CAHSEE Practice: Problems 4-9 CAHSEE Practice: Problems 4-9
CAHSEE Practice: Problems 4-9
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- 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.
- 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
- 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.
- 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.
- 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.
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