Hi I’m Rob. Welcome to Math Antics!
In our previous lesson, we learned how to add mixed numbers,
and in this lesson we’re gonna learn how to subtract them.
Subtraction is always a little trickier than addition
because subtraction does NOT have the commutative property.
That means that the order of the numbers you’re subtracting matters
and you can’t switch it like you can with addition.
Subtraction is also tricky because if you try subtracting a number from a smaller number,
like 1 minus 5, you get a negative answer.
Fortunately, we’re not gonna cover any problems that give negative answers in this video.
But as you’ll see in a few minutes, there are cases where
it SEEMS like you need to subtract a bigger number from a smaller one at first.
All of this means that subtracting mixed numbers
can be quite a bit more complicated than adding them.
But don’t worry, we’ll take it one step at a time.
And it’ll help a lot if you make sure that you have
a good understanding of topics like Adding Mixed Numbers, Subtracting Fractions,
Order of Operations and Multi-Digit Subtraction,
because this lesson builds on all of those concepts.
Okay, to see why subtracting mixed numbers is trickier than adding them,
let’s have a look at two problems side by side:
5 plus 2 and 1/3 and 5 minus 2 and 1/3.
We learned how to do problems like the first one in the previous video about adding mixed numbers.
Since 2 and 1/3 just means 2 plus 1/3 we learned that we could re-write the problem
like this and simply add up the whole number parts to get the whole number part of the answer.
5 plus 2 is 7, so the answer is 7 and 1/3.
But, if we try to do the same thing with the subtraction,
we’re gonna get the wrong answer.
If we re-write 2 and 1/3 as 2 plus 1/3,
it looks like we could just subtract the whole numbers; 5 minus 2 is 3,
which leaves 3 plus 1/3 (or 3 and 1/3) as the answer.
But it turns out that’s NOT correct!
So what did we do wrong?
Well… when we subtract a mixed number,
we need to subtract the ENTIRE mixed number.
We need to subtract the whole number part AND we need to subtract the fraction part.
But we didn’t subtract the 1/3… we added it.
That’s because when we re-wrote the mixed number as a sum of a whole number and a fraction,
we forgot (or we didn’t realize) that they form a group.
We just put the 2 plus 1/3 right after the minus sign and
it made it look like we should subtract the 2 and add the 1/3,
but we actually should have subtracted BOTH of them.
One way to avoid that mistake is to use parentheses around the mixed number you’re subtracting
so you remember that its parts form a group and you need to apply the minus sign to the whole group.
And you can only get rid of the parentheses after you apply (or distribute)
the subtraction to each member of the group like this;
minus 2, minus 1/3.
Now that we have the problem written correctly,
we’ll get the right answer if we do these math operations.
Going from left to right we have 5 minus 2 which is 3.
But then for the last step we need to subtract 1/3 from 3.
That might seem hard to do,
unless you remember what we learned about “whole fractions” in the last video.
When adding mixed numbers, sometimes we’d get an answer like this;
2 and 3/3.
But since 3/3 is a what I call a “whole fraction” its value is just 1.
And we learned that we could simplify an answer like that.
2 and 3/3 is 2 + 1 which is 3.
Well, how about if we just do that same process in reverse?
Instead of simplifying, let’s “un-simplify”.
We’re trying to take 1/3 away from 3, right?
So let's re-write 3 as 2 + 1, and then let’s re-write 2 + 1 as 2 + 3/3.
Does the problem look easier to do now?
We converted part of the whole number into the whole fraction 3/3
and we already know how to subtract fractions.
3/3 minus 1/3 is 2/3, so that means our final answer is 2 and 2/3.
So as you can see, subtracting mixed numbers can be a little complicated,
but not all problems are that hard.
Some are pretty easy, like this one:
6 and 4/5 minus 1 and 3/5
We still need to remember that the mixed numbers form groups
so we have to subtract both parts,
but that’s where the that stacked format that we learned about in the last video
can really help us out.
Since the order of a subtraction problem is important,
we need to be sure to put the mixed number that we’re starting with on the top
and the mixed number that we’re taking away on the bottom.
And doing that helps us remember that BOTH parts of a mixed number are being subtracted,
even if we don’t put parentheses around them to remind us.
That’s because we’ll subtract the parts column by column
just like we would in multi-digit subtraction.
We’ll subtract the fraction on the bottom from the fraction on the top,
and then we’ll subtract the whole number on the bottom from the whole number on the top.
4/5 minus 3/5 is 1/5 and 6 minus 1 is 5.
So the answer is just 5 and 1/5.
See… that really was easy!
And that’s why I recommend using stacked form whenever you can for subtracting mixed numbers.
It’ll help you remember that both parts of a mixed number are being subtracted
without having to worry about groups or parentheses.
But… even though the stacked form is a great way to keep from getting confused about groups,
it doesn’t make every problem quite that simple.
For example, let’s try the problem:
5 and 1/7 minus 2 and 3/7
and let’s use the stacked form again.
We’re starting out with 5 and 1/7 so it goes on top,
and 2 and 3/7 goes on the bottom.
But when we try to subtract the fractions column,
you’ll see we have a little problem.
1/7 is less than 3/7,
so we can’t subtract it without getting a negative number, which we’d really like to avoid.
So what do we do now?
Well, do you remember what you do in multi-digit subtraction
when you have to subtract a bigger digit from a smaller one?
Yep, you “borrow” or “re-group” from the column to the left.
And we can do something really similar to that with mixed numbers.
We can make the fraction part of a mixed number bigger by borrowing from the whole number part.
In the video about adding mixed numbers,
we learned that a mixed number with a fraction part that’s improper is “bad form”
because you can simplify out a whole fraction and then add it to the whole number part.
Well, we can do that process in reverse also.
We can subtract out 1 (as a whole fraction) from the whole number part
and add it to the fraction part to get an improper fraction.
And even though that’s considered bad form, it’s okay
because it’s not our final answer and it helps us subtract the fraction column
without getting a negative fraction.
In this example, we do that by changing the whole number part (5) into 4 + 1,
and then changing 4 + 1 into 4 + 7/7.
And finally, we can combine that 7/7 with the 1/7 to get 8/7.
Now the fraction on top is big enough to subtract 3/7 from.
8/7 minus 3/7 is 5/7.
And then we just need to subtract the whole number parts.
The top whole number used to be 5, but we borrowed from it, so now it’s 4.
4 - 2 = 2, so our answer is 2 and 5/7.
Let’s try another example to make sure you’ve got it.
And let’s make it a story problem this time.
Suppose there’s this guy who’s a builder, named “Rob”
…and suppose he needs to fix something by subtracting the mixed number 1 and 5/8 from 6 and 1/8.
Can he fix it?
Yes I can!
Well.. at least with the help of my trusty cat, Richard.
Where did that rascal run off to?
Richard… Richard… [purring sound in the distance]
Oh well… the first thing we need to do is put these in stacked form,
with the number I’m taking away on the bottom.
Then I need to subtract the fractions,
but I notice that the fraction on top is less than the fraction on the bottom,
so I need to make the fraction on top bigger by borrowing from the whole number part.
I can replace the 6 with 5 + 1
and I can replace 5 + 1 with 5 + 8/8 to get the whole fraction I need to add to the top fraction.
8/8 plus 1/8 is 9/8.
So now I can subtract the fraction column because the fraction on top is bigger.
9/8 minus 5/8 is 4/8.
And then I can subtract the whole numbers.
5 minus 1 is 4, so the answer is 4 and 4/8.
Does that look right to you Richard?
[long meow]
You’re right!
4/8 can simplify to 1/2. So my final answer is 4 and 1/2.
Thanks for your help Richard. [meow]
Good job Rob …the builder.
So now you know how to subtract mixed numbers.
Some problems will be easy, like our second example
where you can just subtract the fraction parts and the whole number parts.
But in other problems, you may need to borrow or re-group using that “un-simplifying” procedure.
And remember that the stacked form is a great way to keep everything straight.
Of course, you may also need to solve problems
where the mixed numbers have un-like fractions
and you’ll need to change them into like-factions before you can subtract.
But that works the same as you saw in the last video so you already know how to handle that.
I know I’m always saying it, but that’s because it’s true.
The way to really learn math is to practice by doing some problems on your own.
As always, thanks for watching Math Antics, and I’ll see ya next time.
Learn more at www.mathantics.com