CAHSEE practice: Problems 23-27

Problem 23. Mario drives 1,500 miles every month. Which line plot correctly represents Mario's total miles driven over a period of 6 months? So it's 1,500 miles every month. So this is per month. So every month-- after 1 month he should drive 1,500 hundred miles, and that looks pretty good there. So after another month, it should be another 1,500. Which goes to 3,000. Right? After 2 months, he should drive 3,000 miles. So far, choice A looks pretty good. So I'm just going to circle that. Now let's just verify that the other ones aren't describing this. So in choice C, it takes him 6 months to go 1,500 miles. That's not what they said, they said 1,500 miles every month. After 1 month here, he only went-- I don't know. He only went 250 miles or 5-- so this is definitely not the case. Choice B shows he started off having traveled 1,500 miles and it never changes. So he doesn't travel anymore. So it's definitely not choice B. And then choice D, after 1 month he travels 9,000 miles. Which is completely wrong. They set it up per month he travels 1,500 miles. Definitely not choice D. So choice A we can feel good about. Let's go to the next page. All right. We have problem 24. The temperature on a mountain peak was 7 degrees Fahrenheit at 6:00 PM. By 8:00 PM, so 2 hours later, the temperature had dropped to 0 degrees Fahrenheit. If the temperature continued to drop at the same rate, which is the best estimate of the temperature at 11:00 PM? So what happened? We went from 6:00 PM to 8:00 PM. That was 2 hours, right? And our temperature went from 7 degrees to 0 degrees Fahrenheit. So over 2 hours, we dropped 7 degrees. Or you could say the rate of droppage of our temperature, if you want to say it that way, so you could say 7 degrees-- let me write it this way. 7 degrees for every 2 hours. Or you could say minus 7 degrees for every 2 hours, we're dropping. Or that's the same thing as saying we're going minus 3.5 degrees per hour. That's the rate at which it dropped from 6:00 PM to 8:00 PM. Minus 3.5 degrees per hour. So they say, what is the best estimate of the temperature at 11:00 PM? Well, 11:00 PM is going to be another 3 hours after 8:00 PM. And if we're dropping at minus 3.5 degrees per hour, and we're going to do that for 3 hours, we just multiply the two. So you have-- do it over here. You have minus 3.5 degrees per hour, and then times 3 hours. Right? That's how long we're going to go from 8:00 PM to 11:00 PM. The hours cancel out, if you want to make sure your units work out. But what's minus 3.5 times 3? Well, let's just write it out. So 3.5 times 3. I could put a minus there. We'll worry about it in a second. 5 times 3 is 15. 3 times 3 is 9, plus 1 is 10. We have exactly 1 number behind the decimal point. We have no decimals here. We have 1 number behind the decimal point in our answer, and then we have a negative times a positive, which is going to be a negative. So from 8:00 PM to 11:00 PM, we're going to drop by 10.5 degrees. Now, they say that the temperature was 0 at 8:00 PM. So if we drop 10.5 degrees from that, we're going to be approximately at minus 10 degrees Fahrenheit. They say, what is the best estimate? If they said exactly, we would say, oh, minus 10.5 degrees. If you wanted to do this-- you could have done this in your head if you like, but I think it's nice to say, oh, we're dropping 3.5 degrees per hour, 3 hours later it's going to be a little over 10 degrees, or 10.5 degrees, that we dropped from the 0 degrees at 8:00 PM. Problem 25. Do it in-- maybe orange. Brad bought a $6 binder and several packs of paper that cost $0.60 each. If his total was $13.20, how many packs of paper did Brad buy? All right. Let's say he bought p packs of paper. So how much did he spend? He bought a $6 binder, so he spent $6, plus $0.60 per pack times the number of packs he bought. Right? So this is the total. $6 for the binder plus $0.60 for each of the packs. If you bought 10 packs, we can do 10 times $0.60 per pack, but we don't know what it is, so we just leave it as p. And the total cost was $13.20. So we just have to solve for p. How many packs of paper did Brad buy? So if we subtract 6 from both sides of this equation-- do it in blue-- just subtract this 6 from both sides. You get 0.6p-- maybe I'll write it like this. Minus 6, plus there, and then minus 6 right there. Just to show you I'm doing the same thing to both sides of this equation. This minus 6 and that positive 6 cancel out, and so you're just left with a 0.6p on the left-hand side. Is going to be equal to 13.2, or $13.20 minus $6. You can write that way if you like. 13 minus 6 is 7. So it's going to be $7.20, or 7.2. And then to solve for p, we just divide both sides by 0.6. So if we divide both sides by 0.6, the 0.6's is cancel out on the left-hand side, so you're just left with a p. So you're left with-- do it over here. p is equal to 7.2 divided by 0.6. So let's work this out. 0.6 goes into 7.2-- I'll throw in some trailing 0's. Now, division by decimals. This is sometimes a little bit confusing, but you just have to remember, you take the number that we're dividing into the other number, and you make it into a whole number. So you shift its decimal point, in this case 1 to the right. So it just becomes a 6. Since you shifted its decimal point 1 to the right, you have to shift this guy's decimal point 1 to the right. So it'll go right there. So 0.6 into 7.2 is the same thing as 6 into 72. Or you could just do this as a fraction. You multiply the top and the bottom by 10, is the same thing as 72 divided by 6. And that's just exactly what we did here. So 6 goes into 72-- well, actually, even better, 6 goes into 7 one time. 1 times 6 is 6. And then 7 minus 6 is 1. Bring down the 2. 6 goes into 12 two times. 2 times 6 is 12. You get a 0. All 0's left. So you get exactly 12 times. And if you know your 6 times tables, you knew that. 72 divided by 6 is 12. So p is equal to 12. He bought 12 packs of paper. Next problem. What is the value of this thing, 3 plus 5 squared, divided by 4 minus x plus 1, when x is equal to 7? So let's write it down when x is equal to 7. So this becomes 3 plus 5 squared-- so let me just write it out. 5 squared is 25. Divided by-- and so they want us to do a little bit of order of operations here. So parentheses are the dominant role here. So this thing, 3 plus 25, this is going to be 28. So parentheses take the dominant role here, so this is going to be a 28 right there. So let me write it down. Parentheses are the most important, then exponents, then multiplication and division, and if there's a tie you go left to right, and then you do addition and subtraction. So they're clearly trying to make sure that we understand our order of operations. So let's do the things in parentheses first. So it's 3 plus 5 squared is 3 plus 25, that's a 28. Then you have an x plus 1. They're telling us that x is 7. This is a 7 plus 1, so that's going to be an 8. And then we do the division before we do the subtraction. So this becomes 28-- let me rewrite it. 28 divided by 4, minus 8. And so order of operations, we do this first. So 28 divided by 4 is 7. And then we have the 7 minus that 8 right there, which is equal to minus 1. So that is choice B. Next problem. Problem 27. What is the equation of the graph shown below? So we just have to figure out its slope and its y-intercept. In general, these are all in kind of the classic mx plus b form. y is equal to mx plus b, where m is the slope, b is the y-intercept. The slope here, our rise over run. For every 1 we run, we rise up 1. For a change in x of 1, we have a change in y of 1. So slope is equal to change in y over change in x, or rise over run. We rise by 1 when we run by 1. So it's equal to 1/1 which is equal to 1. So m is equal to 1, and in every situation here our slope is 1. So that didn't help us so far. But then b is the y-intercept, where we intersect the y-axis. And you could see, just from inspection, that's it. y is equal to 3. So we look for b is 3. So our equation is going to be y is equal to 1 times x, which is just x, plus the y-intercept of 3. So y is equal to x plus 3, which is right there. No more problems on that page.