How to convert a decimal to a fraction
To convert a decimal to a fraction, write the decimal as a numerator over a denominator based on how many decimal places the number has, then reduce the fraction to its simplest form. For example, 0.5 becomes 5/10, which reduces to 1/2. The process works the same way whether you have one decimal place or five — you just adjust the denominator accordingly.
The key is understanding that each decimal place represents a power of 10. One decimal place means tenths (10), two decimal places mean hundredths (100), three mean thousandths (1000), and so on. Once you place the decimal number over the correct denominator, you find the greatest common factor of both numbers and divide both by it to get the simplest fraction.
Key Takeaways
- Count the number of digits after the decimal point to determine your denominator: one digit = 10, two digits = 100, three digits = 1000.
- Write the decimal number without the decimal point as your numerator, and use the denominator you found in step one.
- Find the greatest common factor (the largest number that divides evenly into both the numerator and denominator) and divide both by it.
- The result after dividing by the greatest common factor is your fraction in simplest form.
Step-by-step conversion process
Step 1: Count the decimal places. Look at your decimal number and count how many digits appear after the decimal point. If you have 0.75, there are two decimal places. If you have 0.125, there are three decimal places.
Step 2: Create the denominator. Use the number of decimal places to determine your denominator. One decimal place = 10, two decimal places = 100, three decimal places = 1000, four decimal places = 10,000. The pattern continues — each additional decimal place adds a zero.
Step 3: Write the fraction. Take the decimal number and remove the decimal point. This becomes your numerator. Place it over the denominator you found in step 2. For 0.75, you write 75/100. For 0.125, you write 125/1000.
Step 4: Reduce to simplest form. Find the greatest common factor of the numerator and denominator — the largest whole number that divides evenly into both. For 75/100, the greatest common factor is 25. Divide both 75 and 100 by 25 to get 3/4. For 125/1000, the greatest common factor is 125, giving you 1/8.
Finding the greatest common factor
The greatest common factor is the largest number that divides evenly into both your numerator and denominator with no remainder. One way to find it is to list all the factors of each number and pick the largest one they share. For 75, the factors are 1, 3, 5, 15, 25, and 75. For 100, the factors are 1, 2, 4, 5, 10, 20, 25, 50, and 100. The largest factor they both have is 25.
Another method is to use prime factorization — break each number into its prime factors and multiply the ones they have in common. For 75, the prime factors are 3 × 5 × 5. For 100, the prime factors are 2 × 2 × 5 × 5. They share two 5s, so 5 × 5 = 25 is the greatest common factor. Either method works; use whichever feels more natural to you.
Working with repeating decimals
Some decimals repeat forever, like 0.333... (which equals 1/3) or 0.666... (which equals 2/3). These are harder to convert because you cannot simply count decimal places. For a repeating decimal, you use algebra to find the fraction.
Let x equal your repeating decimal. Multiply both sides by 10 (or 100, or 1000, depending on how many digits repeat). Subtract the original equation from the new one to eliminate the repeating part. For example, if x = 0.333..., then 10x = 3.333.... Subtracting gives 9x = 3, so x = 3/9, which reduces to 1/3. This method works for any repeating decimal pattern.
Converting mixed numbers and whole decimals
A decimal like 2.5 contains both a whole number and a decimal part. Convert the decimal part to a fraction first (0.5 becomes 1/2), then write the whole number next to it as a mixed number: 2 1/2. If you need an improper fraction instead, multiply the whole number by the denominator and add the numerator. For 2 1/2, that is (2 × 2) + 1 = 5, so the improper fraction is 5/2.
A decimal like 3.0 is simply the whole number 3, which can be written as 3/1 if you need it in fraction form. Decimals that end in .0 have no fractional part, so the fraction is just the whole number over 1.
Common decimal-to-fraction conversions
| Decimal | Fraction (Simplest Form) |
|---|---|
| 0.5 | 1/2 |
| 0.25 | 1/4 |
| 0.75 | 3/4 |
| 0.2 | 1/5 |
| 0.4 | 2/5 |
| 0.6 | 3/5 |
| 0.8 | 4/5 |
| 0.125 | 1/8 |
| 0.375 | 3/8 |
| 0.625 | 5/8 |
| 0.333... | 1/3 |
| 0.666... | 2/3 |
Frequently Asked Questions
What if the decimal does not reduce further?
Some fractions are already in simplest form. For example, 0.3 becomes 3/10, and there is no number larger than 1 that divides evenly into both 3 and 10, so 3/10 is already fully reduced. Check by finding the greatest common factor — if it is 1, the fraction cannot be simplified.
How do I convert a decimal like 0.05 that has a zero after the decimal point?
Count all the digits after the decimal point, including zeros. For 0.05, there are two decimal places, so the denominator is 100. The numerator is 5 (drop the leading zero), giving you 5/100. The greatest common factor of 5 and 100 is 5, so divide both by 5 to get 1/20.
Can I convert a very long decimal like 0.123456?
Yes. Count six decimal places, so your denominator is 1,000,000. Your numerator is 123,456, giving you 123,456/1,000,000. Then find the greatest common factor and reduce. This works for any decimal length, though the numbers become larger and reduction takes more steps.
What is the difference between a terminating and repeating decimal?
A terminating decimal ends, like 0.5 or 0.125. A repeating decimal goes on forever, like 0.333... or 0.142857142857.... Terminating decimals convert using the standard method. Repeating decimals require the algebra method described in the section above.