Changing a decimal number to a fraction is a lot easier than it looks. Before we jump into the details, you might want to keep this thought in mind: a decimal number already is a fraction, so what we’re really talking about is getting it to look like the kind of fraction that you want.
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Decimals in Thirty Seconds
First, a very quick refresher course in decimal numbers. Take an ordinary whole (non-decimal) number, like 4674. The farther to the left you go, the bigger the numbers get, and the farther to the right you go, the smaller they get: the 4 at the far left means “four thousand” while the 2 at the right means just plain “four”.
What would happen if you went to the right past the ones’ place? That’s what decimal numbers are all about. You put a decimal point to the right of the ones’ place, and every step to the right you go past that is ten times smaller: tenths (0.1), hundredths (0.01), thousandths (0.001), etc. (The zero to the left of the decimal is just there to let you know that there’s a decimal point; it doesn’t change the value of the number, however.)
Just a Fraction
Notice that we said that the numbers to the right of the decimal were tenths, hundredths, etc. Those are all fractions, aren’t they? So 0.1 is the same as 1/10 — and we just converted a decimal to a fraction! Here’s another one: 0.01 is 1/100. And what’s 0.001? It’s one thousandth, which is the same as 1/1,000. Just another fraction!
Everyday numbers usually aren’t that simple. But they’re also not that difficult to turn into fractions, because the same basic rules apply.
We’ll start with something easy. Instead of 0.1, how about 0.5? If 0.1 is 1/10, then 0.5 should be 5/10, and it is. But it looks like we could do a little more with 5/10, doesn’t it? We can simplify, or reduce, it.
To reduce a fraction, you find a number that goes into both the numerator (the number on top) and the denominator (the number on the bottom) without anything left over in either case. You do the division, then look for another number that will divide evenly into the new numerator and denominator. You keep going until you can’t divide both the numerator and the denominator evenly any more.
You can divide both 5 and 10 by 5, so that’s what you do: 5 goes into 5 once, and 5 goes into 10 twice, giving you 1/2, which is as far as you can go. 0.5 is 5/10, which is really 1/2, so 0.5 is just another way of saying “one half.
Try a Few More
OK. Let’s try a few more basic decimal numbers. What about 0.4? That’s 4/10, and since both 4 and 10 can be divided evenly by 2, it reduces to 2/5. Now try 0.3; it turns into 3/10, which can’t be reduced. Some numbers are like that. Once you convert them to fractions, they won’t reduce any more.
Now look at this number: 1.2; how do you convert it? Anything to the left of the decimal is a whole number, so you treat that part as a whole number, and just convert the part to the right of the decimal point to a fraction. so 1.2 is 1 2/10, which reduces to 1 1/5.
Here’s the basic rule for longer decimal numbers:
- Count the number of digits to the right of the decimal.
- Write down that many zeroes.
- Put a 1 in front of the zeroes.
- Place the actual number to the right of the decimal over the number made up of the 1 and the zeroes.
That gives you the raw (unreduced) fraction.
Here’s an easy example. Start with 0.25. There are two digits to the right of the decimal point, so we take 00 and put a 1 in front of them, to give us 100, which is just one hundred. Place 25 over 100, giving us 25/100; you can divide both the top and bottom by 25, resulting in 1/4.
Now let’s try a number that’s a little longer, and not quite as simple; 0.462 will do. How many digits doe it have to the right of the decimal? Three, so we place a 1 in front of three zeroes, giving us 1,000, then put 462 over it: 462/1000. It looks like both the top and bottom should divide by 2, giving us 231/500, and that’s as far as you can divide them. Here’s another one: 0.16224. five digits means 1 and five zeroes as the denominator, with 16224 on top: 16224/100000. That reduces to 8112/50000, and ultimately t0 507/3125 — maybe not the handiest fraction to work with (which is one of the reasons why people use decimals), but there you have it.
There are some numbers that just have to make things difficult. For example, there are some fractions that just don’t behave well when you convert them to decimals. Take 1/3: when you convert it to a decimal, it becomes 0.333333333333…, and those threes just keep going on forever. They get smaller with each place farther to the right, so the total still adds up to 1/3, but that means that you can’t write it out all the way as a decimal — it’s always going to be approximate.
Keep That Number in Line!
So how do you convert 0.333333333333… back to 1/3? You can’t be exact about it, but you can come as close as you need to. Those three dots are there to tell you that the threes go on forever, so you can pick as many digits as you want for accuracy, then convert it to a fraction: 333333/1000000. That doesn’t look like a very good fraction. But we know it’s approximate anyway, so let’s try dividing through by 333333. That gives us 1/3 with a remainder of 1/1000000; the more digits you use, the smaller the remainder gets, so you can just treat 0.333333333333… as 1/3.
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