusion

❌ 0.5 = 5/100 ✓ 0.5 = 5/10 = 1/2

Tips for Success

Tip 1: Memorize common equivalences (1/2 = 0.5 = 50%, etc.)

Tip 2: Decimal to percentage: "move decimal right 2" Percentage to decimal: "move decimal left 2"

Tip 3: Fraction to decimal: always divide numerator by denominator

Tip 4: Always simplify fractions to lowest terms

Tip 5: Use place value correctly when converting decimals to fractions

Tip 6: Verify conversions by converting back to original form

Tip 7: For repeating decimals, use algebraic method or recognize common patterns

Extensions: Ratios and Proportions

Percentages are special ratios with denominator 100.

Example: 60% represents the ratio 60:100, which simplifies to 3:5.

Application: "60% of students passed" means ratio of passed to total is 3:5.

Proportional reasoning:

If 3/5 of students passed in a class of 40:

(3/5) × 40 = 24 students passed

Or using percentage:

60% of 40 = 0.60 × 40 = 24 students

Summary

Key relationships:

• Fraction ÷ = Decimal × 100 = Percentage
• All three forms represent the same value
• Choice of form depends on context and purpose
• Conversions follow systematic procedures

Essential skills:

• Fraction to decimal: divide
• Decimal to percentage: multiply by 100
• Percentage to fraction: write over 100, simplify
• Recognize and apply common equivalences

Practice