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Problem 10. John uses 2/3 of a cup of oats
per serving to make oatmeal. How many cups of oats does he
need to make 6 servings? So he uses 2/3 per
serving, and he's going to have 6 servings. So it's going to be 6 times--
for every serving there's 2/3, so it's going to be
6 times that. So it's going to be 2/3 per
serving, and we can even put units there. We could say 1 over servings. Well, that just confuses it. 2/3 per serving times
6 servings. So that's just 6 times 2/3. And that is equal to-- maybe
it's easier to visualize in your head as multiplying
two fractions. We can write that as 2/3 times
6/1, and that's equal to-- we could simplify it
ahead of time. Or we can just write that as 2
times 6 in the numerator-- which is 12-- and the
denominator is 3 times 1-- which is 3. Multiplying fractions is far
easier than adding or subtracting. Because literally the new
numerator of the product is just the product of the
numerators, and then the denominator of the product is
just the product of the denominators. 2 times 6 is 12. 3 times 1 is 3. And 12/3-- that's the same thing
as 12 divided by 3-- that just equals 4. Which is choice B right there. Another way you could have
done it is you could have simplified it right here. You could have divided the
6 and the 3 both by 3. So if you had 2/3 times 6/1,
you could divide the 6 by 3 and the 3 by 3. So you get 1, and then you just
get 2 times-- sorry, 6 divided by 3 is 2. So you have 2 times 2 is equal
to 4, over 1 times 1. And the whole reason why you
could do that is this thing right here, instead of solving
it up here, I could have rewritten it as 2 times
6 over 3 times 1. And then you could simplify it
before getting to this stage. You could just divide the
6 by the 3, and you get a 1 and a 2. 6 divided by 3 is
2, you get 4. I'm probably beating a dead
horse, but just in case you need a review multiplying
fractions, you just got it. Which expression represents
0.0000007 in scientific notation? So the easy way to convert
scientific notation, especially when you have a
number behind the decimal point like this, is you
literally just count the digits behind the
decimal point. And we have 1, 2, 3,
4, 5, 6, 7 numbers behind the decimal point. So our answer is
going to be 7. This 7, right here, 7 times 10
to the-- it's going to be a negative exponent because
we're going less than 1. We're going behind the
decimal point. 10 to the minus-- and
I just counted. 1, 2, 3, 4, 5, 6, 7. 10 to the minus 7. Right there. And if you find this problem a
little bit daunting, I have actually two videos where
I go into depth about scientific notation. So you might want to watch
those, just in case you run into the opposite situation. If you had 7,000, and you
want to write that in scientific notation. So in this case we're going
above, we're going into the positive domain. Then, instead of just counting--
when we had a decimal, you counted
the actual digits. You included the 7, right? That's where you got
10 to the minus 7. When you have something like
this, you just count the 0's. So 7,000 would be 7 times
10 to the third. If you want more of the
rationale of why that works, I definitely recommend that you
watch the two videos that I have on scientific notation. Next problem. The Venn diagram below shows
the number of girls on the soccer and track team
at a high school. So this is the soccer team,
this is the track team. And right here, this is the
girls who are on both the soccer and the track team. How many girls are on both the
soccer and the track teams? So I kind of jumped the gun. They're telling us the
answer right there. There are 6 girls. This overlap region-- right? If some girl is, let's say that
some girl right there, she's in the soccer circle,
and she's also in the track circle. And this number tells us that
there's 6 of these girls. So there are 6 girls that are
on both the soccer and the track teams.