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Decimal to Fraction Conversion

Convert decimals to fractions: multiply by 10^n for terminating decimals, or use algebra for repeating decimals.

Concept Fundamentals
0.75 = 75/100 = 3/4
Terminating
x=0.333... โ†’ 10x-x=3 โ†’ 1/3
Repeating
Multiplier for n places
10^n
Simplify result
GCD

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A decimal terminates iff its fraction (simplified) has denominator with only 2 and 5 as prime factors. 0.999... = 1 exactlyโ€”proof: 10x - x = 9. Common equivalents: 0.5=1/2, 0.25=1/4, 0.125=1/8, 0.333...=1/3.

Key quantities
0.75 = 75/100 = 3/4
Terminating
Key relation
x=0.333... โ†’ 10x-x=3 โ†’ 1/3
Repeating
Key relation
Multiplier for n places
10^n
Key relation
Simplify result
GCD
Key relation

Ready to run the numbers?

Why: Converting decimals to fractions gives exact rational form and helps in algebra, geometry, and precise measurements.

How: Terminating: multiply by 10^n (n = decimal places), then simplify by GCD. Repeating: use algebraโ€”let x = decimal, multiply to shift, subtract to eliminate repeat.

A decimal terminates iff its fraction (simplified) has denominator with only 2 and 5 as prime factors.0.999... = 1 exactlyโ€”proof: 10x - x = 9.
Sources:Wikipedia โ€” Repeating DecimalMath is Fun โ€” Decimals to Fractions

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Decimal to Fraction ConverterTerminating and repeating decimals

๐Ÿ“ Examples โ€” Click to Load

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿงฎ Fascinating Math Facts

๐Ÿ›๏ธ

0.999... = 1 exactly. Proof: 10x - x = 9, so x = 1.

๐Ÿ“

1/7 = 0.142857142857... has a 6-digit repeating block.

1. Key Takeaways

  • โ€ข Terminating decimals: multiply by 10^n (n = decimal places), then simplify by GCD.
  • โ€ข Repeating decimals: use algebraic methodโ€”let x = decimal, multiply to shift, subtract to eliminate repeat.
  • โ€ข 0.5 = 1/2, 0.25 = 1/4, 0.125 = 1/8, 0.333... = 1/3, 0.666... = 2/3.
  • โ€ข Integers can be written as fraction with denominator 1 (e.g., 5 = 5/1).
  • โ€ข Always simplify the final fraction to lowest terms.

2. Did You Know?

Terminating Rule

A decimal terminates iff its fraction (simplified) has denominator with only 2 and 5 as prime factors.

0.999... = 1

The repeating decimal 0.999... is exactly equal to 1. Proof: 10x - x = 9.

Common Equivalents

0.5=1/2, 0.25=1/4, 0.75=3/4, 0.2=1/5, 0.125=1/8, 0.375=3/8.

Repeating Pattern

1/7 = 0.142857142857... has a 6-digit repeating block.

Mixed Numbers

1.5 = 1 + 0.5 = 1 + 1/2 = 3/2 or 1ยฝ.

Negative Decimals

-0.25 = -1/4. Apply sign to the numerator of the simplified fraction.

3. How It Works

Converting decimals to fractions involves two main cases. For terminating decimals (e.g., 0.75), count the decimal places (2), multiply by 10^n (100), giving 75/100, then simplify by GCD to get 3/4. For repeating decimals (e.g., 0.333...), use algebra: let x = 0.333..., then 10x = 3.333..., subtract to get 9x = 3, so x = 1/3.

Inputs

A decimal number (e.g., 0.75, 0.333, 1.5)

Outputs

Simplified fraction (numerator/denominator), mixed number if applicable, step-by-step solution

4. Expert Tips

Count decimal places first

For 0.375, there are 3 places โ†’ multiply by 1000 โ†’ 375/1000 โ†’ simplify to 3/8.

Recognize common patterns

0.5โ†’1/2, 0.25โ†’1/4, 0.2โ†’1/5, 0.125โ†’1/8. Memorize these for speed.

Repeating: shift and subtract

For 0.666..., 10x=6.666..., 10x-x=6, 9x=6, x=2/3.

Verify by dividing back

Check: 3/4 = 3รท4 = 0.75. Always verify your conversion.

5. Comparison Table

DecimalFractionMethod
0.51/2Terminating: 5/10 โ†’ simplify
0.753/4Terminating: 75/100 โ†’ simplify
0.333...1/3Repeating: algebraic
0.1251/8Terminating: 125/1000 โ†’ simplify
1.53/2Mixed: 1 + 1/2

6. FAQ

How do I convert 0.75 to a fraction?

0.75 has 2 decimal places. Multiply by 100: 0.75ร—100=75. So 75/100. GCD(75,100)=25. Simplify: 75รท25=3, 100รท25=4. Result: 3/4.

How do I convert repeating decimals like 0.333...?

Let x=0.333... Multiply by 10: 10x=3.333... Subtract: 10x-x=3, so 9x=3, x=1/3.

What is 0.125 as a fraction?

0.125 = 125/1000. GCD(125,1000)=125. 125รท125=1, 1000รท125=8. Result: 1/8.

Can all decimals be converted to fractions?

Rational numbers (terminating or repeating decimals) can. Irrational numbers like ฯ€ or โˆš2 cannot be exact fractions.

How do I convert 1.5 to a fraction?

1.5 = 1 + 0.5 = 1 + 1/2 = 2/2 + 1/2 = 3/2. Or: 1.5 = 15/10 = 3/2.

What about negative decimals?

Convert the absolute value, then apply the negative sign to the numerator. E.g., -0.25 = -1/4.

7. Quick Stats

10^n

Multiplier (terminating)

GCD

Simplify factor

a/b

Result format

Algebra

Repeating method

8. Sources

Wikipedia โ€” Repeating DecimalKhan Academy โ€” FractionsMath is Fun โ€” Decimals to FractionsPurplemath โ€” Percent & DecimalWolfram โ€” Repeating DecimalCommon Core Math Standards

9. Disclaimer

โš ๏ธ Warning: This calculator is for educational purposes. Decimal-to-fraction conversion uses standard mathematical methods. For repeating decimals, the result may be an approximation depending on input precision.

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