GMAT: Math 15

We're on problem 81. They give us this equation, y is equal to 248 minus 398x. Which of the following values of x gives the greatest value of y in the equation above? To get as large a possible y, y is 248 minus this thing. So what we want to do is, since we're subtracting this from 248, we want to minimize this. As small as this can be. And even if we can make what we're subtracting negative, that'd be awesome, because when you subtract a negative, then that'll get us above 248. And actually, if we look at all of the choices, they have one negative number there. You might say if we put a 0 here, then y is going to be 248. But what if we put a negative number in for x, they have as choice E, negative 1. What happens when we put negative 1 there? Then y is equal to 248 minus negative 1 times 398, so it's minus minus negative 398, which is the same thing as plus 398. So that's actually the maximum point, because we're subtracting a negative. That's even better than subtracting 0, and it's way better than subtracting a positive, if we're trying to maximize y. So the choice is E. 82. Machine A produces bolts at a uniform rate of 120 bolts every 40 seconds. So that's the same as-- 120 bolts every 40 seconds is the same thing as 3 bolts per second. OK, that's A. Let me write that down, that's A. Machine B produces bolts at a uniform rate of, so this is B, of 120 bolts every 20 seconds. Well that's equal to 5 bolts per second. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts? So if you're running simultaneously, what's your speed per second? This one's going to do 3 per second, this one's going to do 5 per second. So if they run simultaneously, you're going to be producing 8 per second, 3 plus 5. So 8 bolts per second is their combined rate. And we want to know how many seconds will it take to do 200 bolts. So we just take 200 bolts divided by 8 seconds. Another way with distance, distance is equal to rate times time, I guess you could view this as production is equal to rate times time. Normally rate is how fast you're moving per second, now it's how much you're producing per second, so your total production is going to be your rate times time, and then your time is going to be the production divided by the rate. But sometimes when you think too much in terms of formulas, you lose the intuition. But I just wanted to show you that there is a formula for it, if you need one. But the easiest way to think it, I have to produce 200 bolts, if I said I produce 100 every second, it'll take me 2 seconds. So if I have to produce 8 every second, I take 200 divided by 8. And I can even do the units, 200 bolts divided by 8 bolts per second. That's equal to 200 over 8 times bolts times-- if this isn't the denominator, you just inverse it when you go out-- seconds per bolt, the bolts can't [UNINTELLIGIBLE]. So you know you got the units right, so to some degree that should give you conviction that you got the formula right. And 200 divided by 8 is 25. So it's 25 seconds. It's choice B. Problem 83. What is the decimal equivalent of 1/5 to the fifth power? So let's think about this. Let's write 1/5 as a decimal, that's 0.2. So it's 0.2 to the 5th power. Let's just multiply it out. 0.2 times 0.2 is equal to what? 2 times 2, and we have two digits behind the decimal, so it's 0.04. And you multiply that times 0.2. 2 times 4 is 8, and now we have three numbers behind the decimal point, that's going to be equal to 0.008 One, two, three. Three numbers behind the decimal. So this is 2 to the third. And then 0.008 times 0.2, let me write this is to the third power. That's equal to 2 times 8 is 16, and then how many numbers do we have behind the decimal point? One, two, three, four. So we're going to have point one, two, three, four. So that's to the 4th power. We have one left, so 0.0016 times 0.2 is equal to, 2 times 16 is 32. And now we have one, two, three, four, five numbers behind the decimal point. So one, two, three, four, five. That's the answer, and that is choice A. Actually the easiest way to think about it is, without even having to do this, you could've said, what's 2 to the fifth power? 2 to the fifth power is 32. And actually that alone, because there's only one choice that even has a 32 behind the decimal point. And you can say, every time I take to a power, I'm adding one digit behind the decimal point, so it's going to be 32, but it's going to have five numbers behind the decimal point, so point one, two, three, four, five. I did it the slow way. Next question, 84. 90 minus 8 times 20-- I'll write it the way they did-- 20 divided by 4, all of that over 1/2. This is just a fast simplification problem. That becomes 90 minus-- what's 20 divided by 4? 20 by 4 is 5. 5 times 8 is 40, all that over 1/2. That becomes 50 over 1/2, which is the same thing as 50 times 2 over 1, which is equal to 100. That's choice C. That's just one you just have to do quickly and not make a careless mistake. I did it quickly, I don't know if I did a careless mistake. Let's see, 85. A dealer originally bought 100 identical batteries at a total cost of q dollars. So q is equal to 100 batteries. If each battery was sold at 50% above the original cost, so b is the cost each battery was sold at, so b is going to be 50% above the original cost. So it's going to be 1.5 times the original cost. This is 50% above the original cost per battery. Then in terms of q, for how many dollars was each battery sold? A dealer originally bought 100 identical batteries at a total cost of q dollars. If each battery was sold-- oh I'm sorry. So this is the cost. So q is equal to 100 times what the dealer paid for it. A dealer originally bought 100 identical batteries at the total cost of q dollars. If each battery was sold at 50% above the original cost per battery, this is the original cost, is equal to 1.5 times the cost. In terms of q, for how many dollars was each battery sold? So let's write q in terms of the cost. So cost is equal to q divided by 100. Just divided both sides of that equation by 100. And then we could just substitute c, or the cost here, the selling price is 150% of this. So selling price is equal to 1.5 times c, which is q over a 100. And let's see how they did it. So selling price is equal to 1.5q over 100. I don't expect to see that as a choice. But if we multiply the top and the bottom by 2, that equals 3q over 200. That's choice A. Just need to make sure you get your variables right on that one. Problem 86. In an increasing sequence of ten consecutive integers, the sum of the first five integers is 560. What is the sum of the last five integers in the sequence? OK, ten consecutive integers. So x plus 1 plus x plus-- let me write it this way-- x plus 1 plus x plus 2, well actually the first one, let's call it x, x plus 3, x plus 4. This right here is going to be the first five consecutive integers if we start at x. When you take the sum of those-- well the sum of those is 1, 2, 3, 4, 5x. Let's see, 1 plus 2 is 3, 3 plus 3 is 6, 6 plus 4 is 10. 5x plus 10 is going to be equal to 560. Fair enough. Or we could say 5x is equal to 550. I think we can use that information. Now what is the next five integers going to be? It's going to be x plus 5, x plus 6, x plus 7, x plus 8, and x plus 9. These are five, and what do those add up to be. That's 5x plus-- 5 plus 6 is 11 plus 7 is 18 plus 8 is 26 plus 9. 26 plus 9 is 35. So that's 5x plus 35. And they want to know what this is equal to. That's the next five. Well we know what 5x is, it's equal 550. From this set up, so it's going to be equal to 550 plus 35, which is equal to 35 plus 550, which is 585. That's choice A. And the trick here to being able to do it fast is just to recognize that you didn't have to solve all the way for x. That you could reuse this 5x when you took the sum of the next five. And I'm all out of time, see you in the next video.