CAHSEE practice: Problems 4-9

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.