GMAT: Math 29

We're on problem 148. What is the lowest integer that is the sum of three different primes, each greater than 20? So it's the lowest integer, so we essentially just have to find the three primes above 20. So let's see, 21 isn't a prime, 22 isn't a prime, 23 is a prime. 24, no, 25, no, 26, no, 27, 28, nope. 29, that's prime. 30 isn't, 31 is. 31 plus 29 is 60 plus 23 is 83, and that is choice E. I hope I haven't missed any primes, no, I don't think I have. 24, 25, 26, 27, 28, I think I got all the primes in there. What is the lowest integer of the sum of three different primes, each greater than 20? Yep, that's the answer, E. Problem 149. The average of 6, 8, and 10 equals-- so the average of 6, 8, and 10, so first of all, what's the average of 6, 8, and 10? Well, let's just do it. 6 plus 8 plus 10. You could immediately see the average is going to be 8, because you have 8 in the middle, and both of these numbers are 2 away from it. But that divided by 3 is equal to 24 divided by 3 is equal to 8. And that's equal to the average of 7 plus 9 plus x over 3, so that tells us to multiply both sides of this equation by 3, that says that 24 is equal to-- what's 7 plus 9? That's 16 plus x. Subtract 16 from both sides, you get x is equal to 8. Choice C. And you could have said, OK, 7 and 9 or both half-- 8 is in the middle, and they have to average 8, so you might have just been able to eyeball it and say 8, but it didn't take that much math, it's not that much of a shortcut. Problem 150. If x is equal to minus 1, then x to the fourth minus x to the third plus x squared, all of that over x minus 1 is equal to what? x to the fourth is 1. x to the third is minus 1, but then we're going to minus that, so this whole thing is going to be plus 1, and x squared is plus 1. Negative 1 squared is 1. And you have minus 1 minus 1. So in the numerator you have 1, 2, 3, in the denominator you have minus 2. Let's put the minus out front, minus 3/2. That's choice A. Problem 151. A toy store regularly sells all stock at a discount of 20% to 40%. If an additional 25% were deducted from the discount price during a special sale, what would be the lowest possible price of a toy costing $16 before any discount? So we want to discount the $16 as much as possible. So they say they're normally discounting 20% to 40%, so if you want to discount this as much as possible, you want to discount this by 40%. So if you discount this by 40%, that's the same thing as saying it's going to be equal to 60% of its original price. So this would be the price after being discounted by 40%, and then we would discount by another 25%. Actually, I think it's easier to go the other way. Well, let's just do it this way. So we discount it by another 25%, so we multiply it by 3/4 again. If you discount by 25%, the price is going to be 3/4 of its original price. Actually, let's do it as fractions. Fractions are always easier. 60% percent of your price is 3/5. If you discount by 2/5, or 40%, that's the same thing as multiplying by 3/5, and then we're going to multiply by 3/4. Set 25%. If you're discounting by 25%, that's the same thing as being worth 75% of your original price. So what does that get us? 16 divided by 4, 4, you get a 1, and then you have 4 times 3 times 3, 4 times 9 is 36, over 5, which is equal to-- 5 goes into 36 7 and 1/5. So $7.20. And that's choice B. Problem 152. The shaded portion of the rectangular lot shown above represents a flowerbed. OK, so let me draw this rectangular lot. They drew this shaded portion something like that, and I can even color it in. So they say the shaded portion of the rectangular lot shown above represents a flowerbed. If the area of the bed is 24 square yards, and x is equal to y plus 2, so they tell us this side right here is x, which now they're telling us is equal to y plus 2. And this right here is going to be equal to y. Then z equals what? z is this side right there. So if this area is 24, that tells us that x times y times 1/2 is 24. That's the area of that. So 1/2xy is equal to 24. Or you can even say that xy is equal to 48. We know what xy is. We can just substitute for x, so we get y plus 2, that's x, right? X is equal to y plus 2 times y is equal to 48. And so we get y squared plus 2y is equal to 48. Or y squared plus 2y minus 48 is equal to 0. We could factor this quadratic. What two numbers when you multiply by minus 48, and you add you get plus 6? So y plus 8 times y minus 6 is equal to 0, 8 times minus 6 is minus 48, and then 8 minus 6 is plus 2, so y could be equal to minus 8 or plus 6, right? Whatever numbers make either of these equal to zero. Well, we can't have a negative distance, so y is not equal to minus 8, so y is equal to 6. y is equal to 6, x is equal to 8. Now, we can just confirm, 6 times 8 is 48 times 1/2 is equal to 24. And now we can use the Pythagorean theorem to figure out z. So we'll get z squared is equal to 36. That's y squared plus 64, which is x squared, so z squared is equal to 100. z is equal to 10. Choice E. Problem 153. Jack is now 14 years older than Bill, so Jack is equal to Bill plus 14. If in 10 years Jack will be twice as old as Bill, so 2 times Bill's age, how old will Jack be in 5 years? So they want to know j plus 5. That's how old Jack will be in 5 years. So let's just try to solve for Jack. This top equation tells us that Bill is equal to Jack minus 14, and then we can substitute this in for Bill here. So then this equation will become Jack plus 10 is equal to 2 times Jack minus 14. So Jack plus 10 is equal to 2 times Jack minus 28. If we subtract Jack from both sides, you get 10 is equal to Jack minus 28. We can add 28 to both sides, and you get 38 is equal to Jack. So that's how old Jack is now. He's 38 years old. So how old is Jack going to be in 5 years? 38 plus 5 is equal to 43 years old. And I've clearly made a mistake, because that's not one of the options. So let's see where I'm made an error. So Jack is now 14 years older than Bill. Fair enough. So in 10 years, Jack will be twice-- oh, I see my mistake. He's going to be twice as old as Bill is going to be in 10 years. That was a bone-headed mistake. So let me clear all of this. That was a bad mistake. I shouldn't have made it. I'm lucky that they didn't have what I put down as one of the choices, and I would've gotten it wrong. All right. So we know that in 10 years, Jack is going to be 2 times what Bill is going to be in 10 years. That was my mistake. So now let's see what we can do. We'll use that same information up there, so Bill is equal to Jack minus 14. Substitute that information here, so we get Jack plus 10 is equal to 2 times Jack minus 14, instead of Bill, plus 10. So you get Jack plus 10 is equal to 2 times Jack. Let's see, minus 14 plus 10, that's minus 4, so Jack plus 10 is equal to 2 times Jack minus 8. Let's subtract Jack from both sides. 10 is equal to Jack minus 8. Let's add 8 to both sides. You get 18 is equal to Jack. So that's how old Jack is right now, he's 18. So in five years, Jack is going to be 18 plus 5, which is 23 years old, choice D. And I'm out of time. See you in the next video.