GMAT: Math 19

Problem 100. They draw this graph, which I guess I should also draw. So they draw it pretty big. So I'll draw it pretty big as well. They have a line that goes from the center to there, it's a pie graph. Then they call that category like that, and another category that looks something like that, and then they have a final category that looks something like that. They say that this is the highway trust fund, which is 72%. They say x% is a tire tax. This is 12%, which is a gas tax. Then you have x plus 4%, which is a tax on trucks. According to the graph above, what percent of the funds for highway maintenance-- oh, and this title says sources of funds for highway maintenance. So that's what this whole thing is. According to the graph above, what percent of the funds for highway maintenance came from the tax on tires? So we need to solve for x. So let's think about it. x plus 12 plus x plus 4, that's this, plus 72 have to equal 100%. We have two x's, get those out of the way, and then what is this equal to? 12 plus 4 is 16. 16 plus 72-- let me write that down. I always make careless mistakes. That's 16. 16 plus 72 is 88. 2x plus 88 is equal to 100. Then you have 2x is equal to, subtract 88 from both sides, 12. x is equal to 6. That's exactly what we're looking for, the tire tax. So that's 6%. That's choice B. Drawing it was harder than solving it. Problem 101. A poll reveals that the average arithmetic mean income of 10 households is $25,000. So we could say, average of 10 is equal to 25K. If 6 of the households have income of $30,000 each, what is the average of the other 4 households? So if we were to average all of them, we could say it would be 6 times the households that are 30,000, so 6 times 30K, plus 4 times the average of the other 4. So let's say that's what we're trying to solve for, 4 times the average of the other 4 households. This would be the sum of all of the households. If we were to divide that by 10, we're going to get the total average of all 10, which is 25. Hopefully that makes sense. You have 6 households making 30K, so the sum of those households is going to be 6 times 30. Then the sum of the other 4 households is going to be 4 times their average. That's what they're asking us for, what is the average income? If you take the sum of all the households, all 10, you divide it by 10, you get their average. So you get 6 times 30-- I don't want to make this look like a variable, that's 30,000. We know we're dealing in thousands. So 6 times 30 is 180 plus 4x is equal to 10 times 25, is equal to 250. 4x is equal to, what's 250 minus 180? That's 50, and you have another 20, it's equal to 70. Then you have x is equal to, let's see, 4 goes into 70, 130, goes into it 7, 7 times 4 is 28. Goes into 20 5 times, and the decimal. So 17.5, and we're dealing with thousands, so $17,500. That's choice C. Problem 102. If T is equal to 5/9 times K minus 32, and T is equal to 290, then what is K equal to? This is kind of one of those speed problems. So we have 290 is equal to 5/9 times K minus 32. So multiply both sides by 9/5, you get 9/5 times 290 is equal to K minus 32. So what's 290 divided by 5? 5 goes into 29 5 times, 25, 40, 58 times. So it's 9 times 58 is equal to K minus 32. 58 times 9, 8 times 9 is 72. 5 times 9 is 45 plus 7 is 52. 522. So 522 is equal to K minus 32. So K is equal to 522 plus 32, which is equal to 554, Which is choice D. After doing all that math, I was afraid that I'd made a careless mistake. But it's reassuring to see that 554 is one of the choices. Problem 103. The water from one outlet flowing at a constant rate can fill a swimming pool in 9 hours. So you could say 1 pool per 9 hours from 1 outlet. The water from a second outlet flowing at a constant rate can fill the same pool in 5 hours. So 1 pool in 5 hours, that's the second one. If both outlets are used at the same time, approximately what is the number of hours required to fill the pool? So essentially, we can just add their two rates. This one does 1/9 pool per hour, plus 1/5 pool per hour. I just rewrote those two. So what's 1/9 plus 1/5? Let's find a common denominator. 45 I think is the first common denominator. 1/9 is 5 over 45. 1/5 is 9 over 45. So together, they fill 5 plus 9 is 14/45 pool per hour. If you want to fill exactly 1 pool, so it's just like production is equal to rate times time. So now your production is, you want to fill 1 pool, and you can fill it at 14 over 45 pools per hour, what is the time? Well the time is going to be equal to 1/1 pool divided by 14/45 pools per hour, which is equal to, 45/14-- 1 over a fraction is just the inverse of that fraction-- so 45/14 hours. How many times does 14 going into 45? 14 goes into 45, is it 3 times? 3 times 14 is 12, is 42. Then you have another 30. Well, I already see one of the choices, 3 point something. But 14 goes into 30 2 times. 2 times 14 is 28. Let's see if one of our choices is already-- yes, choice D is already close enough to what we did. We don't have to do any more math. So we get 3.2 hours. If we'd kept dividing, we'd get 3.21. That's choice D. It's just a little confusing. The important thing is to always get it in terms of rate. Add the rate, and just remember distance is equal to rate times time, or production is equal to rate times time. Don't let all these fractions daunt you too much. Next problem. I'm almost out of space. Problem 104. Diana bought a stereo for $530, which was the retail price, plus a 6% sales tax. So that is equal to 1.06 times the retail, 6% percent more than the retail. How much could she have saved if she had bought the stereo at the same retail price in a neighboring state, where she would have paid a sales tax of 5%? So essentially we want to figure out how much she would've saved. So let's figure out what the retail price was. So retail is equal to 530 divided by 1.06. 1.06 goes into 530, it's a decimal, add two 0's, shift the decimal over to the right twice. 106 goes into it 5 times, right? 106 goes into 530 5 times. 5 times 106, it's 530. OK, so the math here was simple. so you have two 0's. So the retail price is $500. So she paid $30 in tax, fair enough. If she went to the neighboring state and paid 5%, how much would she have paid in taxes? So times 0.05, 5 times 5, well 5 times 500 is 2,500, plus two decimals, so it's $25 in tax. So in the neighboring state, she would've paid $25 in taxes. She paid $30 in taxes. So she would have saved $5. So the choice is D. Let's just try to fit one more problem in here, just so we finish this page. They say, if a square mirror has a 20 inch diagonal, what was the approximate perimeter of the mirror in inches? Let's try to do that. That's not what I want to do. I have a square mirror, it has a diagonal. That diagonal, they're telling us, is 20 inches. It's a square, so all the sides are the same. They want to know its perimeter. So they want to know essentially what 4x is equal to if we add up all the sides. So what is x? Well we just use Pythagorean theorem. This is a right angle. So we know that x squared plus x squared is equal to 20 squared, is equal to the hypotenuse squared, is equal to 400. 2x squared is equal to 400. x squared is equal to 200. x is equal to the square root of 200. They want to know the perimeter. Well, it's going to be 4 times that. So the perimeter is going to be equal to 4 times the square root of 200. Now this becomes an approximation problem. So let's think about that. The square root of 200 is about 15. 15 times 15 is 225. Yeah, I think 15 would be a pretty good approximation. 14 would also work. 14, when you square it, is 196. 14 times 14, 4 times 16 is 56. Then you have 140. So 14 is about as close as you're going to get to 200. It's going to be someplace between 14 and 15. So 14 times 4 is 40. So 4 times 4 is 16, it's 56. So it's going to be something higher than 56. If we pick 14, it's 56. If we pick 15, it's 60. So someplace between 56 and 60 is going to be the perimeter of that mirror. So it's one of the choices B, 60. So that's going to be the closest one. So I'd go with B. I'll see you in the next video.