Hi, I’m Rob. Welcome to Math Antics!
In this lesson, we’re gonna learn about Mixed Numbers and how they relate to Improper Fractions.
Mmmm… mixed numbers. I love mixed numbers!
…so full of protein and vitamins.
Do you want some?
Uh… those are Mixed Nuts… and no thanks,
I’m trying to explain Mixed NUMBERS.
Ohhhh… Mixed NUMBERS.
You mean like when I get a whole bunch of numbers and put ‘em in a bag and shake it up
and pull out one to see who wins? …wanna play?
Um… that’s a raffle and that’s not what mathematicians mean when they say Mixed Numbers…
[sigh] Well you sure are hard to please today.
As I was saying… we’re gonna learn how Mixed Numbers relate to Improper Fractions,
but first we need to make sure that you understand what an Improper Fraction really is.
To do that, let’s start with a short number line that counts whole numbers from 0 to 3.
And let’s sub-divide the spaces in between each whole number into 4 equal parts (or fourths).
So a block this size represents the amount 1 because it covers the distance from 0 to 1.
While a block this size represents the fractional amount 1/4
because it only covers the distance from 0 to the first sub-division that we made
(which is 1 out of 4 or one-fourth).
But suppose we add another fourth to the fourth we already have…
That would give us TWO-fourths.
And adding another fourth would give us THREE-fourths,
and adding another fourth would give us FOUR-fourths.
What’s four-fourths?
Yep… it’s what I like to call a “Whole Fraction” because its value equals 1 whole.
Oh, and by the way… we’re gonna be using the term “Whole Fraction” a lot in this video.
It’s not an ‘official’ math term so your teacher might not use it.
But just remember that whenever I say “Whole Fraction”
I’m talking about ANY fraction that has the exact same number on the top and on the bottom.
two over two,
eight over eight,
one hundred over one hundred,
Those are all what I call “Whole Fractions”.
Since all Whole Fractions equal 1, you can replace any Whole Fraction with 1
and… you can replace 1 with any Whole Fraction.
And you can see how that works on our number line.
The four-fourths all combine to cover the same distance as 1.
The fraction 4 over 4 is equivalent to 1 and vice versa.
But now, what if we get a little crazy and add one more fourth?
Now we have FIVE-fourths and we’ve gone past 1 on the number line.
Five-fourths (or 5 over 4) is what we call an Improper Fraction
because the Numerator (the top number) is greater than the Denominator (the bottom number).
That means its value is greater than 1.
In this particular case, how much greater than 1 is it?
Well… on the number line, we’ve gone past 1 by the fractional amount one-fourth.
So five-fourth turns out to be equivalent to 1 and one-fourth.
Have a good look at this diagram for a minute because it shows us something really important
about how Proper Fractions, Improper Fractions, Whole Fractions and Mixed Numbers all relate to each other.
First, on this side of the number line we have five-fourths, which we know is an Improper Fraction.
But notice how we got it…
we added a Proper Fraction to a Whole fraction.
When we just had four-fourths, we had a Whole Fraction.
But when we added one more fourth, it became the Improper Fraction 5 over 4.
So one way to think of an Improper Fraction is that it’s a combination
of one or more Whole Fractions and a Proper Fraction.
That’s helpful because we know that ALL Whole Fractions can be simplified to the whole number 1.
And if we did that, we’d get what’s shown on the OTHER side of the number line.
We’d get the combination of the whole number ‘1’ and the proper fraction one-fourth.
In other words, we’d get a “Mixed Number”.
That’s all a Mixed Number is!
It’s the sum of a Whole Number and a Proper Fraction.
It’s an alternate way to write an Improper Fraction
where all the whole fractions that are inside the Improper Fraction
have been simplified out into a whole number.
To help that sink in, let’s look at another Improper Fraction:
8 over 3 (or eight-thirds)
We’ll use the same number line that goes from 0 to 3,
but this time, let’s subdivide each whole number into 3 parts to make counting thirds easier.
So this would be one-third,
this is two-thirds,
and this is three-thirds.
Oh… there’s a Whole Fraction already.
Let’s make note of that while we continue adding thirds.
Next we have four thirds…
five thirds…
six thirds.
Oh… we just formed another group of three-thirds.
That would equal a Whole Fraction also.
Let’s note that and continue…
seven-thirds…
and eight thirds.
As we noted, eight-thirds actually contained TWO Whole Fractions.
Each of those Whole Fractions simplifies to 1,
so that means our Mixed Number form would have 1 plus 1
(or 2) as the whole number part.
And then the fraction that’s left over after simplifying all the Whole Fractions is two-thirds.
So the Improper Fraction eight-thirds is equivalent to the Mixed Number 2 and 2/3.
Pretty cool, Huh? And to show that you can go back and forth between these two forms, check this out.
In the mixed number 2 and 2/3 the whole number (2) and the fraction (2/3) are being added together.
That’s really important to know even though the plus sign isn’t usually shown.
So 2 and 2/3 is really 2 PLUS 2/3.
And if we wanted to, we could expand the 2 into 1 plus 1, right?
That gives us 1 + 1 + 2/3.
And then… we could replace each of those ‘1’s with the Whole Fraction 3/3 (since 3/3 equals 1).
That gives us 3/3 + 3/3 + 2/3.
Since these fractions all have the same denominator, we can add them easily.
The denominator of the answer will stay the same (3)
and the numerator will be the sum of the other numerators
3 + 3 + 2 = 8.
And there we are, back to our original Improper Fraction 8/3.
You can convert any Mixed Number into an Improper Fraction using that procedure.
You can change the whole number part into a sum of Whole Fractions and then add everything up.
For example:
2 and 1/8 could be changed into 8/8 + 8/8 + 1/8,
which all add up to 17/8
3 and 4/5 could be changed to 5/5 + 5/5 + 5/5 + 4/5,
which all add up to 19/5
4 and 2/3 could be changed to 3/3 + 3/3 + 3/3 + 3/3 + 2/3,
which all adds to 14/3.
Notice that we always chose Whole Fractions
with the same denominator of the fraction part of the Mixed Number
so they’re all ‘like’ fractions that can be added easily.
And some of you might see the short cut here.
In each case, did you notice how many of the Whole Fractions we needed to add together?
Yep - it’s the same as the whole number part of the Mixed Number.
If it’s 2, then we need to add 2 Whole Fractions.
If it’s 3, we need to add 3 Whole Fractions.
If it’s 4, we need to add 4 Whole Fractions, and so on…
The whole number tells us how many times to repeat the addition,
and since multiplication is repeated addition, we can multiply instead of adding.
2 times 8/8 gives us 16/8
And then we add that result to 1/8 to get 17/8
3 times 5/5 gives us 15/5
And then we add that result to 4/5 to get 19/5
4 times 3/3 gives us 12/3
And then we add that result to 2/3 to get 14/3
That short cut really helps when the whole number part of the mixed number is big!
Like, what if we needed to convert the mixed number 15 and 1/4 into an Improper Fraction.
Instead of having to add 15 Whole Fractions together,
we can just multiply 15 by the Whole Fraction 4/4 which gives us 60/4.
Then we add that result to the fraction 1/4 to get 61/4.
That’s the Improper Fraction form of 15 and 1/4.
Alright then… but what if we need to go the other way?
What if we start with an Improper Fraction and need to convert it into a Mixed Number?
Well, whenever we have an Improper Fraction,
we know there’s at least 1 Whole Fraction ‘hiding’ in there that we could simplify out.
The question is how many?
To see what I mean,
let’s try converting the Improper Fraction 7/2 into a Mixed Number
using a little trial and error.
First, let’s try subtracting out just 1 Whole Fraction.
7/2 - 2/2 = 5/2
That means we can write 7/2 as the Mixed Number 1 and 5/2
since we subtracted out 1 Whole Fraction and had 5/2 left over.
And even though that’s true, it’s bad form because 5/2 is STILL an Improper Fraction,
which means there’s at least 1 more Whole Fraction ‘hiding’ in there that we could have subtracted out.
So let’s try again, but this time let’s subtract out 2 Whole Fractions.
7/2 - 2/2 - 2/2 = 3/2.
That means we can write 7/2 as the Mixed Number 2 and 3/2
since we subtracted out 2 Whole Fractions and had 3/2 left over.
But that’s STILL bad form because the fraction part is still improper!
We could have subtracted out ANOTHER Whole Fraction.
So, let’s try again subtracting 3 Whole Fractions this time…
7/2 - 2/2 - 2/2 - 2/2 = 1/2.
That means we can write 7/2 as the Mixed Number 3 and 1/2
since we subtracted out 3 Whole Fractions and had 1/2 left over.
And that’s the ‘proper’ Mixed Number form of 7/2 because it’s a whole number and a Proper Fraction,
so there’s no more Whole Fraction that we could be simplify out.
That process makes sense, but it’s kinda messy having to subtract out so many Whole Fractions.
It turns out there’s a short-cut we can take here too.
Just like multiplication is repeated addition, division is basically repeated subtraction.
That means we can figure out how many Whole Fractions we can subtract out of an Improper Fraction
by just DIVIDING the top number by the bottom number.
Let’s do that with our example 7/2.
If we divide 7 by 2, we find out that 2 will divide into 7 three times, leaving a remainder of 1.
That remainder is actually important as we’ll see in a minute.
Notice that the answer to our division problem
is exactly how many Whole Fractions we were able to subtract out of the Improper Fraction: 3
So the answer to the division tells us what the whole number part of the Mixed Number will be.
And here’s the really cool part…
the remainder of the division tells us what the left over fraction will be.
The remainder is the Numerator (the top number) of the left-over fraction.
In this case, since the remainder is 1, we’ll have 1 over 2 left over in our Mixed Number.
Let’s do one more example to make sure you’ve got that.
Let’s convert 22/5 into a Mixed Number.
If we divide 22 by 5, we see that 5 will go into 22 four times with a remainder of 2.
That means that the whole number part of the Mixed Number will be 4
and the fraction part will be 2 over 5 because our remainder was 2.
…that’s how many fifths will be left over.
So 22/5 is the same as 4 and 2/5.
Alright… so now you know what Mixed Numbers are.
They’re a combination of a whole number and a Proper Fraction.
And you know that those two parts are actually being ADDED together,
even though the plus sign is usually not shown.
You also know that a Mixed Number is basically a simplified form of an Improper Fraction
and that you can use the procedures we learned to convert back and forth between the two forms.
But the way to make sure that you REALLY understand Mixed Numbers, Improper Fractions
and how to convert between them is to practice!!
You need to do some exercises on your own
and check your answers to make sure you’re doing the procedures right.
Oh - I’m doin’ it right… In fact, I found an even quicker short cut!…
[grinding noise]
How’s that for Mixed Numbers!?
[Slurp...]
As always… thanks for watching Math Antics…
[in unison] ...and I’ll see ya next time!
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