Hi! Welcome to Math Antics. In our last video, we learned the basics of multi-digit multiplication.
We learned how to multiply a multi-digit number by a one-digit number.
But in this video, we’re gonna take it to the next level.
We’re gonna learn how to multiply a multi-digit number by another multi-digit number.
You remember the procedure for multiplying when we have a one-digit bottom number, right?
You break up the problem into a series of multiplication steps, one for each of the top digits.
And in each step, you just multiply the bottom digit by a top digit,
starting with the ones place and working your way left until you’ve multiplied all of the digits.
Well, when the bottom number of your multiplication problem also has more than one digit,
you have to do that same procedure we learned for EACH bottom digit.
For example, if you have a two-digit bottom number,
you multiply the first digit by each top digit, and then you multiply the second digit by each top digit.
That means you’re gonna have twice as many steps to do,
AND it means you’re gonna get two different answers!
What?!! - How can we have two different answers for the same problem?
Now don’t panic.
The different answers are just what you get from doing the multiplication procedure for each digit of the bottom number separately.
In fact, it’s kind of like we’re pretending we have two SEPARATE multiplication problems that each have a one-digit bottom number
…which is nice, cuz I kinda like pretending…
No Luke… I AM your father!
NO! That’s IMPOSSIBLE!
Ha Ha Ha!
Ahh… wait…no…
Nooooooooooo…..
Ha Ha Ha Ha Ha…
Err… well… I mean… it’s nice because we already know how to multiply when we have a one-digit bottom number.
But then what will we do with the two different answers we’re gonna get?
Well, it turns out that all we have to do is add them together once we’re finished doing all of our multiplication steps.
Are you ready to see an example? It should make a lot more sense when you see the procedure in action.
So let’s multiply 324 (a three-digit number) by 46 (a two-digit number).
Now remember, we’re gonna do the same procedure that we did in the last video for each of the bottom digits.
And since our top number has three digits, that means there’ll be
three multiplication steps for the first digit AND three steps for the second digit.
Fortunately, we can start the same way we would if the bottom had only one digit
by just ignoring the second digit until we finish the first three steps.
Alright, so our first multiplication step is 6 × 4 which is 24
And since 24 has two-digits, we can leave the ‘4’ in our answer line,
but we need to carry the ‘2’ and put it above the next top digit that we’re gonna multiply.
So the next step is 6 × 2 which is 12.
But we have to add in the ‘2’ that we carried, so 12 + 2 gives us 14.
That’s another two-digit answer,
so we leave the ‘4’ in our answer line, and carry the ‘1’ up above the next digit that we’re gonna multiply.
And our third step is 6 × 3 which is 18. And then we add in the ‘1’ that we carried and we get 19.
This time we can leave both digits of the 19 in our answer line, because there’s no more multiplication steps to do.
Well… at least there’s no more steps for the FIRST digit.
Remember, we still have that other bottom digit that we’ve been ignoring.
NOW we have to multiply IT by each of the top digits also, which means we have three more steps to do.
It also means that we’ll get a second answer.
And because we’ll get a second answer, we need to start a second answer line for the next set of steps.
We’re gonna put our new answer just below the first one.
So this answer line comes from our first bottom digit, and this answer line will come from our second bottom digit.
At the very end, after we’re all done multiplying, we’re gonna add the two answers together.
But for now, let’s continue with the second set of multiplication steps.
Oh… and I almost forgot to tell you…
there’s something VERY important that you need to do when you start the second set of steps.
Because the second digit of the bottom number is in the TENS place,
that means that even though the digit is only a ‘4’, it’s value is really 40.
That’s 10 times bigger… so the answer we get should also be 10 times bigger.
So before we start multiplying, we need to put a ZERO in the first spot of our answer line so it’s 10 times bigger.
That means that all the other answer digits we put there are shifted to the next bigger number place.
And (just so you know) if we happened to have a third bottom digit,
we would get a third answer line, and we’d need to shift the third answer over by TWO zeros, cuz it would be 100 times bigger.
And if we had a fourth digit, there would be a fourth answer line shifted over by THREE zeros.
And if we had a fifth bottom digit, there would be a fifth answer line shifted over by FOUR zeros!
See the pattern?
Boy am I glad we’ve only got two bottom digits!!
And now you see why we always put the number with the fewest digits on the bottom when we’re multiplying.
But let’s continue with our problem…
Let’s do the first step for our second bottom digit.
We multiply that digit (4) by the ones place digit of the top number (which is also a ‘4’).
4 × 4 gives us 16, and that goes in our second answer line, right next to the extra zero we put there.
Now remember, because 16 is a two-digit answer, we have to carry.
And we always put the digit we carry above the next top digit that we will multiply.
But before we can carry it up to that place,
we need to cross out all the digits that we carried from our first answer line, because we’ve already used them.
We don’t want to accidentally add them to our second answer line.
There… so now we can carry our ‘1’ to the top of the tens place, which means it will go in the column above this ‘2’,
because that’s the next top digit that we’ll multiply with our bottom digit.
Now we can do the next multiplication step.
4 × 2 gives us 8, and then we’ll add the ‘1’ that we carried and we get 9.
Great! …finally a one-digit answer, so we don’t have to carry this time.
We just write the ‘9’ in the next place of our answer line and move on to the next step.
The next (and last) multiplication step is 4 × 3 which is 12.
And since there’s no more steps, we can write both digits in our second answer line.
All right… we’re finally done with all our multiplication steps.
We multiplied each bottom digit by each top digit, just like we were supposed to.
But, now what do we do? We have two answers lines, but this is just one multiplication problem.
Well, remember…
the reason we have two answer lines is that we’re pretending that we’re doing two separate multiplication problems.
We treated it like it was 6 × 324 and 4 × 324.
But since the ‘4’ was in the tens place, we had to put an extra zero in our second answer since it would really be the answer from 40 × 324.
Now as I mentioned earlier, all we have to do to get the final answer is add those two answers together.
And the great news is that those answers are already stacked up like an addition problem should be,
so we can just draw a line below them and stick a plus sign on the left side.
Now we can add them, column by column, starting from ones place, just like we did in the multi-digit addition video.
4 + 0 = 4
4 + 6 = 10, so we carry the ‘1’
1 + 9 + 9 = 19, so we carry the ‘1’ again.
1 + 1 + 2 = 4, and then our last answer digit is just 1.
There, our final answer is a pretty big number: 14,904.
Let’s double check with a calculator to make sure we got it right.
Yep, that’s exactly what you get when you multiply 324 by 46.
Okay, I know that procedure is kinda complicated, so don’t get frustrated if you don’t get it right away.
It just takes some time and practice to really get the hang of it.
And you can always re-watch this video if you need to.
And if you’re a Math Antics subscriber, be sure to check out the extra problems I work in the “Prob with Rob” video examples.
After that, you can try some of the exercises on your own.
Thanks for tuning into Math Antics and I’ll see you next time.
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