CAHSEE practice: Problems 10-12

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.