The Long Division Method
Any fraction p/q is really just p divided by q. So to get the decimal, you do the division. Yeah, long division. I know. But it's actually not that bad once you see it a couple of times. Even when the numerator is smaller than the denominator, you just place a decimal point after it, add a zero, and divide as you normally would.
Work through 3/8 as an example:
3.000 ÷ 8
8 goes into 30 → 3 times (8 × 3 = 24), remainder 6
8 goes into 60 → 7 times (8 × 7 = 56), remainder 4
8 goes into 40 → 5 times (8 × 5 = 40), remainder 0
3/8 = 0.375
Same process for any fraction. Hit a remainder of zero and the decimal stops right there. But if the remainders start cycling, the decimal just keeps going in a pattern. We'll cover that in a second.
Got a mixed number like 2 3/8? Just convert the fraction part. Leave the whole number alone. So 2 3/8 = 2 + 0.375 = 2.375. That's all there is to it.
Terminating vs. Repeating Decimals
Here's the thing: every fraction gives you one of two types of decimal. Either it terminates (it ends cleanly) or it repeats (a digit or block of digits cycles forever). Which one you get comes down to the denominator, after you've fully simplified the fraction.
You'll get a terminating decimal when the denominator's only prime factors are 2 and 5. That covers denominators like 2, 4, 5, 8, 10, 16, 20, and 25. Those always end.
Any other prime factor in the denominator, a 3, a 7, an 11, and so on, gives you a repeating decimal. No exceptions.
1/4 = 0.25 (denominator = 2²)
7/20 = 0.35 (denominator = 2² × 5)
Repeating examples:
1/3 = 0.333… = 0.3̄
5/6 = 0.8333… = 0.83̄
1/7 = 0.142857142857… = 0.1̄4̄2̄8̄5̄7̄
Repeating decimals get a dot or bar written above the repeating block. For 1/7, all six digits cycle, so the bar stretches over all six of them.
Common Fraction-to-Decimal Conversions
Honestly, just learn this table. It'll save you a ton of time on tests, and once these are in your head, mental estimation becomes way easier. These are the ones that come up constantly.
| Fraction | Decimal | Terminates? |
|---|---|---|
| 1/2 | 0.5 | Yes |
| 1/3 | 0.333… | No - repeats |
| 2/3 | 0.666… | No - repeats |
| 1/4 | 0.25 | Yes |
| 3/4 | 0.75 | Yes |
| 1/5 | 0.2 | Yes |
| 2/5 | 0.4 | Yes |
| 3/5 | 0.6 | Yes |
| 4/5 | 0.8 | Yes |
| 1/6 | 0.1666… | No - repeats |
| 1/8 | 0.125 | Yes |
| 3/8 | 0.375 | Yes |
| 5/8 | 0.625 | Yes |
| 7/8 | 0.875 | Yes |
| 1/9 | 0.111… | No - repeats |
| 1/10 | 0.1 | Yes |
| 1/12 | 0.0833… | No - repeats |
Notice the pattern: halves, quarters, fifths, eighths, and tenths all terminate because their denominators are built from 2s and 5s. Thirds, sixths, ninths, and twelfths? They all repeat because 3 sneaks into the denominator.
Converting a Decimal Back to a Fraction
Going the other way relies on place value. Read the decimal, write it as a fraction over the matching power of 10, and then simplify. That's the whole method.
For a terminating decimal:
0.6 = 6/10 → simplify → 3/5
Convert 0.48 to a fraction:
0.48 = 48/100 → simplify → 12/25
Convert 0.375 to a fraction:
0.375 = 375/1000 → simplify → 3/8
Quick rule: one decimal place means denominator 10, two places means 100, three places means 1000, and it keeps going like that.
For a repeating decimal, you need a slightly different move. Set x equal to the decimal, multiply to slide the repeating part past the decimal point, then subtract to cancel it out.
Let x = 0.333…
10x = 3.333…
10x − x = 3.333… − 0.333…
9x = 3
x = 3/9 = 1/3
Simplifying the Resulting Fraction
Once you've converted a decimal to a fraction, you'll almost always need to simplify it. A fraction is in simplest form when the numerator and denominator share no common factor other than 1.
To simplify, find the Greatest Common Factor (GCF) of the numerator and denominator, then divide both numbers by it.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
GCF = 12
36 ÷ 12 = 3 | 48 ÷ 12 = 4
Result: 3/4
If you're not sure what the GCF is, don't overthink it. Just divide by small primes one at a time: try 2, then 3, then 5. Keep going until nothing divides cleanly anymore. You'll get there.
Want to make sure you got it right? Convert the simplified fraction back to a decimal with long division. If it matches the decimal you started with, you're good.