How to Convert Decimals to Fractions
Terminating Decimals
For decimals that end (like 0.75 or 0.125):
- Write the decimal as a fraction over a power of 10
- Simplify the fraction by dividing numerator and denominator by their GCD
Repeating Decimals
For decimals with repeating patterns (like 0.333...):
- Let x = the repeating decimal
- Multiply by powers of 10 to align the repeating parts
- Subtract to eliminate the repeating part
- Solve for x
Examples
Example 1: Convert 0.75 to Fraction
Problem:
0.75 = ?
Step 1: Write as fraction over power of 10
0.75 has 2 decimal places, so 0.75 = 75/100
Step 2: Find GCD of 75 and 100
GCD(75, 100) = 25
Step 3: Divide by GCD
75/100 = (75/25)/(100/25) = 3/4
Result: 0.75 = 3/4
Example 2: Convert 0.125 to Fraction
Problem:
0.125 = ?
Step 1: Write as fraction over power of 10
0.125 has 3 decimal places, so 0.125 = 125/1000
Step 2: Find GCD of 125 and 1000
GCD(125, 1000) = 125
Step 3: Divide by GCD
125/1000 = (125/125)/(1000/125) = 1/8
Result: 0.125 = 1/8
Example 3: Convert 2.5 to Fraction
Problem:
2.5 = ?
Step 1: Separate the whole number and decimal part
2.5 = 2 + 0.5
Step 2: Convert the decimal part
0.5 = 5/10 = 1/2
Step 3: Combine as improper fraction
2 + 1/2 = 4/2 + 1/2 = 5/2
Result: 2.5 = 5/2 (or 2 1/2 as mixed number)
Example 4: Convert 0.333... to Fraction
Problem:
0.333... = ?
Step 1: Let x = 0.333...
Step 2: Multiply both sides by 10
10x = 3.333...
Step 3: Subtract the original equation
10x - x = 3.333... - 0.333...
9x = 3
Step 4: Solve for x
x = 3/9 = 1/3
Result: 0.333... = 1/3
Example 5: Convert 0.666... to Fraction
Problem:
0.666... = ?
Step 1: Let x = 0.666...
Step 2: Multiply both sides by 10
10x = 6.666...
Step 3: Subtract the original equation
10x - x = 6.666... - 0.666...
9x = 6
Step 4: Solve for x
x = 6/9 = 2/3
Result: 0.666... = 2/3