I’m practicing converting fractions to decimals and back, but I keep getting tangled up and I’m not sure what I’m missing. I can do long division, but it’s slow and I lose track of what’s repeating versus what actually stops. For example, 3/8 feels straightforward, but 1/6 or 7/12 make me second-guess whether I should simplify first or just start dividing.
I remember in school I memorized a few conversions (like 1/4 and 1/3), but I never really understood the reasoning. Whenever a denominator like 12 or 40 shows up, I stall. Sometimes I reduce the fraction first and sometimes I don’t, and I’m worried that choice is actually changing whether the decimal terminates-does it?
On the flip side, going from decimals to fractions also trips me up. If I see something like 0.0375, I try to “clear the decimal” and then simplify, but I’m not confident I’m doing it in the best order. And repeating decimals really confuse me: for 0.12(3) or 2.1(6), I’ve tried the trick of setting x equal to the decimal and multiplying by 10 or 100 to line things up, but I’m never certain which power to use or how to handle the non-repeating part cleanly. Also, does 0.30 versus 0.3 matter when turning it into a fraction, or is that a red herring?
I did try factoring denominators into primes and making a little table to spot patterns, but I’m not sure I’m applying that idea correctly. Maybe it’s relevant, maybe not.
Could someone explain, step by step, how to: (1) tell in advance if a fraction’s decimal will terminate (without doing all the long division), and whether simplifying first changes that; (2) convert a repeating decimal like 0.12(3) or 2.1(6) into a fraction reliably; and (3) handle decimals like 0.0375 or 0.300 so I end up with a fraction in lowest terms without skipping steps? I’m especially interested in the “why” behind the steps so I can stop guessing.
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3 Responses
Here’s my go-to litmus rule: after reducing, a/b terminates exactly when the denominator’s primes are only 2s and 5s-money primes, like nickels and dimes clicking neatly into a dollar-so 3/8 stops but 1/6 and 7/12 repeat; simplifying first doesn’t change that.
To go decimal→fraction, line up with powers of 10 and subtract to trap the repeat (x=0.12(3): 100x=12.333…, 10x=1.233…, so 90x=11.1 ⇒ x=111/990), and plain ones like 0.0375 = 375/10000 just cancel 2s/5s; 0.30 and 0.3 start as 30/100 vs 3/10 but simplify to the same slice-like cutting the same pizza into more pieces.
Here’s the no-drama rule for when a fraction’s decimal terminates: reduce the fraction first. Then look at the (reduced) denominator’s prime factors. If it’s only 2s and 5s, the decimal ends; if any other prime shows up, it repeats. Why? Because decimals are base 10, and 10 = 2 × 5. A fraction terminates exactly when the denominator divides some power of 10. Examples: 3/8 → denominator 8 = 2^3, so 0.375. 1/6 → denominator 6 = 2 × 3 (that 3 ruins it), so it repeats. 7/12 → 12 = 2^2 × 3, so it repeats. And yes, simplifying can change the verdict, because you cancel “bad” primes: 6/15 reduces to 2/5, and that does terminate (0.4), even though 15 had a 3 in it. If you already know the denominator is only 2s and 5s, you can skip long division: scale to a power of 10. For 3/8, multiply top and bottom by 125 to get 375/1000 = 0.375.
Repeating decimals to fractions: split into “non-repeating” (k digits) and “repeating” (r digits). The reliable move is: take all digits up to the end of the first repeat as one number, subtract the non-repeating block as another number, and divide by a denominator made of r nines followed by k zeros. Example: x = 0.12(3). Non-repeating = 12 (k = 2), repeating = 3 (r = 1). So x = (123 − 12) / (9 followed by 2 zeros) = 111/900 = 37/300. For 2.1(6), peel off the 2 first: 2 + 0.1(6). Now k = 1, r = 1 on 0.1(6): (16 − 1)/(90) = 15/90 = 1/6, so the whole thing is 2 + 1/6 = 13/6. That’s the same algebra trick you learned (multiply by powers of 10 to line things up), just packaged so you don’t guess the powers.
Terminating decimals back to fractions: write the decimal as an integer over a power of 10, then simplify. 0.0375 = 375/10000 = 3/80 (divide by 125). The “why” is the same 2s-and-5s story: 10^n only has 2s and 5s, so you’re just cancelling those. And 0.30 vs 0.3? Red herring. 0.300 = 300/1000 = 3/10, same as 0.3. Trailing zeros don’t change the value; they just mean extra 2s and 5s that get cancelled when you simplify.
– A/b in lowest terms has a terminating decimal iff b’s prime factors are only 2s and/or 5s (simplifying first doesn’t change this); to go decimal→fraction, write it over a power of 10 and reduce: 0.0375 = 375/10000 = 3/80, 0.30 = 3/10, and 3/8 = 0.375 while 1/6 or 7/12 repeat because of a factor 3.
– For a repeating decimal with a nonrepeating part, multiply to shift past both parts and subtract: 0.12(3) = (123−12)/900 = 37/300 and 2.1(6) = 13/6; see https://www.mathsisfun.com/numbers/recurring-decimals.html for a clear walkthrough.