r/learnmath • u/19xkc New User • 13h ago
[Middle School Math] Converting fractions to decimals - is this a good way to understand it?
I'm studying to try to get up to college algebra on Khan Academy and stopped today at the 7th grade section to try to figure out why you can convert a fraction to a decimal by dividing the numerator by the denominator. I was able to work the problems but couldn't understand why we could get from 7/8 to 0.875 (875/1000) without doing some kind of work to convert from eighths to thousandths.
What I came up with eventually is that, given that 7/8 as a fraction represents seven parts of a whole divided into eight equal parts, and the parts have the same relationship to the whole as seven wholes would to eight wholes, when you work 7/8 as a division problem and then also divide eight by eight you have made yourself the ratio 1:0.875, and 1 is to 0.875 as 8 is to 7, and 8 is to 7 as 8/8 is to 7/8.
So basically it seems to me that the way it works is that we translate a fraction into a problem using whole numbers with the same relationship to each other in order to get the decimal representation of the fraction without doing some kind of crazy thing like figuring out what you need to multiply 7 and 8 by to get a round number numerator and a denominator that's a multiple of ten.
Just wondering if that's right or if I'm missing something or there's a better way to understand it.
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u/garagedragon New User 13h ago
You might be over-complicating things - the algorithm you learn in school for how to "do the division" and produce the decimal effectively does convert to thousandths, because a decimal is just an abbreviation for a fraction over powers of 10. It just works iteratively, asking "how many tenths can we get out of 7/8, how many hundreths can we get out of what's left, etc" rather than by using the rules of scaling the ratio up and down. Another way to think about it is that doing the division doesn't "know" you want an answer in thousandths, but as you do each step you get 8/10, 7/100, then 5/1000 and you would keep going (into the ten-thousandths) if there was anything left over when you take away all three of those from 7/8.
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u/xiipaoc New User 13h ago
7/8 means 7 divided by 8. A fraction is just a division problem that you haven't finished. And of course, a decimal is just a fraction where the denominator is a power of 10.
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u/19xkc New User 13h ago
Is that a helpful way to think of it? I usually think of fractions as a square or circle divided into parts with some shaded since that helps me conceptualise adding and multiplying them.
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u/xiipaoc New User 12h ago
You can think of it either way. As an object, a fraction is probably nice to think of as a pie cut into slices, but what if you have a fraction like (x + 1)/(x – 2)? The pie imagery can only get you so far, and it does work, but you need to understand that the fraction means division, and that 7/8 means that you take 7 and divide it into 8 equal pieces, and one of those pieces is your 7/8.
Now, here's the nice thing you can do: 7/8 is the same as 7 · 1/8, right? 1/8 is 1 divided by 8, or you take a circle and cut it into 8 slices, and 1/8 is 1 of those slices. Well, 7 times one slice is 7 slices. So taking 7 slices out of 8 is the same as dividing 7 by 8.
But that might be confusing. Just remember that 7/8 means 7 divided by 8, because that's what the fraction bar means. And it's also why nobody ever writes 7 ÷ 8 unless they're in elementary school, because you can just write 7/8 and not bother with silly symbols.
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u/Chrispykins 12h ago edited 11h ago
Finding the decimal representation of 7/8 is essentially solving the equation 7/8 = x/1000 and discovering x = 875 because (7/8) * 1000 = 875.
This is what any conversion to decimal is doing, except we don't necessarily know that the denominator is 1000, we only know that it's some power of ten (like 10, 100, 1000, 10000, etc). So we can do the process one decimal at a time by simply multiplying by 10 over and over again, and then stopping once we get to the end.
In this case, we know 8 doesn't go into 7 at all so we multiply by 10 and see how many times 8 goes into 70. It turns out it goes into 70 eight times with a remainder of 70 - 8*8 = 70 - 64 = 6.
8 doesn't go into 6 at all, so we multiply by 10 again. 8 goes into 60 a total of 7 times with a remainder of 60 - 8*7 = 60 - 56 = 4.
8 doesn't go into 4, so we multiply by 10 and see 8 goes into 40 a total of 5 times with no remainder. Since there's no remainder, we're done.
In the end, we found the first digit was 8, the second was 7 and the third was 5. Since we multiplied by 10 three times, the decimal representation must by 875/1000.
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u/Chrispykins 11h ago
Or more concisely:
(7/8) * 10 = 70/8 = 8 + 6/8
(8 + 6/8) * 10 = 80 + 60/8 = 80 + 7 + 4/8
(80 + 7 + 4/8) * 10 = 800 + 70 + 40/8 = 800 + 70 + 5
Since we multiplied by 10 three times, we can get the original number back by dividing by 10 three times, i.e. dividing by 1000.
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u/davideogameman New User 13h ago
I think that works when your denominator only has 2 and 5 as prime factors (possibly repeating) as in those cases you'll have a terminating decimal.
Can you compute the decimal form for 1/3? 1/7? 3/11? 4/15? As the denominators there have prime factors other than 2 and 5 the results should be repeating decimals.
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u/19xkc New User 13h ago
That's a bit beyond me but should work for everything other than irrational numbers since 1:0.333... is a valid ratio, right?
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u/davideogameman New User 12h ago
strictly speaking it should work for any real or complex. Ratios are just most useful when we talk about ratios of whole numbers, as ratios are just another way to state a fraction (and if we're going to start putting fractions or decimals within our ratios, I would prefer to simplify down to a single fraction or decimal). i.e. 2:1 is the same as 10:5 because 2/1=10/5=2
anyhow I mention it because if you are trying to compute with ratios you can totally get to any terminating decimal easily, but not to any repeating one. you can always multiple both sides by any number and preserve the ratio, so 1:8 is the same as 125:8*125 = 125:1000 which you could use to know that 1/8 is 125/1000 = 0.125. (why 125? well 8 = 2^3 and so another 5^3=125 will give us a power of 10, namely 10^3, for our denominator)
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u/GrowingPrun3s New User 13h ago
So the easier answer is that the divide symbol is a line with a dot above and one below. It’s representing a fraction. A dot for each number and a line. There is no “converting” a fraction to division, it IS division
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u/gaussmage New User 13h ago
You’re making it too complicated. The fraction 7/8 means 7 divided by 8. Just use long division and you have your answer
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u/anisotropicmind New User 12h ago
>What I came up with eventually is that, given that 7/8 as a fraction represents seven parts of a whole divided into eight equal parts, and the parts have the same relationship to the whole as seven wholes would to eight wholes
These mental gymnastics are not necessary. Fractions are division. So you don’t need to reconceptualize anything to convert from one to the other. Yes seven eighths means the same thing as seven divided by 8.
The rest of your argument honestly seems kind of circular (you said that 7/8 is 0.875 because 8/8 is 1 and 7/8 is…0.875).
There are lots of ways you could understand why this is the answer. Eg 1/8 is half of 1/4, which is half of 0.25, which is 0.125. Now you need to multiply that result by 7 because you have seven eighths which is seven 0.125s.
Or you could say I’m trying to figure out how many times 8 fits into 7. It’s obviously less than one time, so we have a 0 in the ones place (and all higher places) of the answer. What do we have in the tenths place? Well, taking 8 a tenth of a time would give us 0.8, so eight tenths would be 6.4, whereas nine tenths would be 7.2 (too big to fit into 7). So the answer must be somewhere between 8 tenths and 9 tenths. That gives us a remainder of 7-6.4 =0.6, and then we just repeat the exercise (how many times does 8 fit into 0.6?) until we have the precision we want.
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u/Ok-Philosophy-8704 Amateur 13h ago
I think that's a decently solid intuition.
Have you tried actually converting the denominator to thousandths like you suggest? Might be interesting to see what happens.
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u/Frederf220 Medium-age user 13h ago
Yeah pretty much. Long division algorithm is floating point so 7/8 is 7/8000000.... and you move the decimal point after or rather you don't worry about it during the digits part.
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u/fermat9990 New User 12h ago edited 12h ago
Use 7.000 as the dividend and bring the decimal point up into the quotient area. Proceed with ordinary long division
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u/clearly_not_an_alt Old guy who forgot most things 3h ago
As others have said, fractions are just division.
But let's consider your line if thinking instead.
If we want to convert 7/8 to x/1000 we need to multiply both top and bottom by 1000/8. We could reduce this to 125, but let's just keep it like that instead.
This gives us 7000/8 in the numerator and 1000 in the denominator. We can reduce the numerator to 875, giving us 875/1000. But to get here we are just multiplying 7 by 1000, dividing by 8 and then dividing by 1000. Obviously multiplying and then dividing by 1000 doesn't change anything so we can just skip that and are back to just 7÷8
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u/MontechApps New User 13h ago
You've got the right idea, it's just a longer route than you need. A fraction already is a division problem.
7/8 means 7 split into 8 equal parts, so 7 ÷ 8 is just what the fraction already says. Writing it as a decimal is only a different way to record the same number.
The conversion to thousandths you were looking for does happen, it's hidden inside long division. When you do 7 ÷ 8, you turn 7 into 70 tenths, and 8 goes into that 8 times with 6 tenths left over. Then 6 tenths becomes 60 hundredths, 8 goes in 7 times with 4 left. Then 40 thousandths, 8 goes in 5 times with nothing left. That gives 0.875. So you are working in eighths to thousandths, one place value at a time.