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# CAHSEE practice: Problems 4-9

Video transcript

We're on problem 4. Traditions Clothing Store
is having a sale. Shirts that were regularly
priced at $20 are on sale for $17. What is the percentage of
the decrease in the price of the shirts? So what is just the decrease
in the price of the shirts? Well, they've gone
from $20 to $17. So the decrease is $3. We have a $3 decrease in the
price of the shirts. So what is the percentage
of decrease? Well, we're starting at $20,
and we're decreasing by $3. So the percentage of the
decrease-- let me write it in a darker color. We're starting at $20 and we're
decreasing it by $3, or decreasing it by 3/20. This is the percent decrease,
or the fraction decrease right there. And this is going to
be equal to what? This is the same thing. We can divide it out. You might be able to do
that in your head. But just in case you
can't, let me just write it out like this. If I divide 20 into 3, 20
does not go into 3. Or you could say 20 goes
into 3 zero times. 0 times 20 is zero. And then I put a decimal
point right there. 20 goes into 30-- maybe I should
have done this-- let me do it over here. 20 goes into 3 zero times. 0 times 20 is 0. And then let me put a decimal
point like that. And then I'll do
3 minus that 0. So I'll just say you have
3.0, just like that. Maybe I shouldn't even
write the decimal. Maybe I'll write
a 30 like that. We have the decimal up here. 20 goes into 30 one time. 1 times 20 is 20. 30 minus 20 is 10. And then bring down another
0, this 0 right there. And 20 goes into
100 five times. 5 times 20 is 100. 100 minus 100 is 0. So 3/20 can be rewritten
as 0.15. And another way you could think
about this is 3/20, is the same thing-- if you multiply
the numerator and the denominator by 5-- is the
same thing as 15/100. Right? If I multiply the top and the
bottom by 5, which we can always do, this is the
same thing as 15/100. So that's the way you could have
done that in your head, saying, oh, that's the
same thing as 0.15. And 0.15 expressed as
a percentage is 15%. So the correct answer is B. All right. Problem number 5. Which number equals 2 to
the minus fourth power? So this is a little bit of
review of exponents. 2 to the minus 4 is the same
thing as 1 over 2 to the fourth power. That's all a negative
exponent does. It means 1 over essentially
the base to the positive exponent. And this is just going to
be 1 over 2 times 2 times 2 times 2. And that's what? 2 times 2 is 4, times 2
is 8, times 2 is 16. So it's equal to 1/16. So that is choice C. Problem 6. What is 3/4 minus 1/6? So whenever you add or subtract
fractions, you have to find a common denominator. And a good common denominator
is the least common multiple of these two guys, or the
smallest number that both of these denominators go into. So the smallest number that both
of these denominators go into is 12, right? 4 goes into 12 three times, and
6 goes into 12 two times. So let's rewrite these fractions
with 12 as a denominator. So something over 12 minus
something over 12. So how do we rewrite 3/4
as something over 12? Well, 4 goes into
12 three times. Or we could say 3
times 4 is 12. So 3 times 3 will be the
other numerator. You get 9/12. 3/4 and 9/12 are the exact
same fraction. This is in kind of its simplest
form, when you reduced the numerator
and denominator as much as you can. But this is a completely
equivalent fraction. To go from there to there, you
just multiply the numerator and the denominator by 3. 3 times 3 is 9, 3
times 4 is 12. And the way you can think
about it, 4 goes into 12 three times. So just multiply 3 times
that numerator. Let's do the same
thing with 1/6. 6 goes into 12 two times. So to go from there to
there, we have to multiply the 6 by 2. So to go from there to whatever
this numerator is, we have to multiply that
numerator by 2. So you stick a 2 right there. And now that we have a common
denominator, this becomes a simple problem to work out. It is equal to-- we have 9 out
of 12 pieces, minus 2 out of 12 pieces, or whatever
we're talking about. Or slices of pie. So that's going to
be equal to 7/12. And that is choice C. Next problem. Problem 7. Do it in blue again. A salesperson at a clothing
store earns a 2% commission on all sales. How much commission does
the salesperson earn on a $300 sale? Well, they earn 2% on that. Or you could say 0.02
of the sale. So you could just multiply
300 times 0.02. This is kind of the
way to just do it. And I'll maybe give you a little
intuition on how you could do this in your head if
you're constrained for time. But the easiest way to think
about it is just multiply 2 times 300. Or 2 times 0 is 0. 2 times 0 is 0, and
2 times 3 is 6. You could have done
that in your head. 2 times 300 is 600. And then you worry about
the decimals. So we have two spaces behind
the decimal right here. Right? We have one, two. So we're going to have to have
two spaces behind the decimal in our answer. So the answer right
here is $6. And that is choice A. Now, a way you could have done
this in your head, you could have said, look, he makes
a 2% commission. So for every $100,
he makes $2. Right? 2% is the same thing
as 2 out of 100. So every $100, he makes $2. He sold $300, so that would
also be 3 times $2 per 100, or $6. Whatever is easier for you
to understand, that's what you should do. Problem number 8. I'll do this in green. Some students attend school 180
of the 365 days in a year. About what part of the year
do they attend school? And so the keyword here is
about, which tells me that they don't want an
exact answer. They don't want me to sit
there-- I mean, if I wanted to, they attend 180
out of 365 days. If I wanted the exact
percentage, I would have to take 365 and divide it into 180,
with some decimals, and I'll get some-- I could work
it out, just like I did the decimal division in the past,
but it would take some time, and they just want
to know about. So what's 180/365 roughly? Well, what's 180 times 2? You could say 180/360, and I
picked 360 because that's 180 times 2, that's equal to 50%. This is not that different
than this. We just have a small change
of the denominator, small relative to how large
that denominator is. So the answer here is 50%. If they had a couple of other
choices here like 49% or 51%, then you would have to
work this out in a little bit more detail. But it's pretty clear
that 180/365 is pretty close to 50%. It's nowhere near 18% or 75%,
so you can feel pretty good about this answer. If 49% were one of the choices,
then I would have to do a little bit more
arithmetic right here with the division. Problem number 9. What is the value of 2 to the
sixth times 2 to the fourth, divided by 2 to the fifth? Now, you could solve each
of these powers. 2 to the sixth is what? It's 64. You could work out each of
these exponents, and then multiply them and then divide,
but it would take you forever. So what they really want you
to do here is use your exponent rules. So when you multiply two
exponents and they have the same base-- let me write
it this way. 2 to the sixth times 2 to
the fourth, all of that over 2 to the fifth. What's that equal to? So let's just simplify the
numerator first. So I'll keep the denominator the same. When you multiply two exponents
with the same base, you essentially can just
add the exponents. So the same base is 2. So this is going to be equal to
2 to the 6 plus 4, or 2 to the 10th power. Now, when you divide exponents
that have the same base, you subtract the exponents. So this is going to be equal to
2 to the 10 minus 5 power, which is equal to
2 to the fifth. And that's our answer. And we look at our choices, we
don't see 2 to the fifth. So we're actually going to
have to multiply it out. And so 2 to the fifth
is 2 times 2 times 2 times 2 times 2. 2 times itself five times. That's what 2 to the
fifth means. So what is this? 2 times 2 is 4. 4 times 2 is 8. 8 times 2 is 16. 16 times 2 is 32. So that is our answer. D.