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## Topic G: Repeated addition of fractions as multiplication

## Video transcript

We've already seen that the
fraction 2/5, or fractions like the fraction 2/5, can
be literally represented as 2 times 1/5, which
is the same thing, which is equal to literally
having two 1/5s. So 1/5 plus 1/5. And if we wanted
to visualize it, let me make a hole
here and divide it into five equal sections. And so this represents
two of those fifths. This is the first of the
fifths, and then this is the second of the fifths,
Literally 2/5, 2/5, 2/5. Now let's think about something
a little bit more interesting. What would 3 times
2/5 represent? 3 times 2/5. And I encourage you
to pause this video and, based on what
we just did here, think about what you think
this would be equivalent to. Well, we just saw that 2/5
would be the same thing as-- so let me just
rewrite this as instead of 3 times 2/5
written like this, let me write 2/5
like that-- so this is the same thing as
3 times 2 times 1/5. And multiplication, we can
multiply the 2 times the 1/5 first and then
multiply by the 3, or we can multiply the 3
times the 2 first and then multiply by the 1/5. So you could view this literally
as being equal to 3 times 2 is, of course, 6, so this is
the same thing as 6 times 1/5. And if we were to try
to visualize that again, so that's a whole. That's another whole. Each of those wholes
have been divided into five equal sections. And so we're going to
color in six of them. So that's the first 1/5, second
1/5, third 1/5, fourth 1/5, fifth 1/5-- and that
gets us to a whole-- and then we have
6/5 just like that. So literally 3 times 2/5
can be viewed as 6/5. And of course, 6
times 1/5, or 6/5, can be written as--
so this is equal to, literally-- let me do the
same color-- 6/5, 6 over 5. Now you might have said, well,
what if we, instead of viewing 2/5 as this, as we just
did in this example, we view 2/5 as 1/5 plus
1/5, what would happen then? Well, let's try it out. So 3 times 2/5-- I'll rewrite
it-- 3 times 2/5, 2 over 5, is the same thing as
3 times 1/5 plus 1/5. 2/5 is the same thing
as 1/5 plus 1/5. So 3 times 1/5 plus
1/5 which would be equal to-- well, I just
have to have literally three of these added together. So it's going to be
1/5 plus 1/5 plus 1/5 plus 1/5 plus-- I think
you get the idea here-- plus 1/5 plus 1/5. Well, what's this going to be? Well, we literally
have 6/5 here. We can ignore the parentheses
and just add all of these together. We, once again, have
1, 2, 3, 4, 5, 6/5. So once again, this
is equal to 6/5. So hopefully this
shows that when you multiply-- The 2/5 we saw
already represents two 1/5s. We already saw that,
or 2 times 1/5. And 3 times 2/5 is literally
the same thing as 3 times 2 times 1/5. In this case, that would be 6/5.