We're on problem 70. If x plus 5 is greater than 2, and x minus 3 is less than 7, the value of x must be between which of the following pairs of numbers? Well for this top one, let's just subtract 5 from both sides of this equation. If you subtract 5 from both sides, you get x is greater than, what's 2 minus 5, x is greater than minus 3.
For this bottom one, let's add four to both sides of the equation. So x minus 3-- sorry, lets add 3 to both sides of this equation. So x minus 3 plus 3, that's x, is less than, what's 7 plus 3? 10. So x is greater than minus 3-- I just wrote this the other way around-- and it's less than 10. So that's A.
Problem 71, a gym class can be divided into 8 teams with an equal number of players on each team-- 8, that means you get an equal number-- or into 12 teams with an equal number on each team. Fair enough. What is the lowest possible number of students in the class? So essentially it's saying, that the number of students in the classes is divisible by both 8 and 12.
So what they want to know is, what's the smallest number that's divisible by both 8 and 12. That's the least common multiple of 8 and 12. That might ring a bell from middle school. So what's the smallest number that's divisible by 8 and 12? So let's think about 8's multiples. 8 is indivisible by 12. 16 is indivisible by 12.
What about 24? Well yeah, sure, 24 is divisible by 12. So the least common multiple is 24, which is B. Clearly you can divide a gym class into 3 groups of 8, or 2 groups of 12. 72.
At least 2/3 of the 40 members of a committee must vote in favor of a resolution for it to pass. What is the greatest number of members who could vote against the resolution and still have it to pass? So essentially they're saying, what is the least number that we need to have it pass, and then 40 minus that is the greatest number that could vote against it. So we have to have at least 2/3 of 40. So what's 2/3 times 40? Well that's equal to 80/3. That's 3 goes into 80, let's figure it out. 3 goes into 8 2 times.
2 times 3 is 6, 20. 3 goes into 20 6 times. 6 times 3 is 18. You bring down a 2 and then a 0.
So it's going to just keep repeating. So you have to have at least 26.6 people vote for it. But clearly you can't have a 0.6 person vote for it. 26 people isn't going to be good enough. So you need at least 27 people. So if 27 are going to vote for, how many going to vote against? Well that means 40 minus 27, that means 13 against, which is E. That is the greatest number. Because if a larger number voted against it, you would have less people voting for it.
Problem 73. In the Johnson's monthly budget, the dollar amounts allocated to household expenses, food-- so I'll say h for household expenses, f for food-- and miscellaneous items, are in the ratio of 5:2:1-- fair enough-- respectively. If the total amount allocated to these three categories is $1,800, so $1,800 is the total, what is the amount allocated to food? So we could view it this way. So food is going to be twice times miscellaneous, and household items is going to be equal to 5 times the miscellaneous.
So we could say that household plus food plus miscellaneous is going to be equal to $1,800 This ratio tells us that food is 2 times miscellaneous, household is 5 times miscellaneous. So let's just substitute that in. 5 times miscellaneous plus 2 times miscellaneous plus miscellaneous is equal to $1,800 Let's see, 5 plus 2 is 7 plus 1 is 8. 8 times miscellaneous is equal to $1,800 So m is equal to $1,800 over 8. Let's see, what does that turn into? That is equal to 900 over 4, which is equal to 450 over 2. That's how my brain works, I have to go in steps.
What's 450 divided by 2? That's 225. That's miscellaneous. They want to know how much is allocated to food. Well, food is 2 times miscellaneous. So 2 times 225 is equal to, we're back at 450. And that is choice D. Problem 74. There are 4 more women than men on Centerville's Board of Education.
So 4 more women, so women is equal to men plus 4. If there are 10 members on the board, how many are women? So we know that men plus women is equal to 10. We want to solve for women. So let me solve for men here, then substitute there.
So you get men is equal to 10 minus the women. We could substitute that back there. We get women is equal to m, or the men, which we just figured out is 10 minus the women plus 4. My brain was jumping ahead. So let's add w to both sides.
So you get 2w is equal to 10 plus 4, which is equal to 14. w is equal to 7. There are seven women on the board, so that's D. Problem 75. After this video, I think I'm going to call it for a night and go watch a movie or something. 75.
Leona bought a 1 year $10,000 certificate of deposit that pays interest at an annual rate of 8%. So 1 year, $10,000, and it paid a rate of 8%. Compounded semi-annually, what was the total amount of interest paid on the certificate at maturity? So when you say compound semi-annually, that means you collect half of the interest every half year. That's something I don't know if the GMAT, they should be expecting you to know that, it's something you probably would learn in business school. But anyway, so after 6 months you're going to get 4% on the original amount. So you're going to have $10,000 dollars.
And you're going to 4% percent of this. So they're going to owe you-- compounded semi-annually, what was the total amount of interest paid on the certificate at maturity? So they're going to pay you $10,000 times half of it, because only six months have gone by. 8% is for the full year.
So for half the year, you're going to get times 4%. So let's see, $10,000 times 4%, what is that? $10,000 times 0.04. Let's look at it this way, 4 times $10,000 is $40,000. We have two numbers behind the decimal place, so it's $400. So they're going to pay you $400.
So then you can view it essentially, they owe you $10,400. That's essentially how much they borrowed. Because they're not going to give it to you just then. You're going to get all your money back at the end. So at the end, they're going to have to pay you another 4%.
But this time, it's not going to be off of the $10,000. It's going to be off of the $10,400. So $10,400 times 0.04. 4 times 0 is 0.
4 times 0 is 0. 4 times 4 is 16. 4 times 0 is 0 plus 1 is 1. 4 times 1 is 4. Then we have two numbers behind the decimal point. So now they're going to pay you $416, the second payment. What was the total amount of interest? So it's going to be $416 plus $400, which is $816, which is choice C.
I'm a little surprised that they expect you to know what compounding semi-annually means. But since it's there, I guess you should know it. When it compound semi-annually, it means that each semi-annual point in time, you divide the annual interest rate by 2, and you get that much. But then the next interest you get is off of, not just the initial amount, but the initial amount plus the previous interest, because they haven't paid you yet. But anyway, I hope you don't find that too confusing. You might want to look it up on the internet. You could also compound monthly, or daily, or continuously.
I actually have a couple of videos on the Khan Academy. But anyway, I'm all out of time, so I'll see you in the next video.
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