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# Multiplying challenging decimals

Video transcript

Let's multiply 1.21,
or 1 and 21 hundredths, times 43 thousandths, or 0.043. And I encourage you
to pause this video and try it on your own. So let's just think about a
very similar problem but one where essentially we
don't write the decimals. Let's just think about
multiplying 121 times 43, which we know how to do. So let's just think
about this problem first as kind of
a simplification, and then we'll think about
how to get from this product to this product. So we can start with-- so we're
going to say 3 times 1 is 3. 3 times 2 is 6. 3 times 1 is 3. 3 times 121 is 363. And now we're going to
go to the tens place, so this is a 40 right over here. So since we're in the tens
place, let's put a 0 there. 40 times 1 is 40. 40 times 20 is 800. 40 times 100 is 4,000. And we've already known
how to do this in the past, and now we can just add
all of this together. And we get-- let me do a new
color here-- 3 plus 0 is 3. 6 plus 4 is 10. 1 plus 3 plus 8 is 12. 1 plus 4 is 5. So 121 times 43 is 5,203. Now, how is this useful for
figuring out this product? Well, to go from
1.21 to 121, we're essentially multiplying by 100. Right? We're moving the decimal two
places over to the right. And to go from 0.043 to
43, what are we doing? We're removing the
decimal, so we're multiplying by ten,
hundred, thousand. We're multiplying by 1,000. So to go from this
product to this product or to this product, we
essentially multiplied by 100, and we multiplied by 1,000. So then to go back to this
product, we have to divide. We should divide by
100 and then divide by 1,000, which is equivalent
to dividing by 100,000. But let's do that. So let's rewrite this
number here, so 5,203. Actually let me
write it like this just so it's a little
bit more aligned, 5,203. And we could imagine a
decimal point right over here. If we divide by 100-- so you
divide by 10, divide by 100-- and then we want to
divide by another 1,000. So divide by 10, divide
by 100, divide by 1,000. So our decimal point is
going to go right over there, and we're done. 1.21 times 0.043 is 0.05203. So one way you could think about
it is just multiply these two numbers as if there
were no decimals there. Then you could count
how many digits are to the right of
the decimal, and you see that there are one,
two, three, four, five digits to the right
of the decimal, and so in your product, you
should have one, two, three, four, five digits to the
right of the decimal. Why is that the case? Well, when you ignored the
decimals, when you just pretended that this
was 121 times 43, you essentially multiplied
this times 100,000-- by 100 and 1,000-- and so to
get from the product you get without the decimals to
the one that you need with the decimals, you have to
then divide by 100,000 again. Multiplied by 100,000
is essentially equivalent to moving the decimal
place five places to the right, and then dividing by 100,000
is equivalent to the moving the decimal five
digits to the left. So divide by 10, divide
by 100, divide by 1,000, divide by 10,000,
divide by 100,000. And either way, we are done. This is what we get.