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GMAT: Math 22

115-119, pgs. 168-169. Created by Sal Khan.

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We're on problem 115. Mary's income is 60% more than Tim's income. So that's 1.6 times Tim's income. You could write it as Tim's income plus 60% of Tim's income, but hopefully it's second nature that this is just 1.6 times Tim's income. So if Mary's income is equal to 1.6 times Tim's income, and Tim's income is 40% percent less than Juan's income. So Tim's income, if it's 40% less than Juan's income, that means that it's 60% of Juan's income. You can do that as Juan's income minus 40% percent of Juan's income. That's 0.6 times Juan's income. What percentage of Juan's income is Mary's income? So let's just do a little substitution. Substitute this T over here. So you get Mary is equal to 1.6 times 0.6 times Juan. And then what's 6 times 16? 60 plus 36. 96. We have 2 numbers to the right of the decimal. So it's 0.96 times Juan's income. So what are they asking? What percentage of Juan's income is Mary's income? Mary's income is 96% of Juan's income. And that's choice C. 0.96 is 96% of. Problem 116. Looks like something I have to draw. Each dot in the mileage table above represents an entry indicating the distance between a pair of five cities. Let me try to draw this out quickly. So we have five cities. It's 1, 2, 3, 4, and 5. And then we have to do it the other way. 1, 2, 3, and then 4. They have the same cities on both axes. City A, B, C, D, and E. City A, B, C, D, and E. And they have some dots. That's a dot, a dot, a dot, a dot, then dots down that way. Each dot in the mileage table above represents an entry indicating the distance between a pair of the five cities. If the table were extended to represent the distances between all pairs of 30 cities, and each distance were to be represented by only one entry, how many entries would the table then have? So they're saying, OK, this is essentially the case with five cities. You don't want to care about the distance between A and A. They want the distance between all pairs of 30 cities. So this is the case when you have n is equal to 5. And the reason why you could say the distance between A and B is this dot, you could have also said the distance between B and A is this dot down here. But they didn't draw a dot there, because they said and each distance were to be represented by only one entry. So they don't want to double-- if this was 5, this would also be 5. They only want to write it there once. So relative to 5, how would we figure this out? So if you think about it, what you're going to have is-- no matter how big this table is, the diagonal is going to be empty. You don't have to fill out the distance between the city and itself. And then if you just continue the trend that they did, out of what's left over, you're just going to fill out half of them, because the other half is just redundant information. So for example, if you have 5 cities, you're going to have 5 times 5 is equal to 25 squares. How many squares are going to be in the diagonal? We're going to have exactly 5 squares in the diagonal. 1, 2, 3, 4, 5. So you're going to subtract 5. So you're going to have 20 boxes left that aren't in the diagonal. And then you are going to have to to fill out half of those. So you're going to fill out exactly 10 boxes, so 1, 2, 3 4, 5, 6, 7, 8, 9, 10. So let's do the same exercise with 30. If you have 30 cities, you're going to have 30 times 30 boxes, which is equal to 900. You're going to take out the 30 boxes that are in the diagonal, so minus 30. So that's equal to 870. And then you are only going to have to fill out half of those, because you don't want redundant information. So you are only going to fill out the top half they way they did it in this diagram. So if you divide this by 2, you're going to get to what? 870 divided by 2 is 435. And that's choice B. Next question, 117. Which of the following has a value less than 1? So choice A is 2 times 7/13. 2 times 7 is 14, so that's greater than 13. So it's not A. B, square root of 10/2. The square root of 10, that's going to be 3 point something, which is greater than 2. So that's not right. Choice C is 2 over the square root of 2. The square root of 2 is less than 2. It's 1 point something. So this is going to also have a value greater than 1. So it can't be this. We're looking for less than 1. Choice D says 1 divided by 1/2. That's equal to 1 times 2, which is equal to 2. So I can already tell you the choice is going to be E. Let's look at it. So they have 9/10 squared. That's 81/100, which sure enough is definitely less than 1. So the choice is E. Problem 118. The ratio of the length to the width of a rectangular advertising display is approximately 3.3:2. So the length is 3.3 and the width is 2. At least that's the ratios. If the width of the display is 8 meters, what is the approximate length of the display in meters? So they tell us the length to the width is approximately 3.3:2, which is approximately equal to the length to 8 meters. They say the width of the display is 8 meters. So we just have to solve this as best as we can for L. The easiest way to think about it is to go from 2 to 8, you multiply by 4. So to go from 3.3 to L we have to multiply by 4. So L is equal to 3.3 times 4. What's that? 33 times 4, or 3.3 times 4. 4 times 3 is 12, plus 1 is 13. One number behind the decimal point. 13.2. So they say approximate, and the closest one is choice C, 13. Problem 119. The average salary of 15 people in the shipping department of a certain firm is $20,000. So the average of 15 people is equal to $20,000. The salary of 5 of the employees is $25,000 each, so we can even say the average of those 5 is going to be $25,000 because all of them make $25,000. And the salary of 4 of the employees is $16,000 each. The average of 4 of the employees is $16,000 each. That's the salary, but if you average 4 people making $16,000, the average is going to be $16,000. What is the average salary of the remaining employees? Let's think about this. Let's try to find the average of all of the employees. So if we find the average of all 15 employees, we're going to sum up all their salaries, divide by 15. And we know that that is going to be equal to $20,000. Now we have these 5 guys making $25,000, so the sum of their salaries is going to be 5 times $25,000. And then you have these 4 people making $16,000, so it's plus 4 times 16. I'm just trying to sum up everyone's salaries up here in the numerator, and I'm going to divide by 15. And then I'm going to have what? The average of the remaining people. How many people are remaining? I've already done 9 people, so there's 6 people remaining because there's a total of 15 people. So the average of the 6 remaining people times 6. Whatever their average is, times 6, that'll give you the sum of their salaries. And now we just solve for the average of those 6. Let's see what we can do. So 5 times 25, that's 125. Plus 4 times 16 is 64. Plus 6 times-- let's call it A-- 6A is equal to 20 times 15. So what's 20 times 15? 20 times 10 is 200. 20 times 5 is 100. So it's equal to 300. 125 plus 64 is 189 plus 6A, is equal to 300. Subtract 189 from both sides. You get 6A is equal to 300 minus 189. That's 111. 89, add 11 to it, you get to 200. And then you have to add another 100 to get to 300, so it's 111. 300 minus 189. And then how many times does 6 go into 111? Divide both sides by 6. 6 goes into 111-- 6 goes into 51 eight times. 8 times 48. 18.5 times. $18,500 is the average of their salary. And luckily enough, that's one of the choices. That's choice B. $18,500. We did everything in [? thousandths ?] so we got 18.5, but that's $18,500. See you in the next video.