Decimals are a way to represent fractions. Decimals and fractions both
represent values in-between two whole numbers. You may already know that between
the numbers 2 and 3 there are billions and billions of other numbers. In fact,
the actual amount of numbers between 2 and 3 is endless. This is true for any
other two consecutive whole numbers as well. Between each of these numbers
are countless other numbers that are represented by a fractional or decimal
value. For instance, between 9 and 10 there is: 9.2, 9.22, 9.345, 9.55046, and
9.999012. The list goes on forever!
These numbers allow us to make more accurate
calculations when the need arises. When you have a birthday and your mom makes a
cake, if we only had whole numbers, you could only tell someone when the cake was
gone or when none of it had been eaten yet. You couldn't say, "The cake is half eaten."
because "half" is not a whole number. One half is the same as the decimal ".5".
It allows us to more accurately describe how much of the cake is left. Without
it we would have to say, "There is still some cake, but it's not a whole cake
anymore." In other words, the cake is greater than 0, (or no cake), but less than
1 cake. That isn't a good description. One half of a cake is much more accurate.
Decimals are more "friendly" to use for most people than fractions.
They are easier to enter into a calculator also. Calculators convert fractions
into decimals before they can use them. If you enter "1/3" into most calculators,
it spits out ".33333333" which is the decimal form of "1/3" or one third. One
third is what's known as a "repeating decimal." Those 3's go on and on forever,
because 1/3 can't be completely accurate when represented by a decimal. We have
to round it off somewhere so that we are able to write it down. If we're talking
about a cake, .33 should be accurate enough. If we're talking about a chemical
used in a laboratory, we might need to represent it more accurately like
.33333333.
If you're working on something important or a long problem that
requires lots of other calculations before you get to your final answer, the
accuracy of the decimal place becomes a very important consideration. Take the
problem .3 x .3 x .3. If we multiply these three together we get .027. If
we get more accurate and multiply .333333 x .333333 x .333333 together we get
0.037036925926037037. There is quite a bit of difference in these two answers! It's
all because of the accuracy we used in our decimal representation of 1/3. We'll
never be able to say that 1/3 = .333333, no matter how many
decimal places we add. All we can accurately say is that 1/3 is approximately equal to .333333.
Repeating decimals come in many forms. Some, like .333333 are easy to spot.
Others, like .657657657 are not as easy. We have to look for a pattern that
repeats itself. When we find a pattern like that, we have to round it off somewhere
according to the accuracy we need. Why round it you say? If we don't round it,
we'll never be able to write it down, because repeating decimals always keep
repeating the same pattern over and over forever! So where do we want to round it?
We have to decide how accurate the answer needs to be. In most cases 3 or 4 places
is fine. In other cases you might need to go several decimal places before you
round. It all depends on how accurate you want your final answer to be.
To round .333333 to three decimal places you would write .333. To round .666666 to
three decimal places you would write .667. Why the 7? If the number is 5 or greater,
you round up. If the number is smaller than 5, you round down. Since the third
decimal place was 3 in .333333, we rounded down. Since the third place in .666666
was 6 we rounded up to 7.
Please examine the picture below very carefully. It shows how you can "expand"
a decimal number into an addition problem. It also shows the value of each place
in the number 123.456789. You're probably already familiar with the "hundreds,"
"tens," and "ones" places. To the right of the decimal point you'll notice some
new places. Why do decimals start with the "tenths" place instead of the "oneths"
place? Good question! The reason is because 1/1 is the same as 1, which is in the
ones place. The next place value to the right of that, determined by our "base 10"
numeral system is 1/10. 1/10 is one "tenth" NOT one "oneth." There is no such thing
as "one oneth." For this reason, decimal values beginning to the right of the
decimal place start at the "tenths" place. From there they go to "hundredths,"
"thousandths," "ten thousandths," and so on. With everything we've talked about
in mind, please take a long careful look at the picture below. It reveals many
important things concerning decimals.
To convert a decimal into a fraction, look at the last decimal
place. Determine what value that place is and write it as the
denominator, (or bottom portion of the fraction). Next, take the
numbers to the right of the decimal and write them as the
numerator, (or top portion of the decimal). In the picture below
I've converted .333 to a fraction. I looked at the last decimal
place, which was the "thousandths" and wrote it as the denominator
in step 2. In step 2 I also took those numbers that were to the
right of the decimal and wrote them as the numerator. .333 as a
fraction is 333/1000 or 333 thousandths. Study the picture to
see how it's done.
Converting a fraction to a decimal takes a little more time.
We have to divide the fraction out. In the example below I
converted 1/3 to the decimal .333. In step 2 I tried to divide
1 by 3, but that didn't work very well. A calculator would have
done it without any problems, but since we don't always have
calculators handy, I wanted to do it with long division. In
order to divide this problem out, I needed to add some 0's.
Dividing 10 by 3 looks a lot nicer than dividing 1 by 3.
In step 3 I added 3 zeros because I decided that I wanted my
answer to be accurate to 3 decimal places. I put the decimal
right after the 1 because 1 is the same as 1.0 or 1.00 or so
on. When I did my dividing, I ignored the decimal and divided
10 by 3 instead of 1.0 by 3 since that would have been the
same problem I had before. In step 4 I finished working the
problem out and found that the 3's were going to keep repeating
themselves. I only wanted my decimal to be accurate to 3 places
so .333 was fine.
For the final answer, I brought the decimal
straight up and put it in the same location directly above in
the answer. In step 5 the line over my answer of .333 shows
that this portion of the decimal repeats itself. This is one
way of showing the repeating portion of a decimal value.
Examine the picture below to see how it's done.
Adding with decimals is actually pretty easy. All you do is line up the decimals,
add any needed 0's like I did in step 2 below, and add. It's that simple. Take a look
at the picture below to see what I mean.
Subtracting with decimals is just as easy as adding. You line up the decimals,
add any needed 0's like I did in step 2 below, and subtract. It's not any harder
than that! Again, check out the picture below to see how it's done.
Multiplying with decimals is pretty easy too. Set your problem up like any other multiplication problem,
ignoring the decimals, and multiply everything out like I did in step 2 below. When you have your answer, count
all of the places to the right of the decimal in both numbers you multiplied together. In the answer,
counting from the right, count that many places over and put the decimal there. For the problem I worked
below, 23.4 had 1 place to the right of the decimal. .34 had 2 places to the right of the decimal.
Since I needed the total places in both items I multiplied together, I added the 2 and the 1 together
and got 3. In my answer of 7596, I counted those 3 places from the right and put my decimal there.
My final answer was 7.596. Study the picture below to see how it's done.
We've already worked one problem where we divided using decimals. In this one we'll go a step further and divide
two numbers that both have decimal places. The problem I chose to work below is 9.27/.3. Both numbers we're working
with have decimal places in them. What we need to do is get rid of those decimal places in the .3. We always want
to get rid of the decimal places in the number that we're dividing by. In this case the number is .3. We do that by
moving the decimal however many places over to the right it takes until all of the decimal places are gone. Next we
have to move the decimal in what we're dividing into the same exact amount of places.
In step 2 below I moved the decimal in .3 to the right one place to get rid of it. I had to move the decimal the same
amount of places in 9.27 also. Moving the decimal to the right one place in 9.27 changed it to 92.7, as shown in step 2 also.
I can now go ahead and place the decimal straight up above for my answer. When dividing decimals, the decimal in the answer
will always be directly above the decimal in what we're dividing into, as show in step 2 below. From here we just ignore
the decimals and divide everything out just like we have always done. Study the picture below carefully to see what
I'm talking about.
We'll conclude this lesson talking about percents. Percents are a way of saying, "out of one hundred."
If you wanted to know what 100% of 2 was, you would just multiply 2x1=2. 100% of 2 is 2! If you wanted to
know what 50% of 2 was, you would multiply 2 by .50, (or one half). 50% of 2 is 1. "Out of one hundred" can
be thought of like the fraction 100/100, which would be 100 out of 100. 100 out of 100 is 100%. As a
decimal it is 1.0. The fraction 50/100 would be 50 out of 100. 50 out of 100 is 50%. As a decimal it is .50.
To find 25% of something just multiply it by .25.
As you're starting to see, to write a decimal as a percent
you just move the decimal in a decimal number like .25 two places to the right and add a percent sign.
To write a percent as a decimal you just reverse things. Remove the percent sign and move the decimal two
places back to the left. What would happen if we didn't know "out of one hundred" but wanted to find the percentage
"out of one hundred" of something?
Let's take a boy shooting basketball. We all know that boys can't shoot
basketball as good as girls right? Just kidding guys! The boy makes 12 shots. He misses 3 of the shots but makes
9. He made 9 out of 12 shots. This can be written as 9/12. Notice that this is a fraction. We just learned
how to convert fractions to decimals, so let's convert it. I got 9/12=.75. We already have what we need, but let's
go one step further just to see what happens.
Let's convert .75 back to a fraction. We learned that earlier
also. It was pretty easy really. Converting .75 back to a fraction gives us 75/100 or 75 "out of one hundred."
If you review what we talked about earlier when we converted a decimal to a fraction, it should help explain
how this works. We already know that 75 out of 100 is actually 75%. But wait you say! 75/100 is not the same
as 9/12! Hold on a second. Try reducing 75/100 and see what you get. You should get 3/4. If you reduce 9/12
you should get the same thing. Converting a fraction like 9/12 to a decimal allows us to also convert it to
a number that is "out of one hundred."
Thanks for hanging in there! All we have left is the quiz! Please review everything and make sure you understand
it completely. Decimals aren't really that much different than the numbers you've always worked with. We just have
to keep an eye on that little decimal, making sure that we handle things properly when we work problems where it's
involved. Good luck on the quiz!
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