Now that we know the basics of how decimal numbers work,
let’s see how we can write some special fractions using decimal numbers.
I’m going to call these fractions “Base 10 fractions”
because their bottom numbers are all ‘powers of 10’, like 10, 100, or 1,000.
Let’s start with this fraction: one over ten.
You should recognize that. It’s one of our building blocks.
And this should be easy to write as a decimal number because we have a number place just for counting tenths.
So all we have to do is put a ‘1’ in the tenths place like this: zero point one
Now when you write decimal numbers, it’s important that you always include the ones place.
But since we don’t have any ones, we just put a zero in that spot.
The zero makes the decimal point easier to see.
Alright… so that’s one tenth, but what if we have 2 over 10 instead?
All we do is change the digit in the tenths place to a ‘2’.
So 2 over 10 equals 0.2
In fact, we can keep counting tenths like this…
3 tenths, 4, 5, 6, 7, 8, 9, and finally 10 tenths.
But look what happened when we got to ten tenths.
We don’t have a digit for 10, so we had to use the next number place over: the ones place.
But that makes sense because if you have the fraction 10 over 10, that makes a whole and the value is just ‘1’.
Of course we don’t really need the zero in the tenths place to write ‘1’,
but as long as the decimal point is there, at least we won’t confuse it with 10.
Alright, tenths are pretty easy, but what about hundredths?
Let’s start with the hundredths building block: 1 over 100.
To write that as a decimal, we simply put a ‘1’ in the hundredths place.
We also need to put a zero in the tenths place to act as a place-holder and show that we have no tenths.
And we still need to put a zero in the ones place as usual.
Next let’s try ‘2’ hundredths. For that, we simply put a ‘2’ in the hundredths place.
Let’s keep on counting with hundredths, just like we did for tenths…
3 hundredths, 4, 5, 6, 7, 8, 9, and 10 hundredths.
Ah, but look what happened when we got to 10 hundredths.
Just like before, we have to use the next number place to the left …the tenths place.
This happens because any time you have ten of a building block,
they combine to form one of the next biggest building block.
For example… ten hundredths is a tenth.
ten tenths is one.
ten ones is ten.
and ten tens is a hundred.
Now the next fraction after 10 over 100 is 11 over 100.
If you think about it, you’ll see that eleven-hundredths is really just a combination of ten-hundredths and one-hundredth.
Knowing that will help us write it as a decimal.
Because a group of 10 hundredths is equal to 1 tenth, we put a ‘1’ in the tenths place.
And we still have that 1 hundredth left over, so we put a ‘1’ in the hundredths place.
There, 11 over 100 is just 0.11 as a decimal.
Fortunately, you don’t have to break up the fraction into tenths and hundredths each time.
Any time you have a 2 digit number over 100, all you have to do is put those digits in the tenths and hundredths places of your decimal number.
Let’s look at a few more examples to help you see the pattern.
24 over 100 would be 0.24
32 over 100 would be 0.32
78 over 100 would be 0.78
and 99 over 100 would be 0.99
Now, what do you think will happen if we convert the fraction 100 over 100 into a decimal?
Right, 100 has 3 digits, so we need to use another number place. Now the next one over is the ones place.
That makes sense because 100 over 100 is a whole, and its value is just ‘1’.
Now that we know how to convert hundredths into decimals, let’s try converting thousandths.
That’s fractions that have 1,000 as the bottom number.
Let’s start with 1 over 1,000. Now this should be easy.
All we have to do is put a ‘1’ in the thousandths place.
Notice that this time we need zeros in both the tenths and the hundredths place to act as place holders.
Next, let’s try converting 10 over 1,000.
Remember that 10 thousandths is the same as 1 hundredth,
so we we will put a ‘1’ in the hundredths place and we’ll put a zero in the thousandths place.
We don’t really need the ‘0’ at the end, but it helps us see that this was 10 thousandths.
Alright, what if we have 100 over 1,000.
Now that’s a three digit number on top, so we’re going to need to use three number places:
the thousandths place, the hundredths place, and the tenths place.
So as you can see, 100 over 1,000 is just the same as one tenth.
Let’s see a few more examples…
58 over 1,000 is 0.058.
73 over 1,000 is 0.073
365 over 1,000 is 0.365
and 999 over 1,000 is 0.999
And finally, what do you think we’d get if we converted 1,000 out of 1,000 ?
…right again! 1,000 over 1,000 is just a whole, so its value would be ‘1’.
Okay, so we’ve learned how to convert base 10 fractions into decimals,
but we can go the other way too. We can start with a decimal and convert it into a fraction.
Let’s say we want to convert a decimal number into a fraction.
All we have to do is take the decimal digits and make them the top number of a base 10 fraction.
The bottom number will be determined by the smallest number place used in our decimal.
For example, to convert 0.8 into a fraction, we put an 8 on the top, and a 10 on the bottom,
because the smallest number place in our decimal was the tenths place.
And to convert 0.29 into a fraction, we put a 29 on top and we put 100 on the bottom,
because the smallest number place in our decimal was the hundredths place.
And finally, to convert 0.568 into a fraction, we put 568 on top and 1,000 on the bottom,
because the smallest number place in our decimal was the thousandths place.
Okay… so far, all of the fractions that we’ve converted to decimal numbers (and vice-versa) have bottom numbers like 10, 100, or 1,000.
Those fractions are easy to convert, because our number system is based on ‘powers of 10’.
We have number places specifically for counting those.
But what if we want to take fractions with different bottom numbers
like 1/2, 3/4, or 8/25, and write those as decimal numbers?
We don’t have special number places for halves, fourths or twenty-fifths,
so what are we going to do?
Well, you’re going to have to watch the next section to find out.
But first let’s take a minute and review all this.
If a fraction has a bottom number that is a power of 10,
then it’s easy to convert it into a decimal number because there are number places just for counting base 10 fractions.
To convert tenths, all you have to do is put the top number in the tenths place.
To convert hundredths, you have to use both the tenths and hundredths place together.
To convert thousandths, you have to use three number places, and so on…
You can also convert from a decimal number to a fraction just by making the decimal digits the top number of the fraction,
and by using a bottom number that’s based on the smallest number place from our decimal.
Be sure to do the exercises so you get really good at converting base 10 fractions.
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