Mixed numbers and improper fractions

Converting mixed numbers to improper fractions and improper fractions to mixed numbers

We're now going to learn how to go from mixed numbers

to improper fractions and vice versa.

So first a little bit of terminology.

What is a mixed number?

Well, you've probably seen someone write,

let's say, two and one half.

This is a mixed number.

So you're saying why is it a mixed number?

Well, because we're including a whole number and a fraction.

So that's why it's a mixed number.

It's a whole number mixed with a fraction.

So two and one half.

And I think you have a sense of what two and one half is.

It's some place halfway between two and three.

And what's an improper fraction?

Well an improper fraction

is a fraction where the numerator is larger than the denominator.

So let's give an example of an improper fraction.

I'm just going to pick some random numbers.

Let's say I had twenty-three over five.

This is an improper fraction.

Why?

Because twenty-three is larger than five.

It's that simple.

It turns out that you can convert an improper fraction into a mixed number

or a mixed number into an improper fraction.

So let's start with the latter.

Let's learn how to do a mixed number into an improper fraction.

So first I'll just show you kind of just the basic systematic way of doing it.

It'll always give you the right answer,

and then hopefully I'll give you a little intuition for why it works.

So if I wanted to convert two and one half into an improper fraction,

or I want to unmix it you could say,

all I do is I take the denominator in the fraction part, multiply it by the whole number,

and add the numerator.

So let's do that.

I think if we do enough examples,

you'll get the pattern.

So two times two is four plus one is five.

So let's write that.

It's two times two plus one,

and that's going to be the new numerator.

And it's going to be all of that over the old denominator.

So that equals five halves.

So two and one half is equal to five halves.

Let's do another one.

Let's say I had four and two thirds.

This is equal to -- so this is going to be all over three.

We keep the denominator the same.

And the new numerator is going to be three times four plus two.

So it's going to be three times four, and then you're going to add two.

Well that equals three times four--

order of operations, you always do multiplication first,

and that's actually the way I taught it-- how to convert this, anyway.

three times four is twelve plus two is fourteen.

So that equals fourteen over three.

Let's do another one.

Let's say I had six and seventeen eighteenths.

I gave myself a hard problem.

Well, we just keep the denominator the same.

And then new numerator is going to be eighteen times six

or six times eighteen, plus seventeen.

Well six times eighteen.

Let's see, that's sixty plus forty-eight it's one hundred eight,

so that equals one hundred eight plus seventeen.

All that over eighteen.

One hundred eight plus seventeen is equal to one hundred twenty-five over eighteen.

So, six and seventeen eighteenths is equal to one hundred twenty-five over eighteen.

Let's do a couple more.

And in a couple minutes I'm going to teach you how to go the other way,

how to go from an improper fraction to a mixed number.

And this one I'm going to try to give you a little bit of intuition for why what I'm teaching you actually works.

So let's say two and one fourth.

If we use the-- I guess you'd call it a system that I just showed you--

that equals four times two plus one over four.

Well that equals, four times two is eight plus one is nine. Nine over four.

I want to give you an intuition for why this actually works.

So two and one fourth, let's actually draw that,

see what it looks like.

So let's put this back into kind of the pie analogy.

So that's equal to one pie.

Two pies.

And then let's say, one fourth of a pie. Oh, sorry.

One fourth is like this. A fourth of a pie, right?

Two and one fourth, and ignore this, this is nothing.

It's not a decimal point-- actually, let me erase it

so it doesn't confuse you even more.

So go back to the pieces of the pie.

So there's two and one fourth pieces of pie.

And we want to rewrite this as just, how many fourths of pie are there total?

Well if we take each of these pies--

oh, whoops! I need to change the color--

if we take each of these pies,

and we divide it into fourths,

we can now say how many total fourths of pie do we have?

Well we have one, two, three, four, five, six, seven, eight, nine fourths.

Makes sense, right?

Two and one fourth is the same thing as nine fourths.

And this will work with any fraction.

So let's go the other way.

Let's figure out how to go from an improper fraction

to a mixed number.

Let's say I had twenty-three over five.

So here we go in the opposite direction.

We actually take the denominator,

we say how many times does it go into the numerator?

And then we figure out the remainder.

So let's say five goes into twenty-three--

well, five goes into twenty-three four times.

Four times five is twenty.

And the remainder is three.

So twenty-three over five, we can say that's equal to four,

and in the remainder, three over five.

So it's four and three fifths.

Let's review what we just did.

We just took the denominator

and divided it into the numerator.

So five goes into twenty-three four times.

And what's left over is three.

So, five goes into twenty-three four and three fifths times.

Or another way of saying that is twenty-three over five is four and three fifths.

Let's do another example like that.

Let's say, seventeen over eight.

What does that equal as a mixed number?

You can actually do this in your head,

but I'll write it out just so you don't get confused.

Eight goes into seventeen two times.

Two times eight is sixteen.

Seventeen minus sixteen is one.

Remainder, one.

So, seventeen over eight is equal to two-- that's this two-- and one eighth.

Right? Because we have one eight left over.

Let me show you kind of a visual way of representing this too,

so it actually makes sense how this conversion is working.

Let's say I had five halves, right?

So that literally means I have five halves,

or if we go back to the pizza or the pie analogy,

let's draw my five halves of pizza.

So let's say I have one half of pizza here,

and let's say I have another half of pizza here.

I just flipped it over.

So that's two.

So it's one half, two halves.

So that's three halves.

And then I have a fourth half here.

These are halves of pizza,

and then I have a fifth half here, right?

So that's five halves.

Well, if we look at this, if we combine these two halves,

this is equal to one piece, I have another piece,

and then I have half of a piece, right?

So that is equal to two and one half pizzas.

Hopefully that doesn't confuse you too much.

And if we wanted to do this the systematic way,

we could have said two goes into five--

well, two goes into five two times,

and that two is right here.

And then two times two is four.

Five minus four is one, so the remainder is one,

and that's what we use here.

And of course, we keep the denominator the same.

So five halves equals two and one half.

Hopefully that gives you a sense of how to go from a mixed number to an improper fraction,

and vice versa,

from an improper fraction to a mixed number.

If you're still confused let me know,

and I might make some more modules.

Have fun with the exercises!