1. LA EDUCACIÓN ESPECIAL EN LA HISTORIA RECIENTE DE ESPAÑA
1.3. LA DEMOCRATIZACIÓN DE LA EDUCACIÓN ESPECIAL: DE LA LODE A
Distance
Time or Rate =
Amount of Work Time
• Use the Rate Pie to organize your information.
Median, Mode, and Range
• The mean is the average of all the numbers in a set.
• The median is the middle number in a set.
• The mode is the number (or numbers) that appear(s) most frequently in a set.
• The range of a set is the difference between the highest number and the lowest number in the set.
Standard Deviation
• On the GMAT, problems dealing with standard deviations, or normal distributions, are about the bell curve.
• The mean, or average, is the center of the bell curve.
• A standard deviation is a specific distance away from the mean.
• The percentages 34%, 14%, and 2% correspond to the 1st, 2nd, and 3rd standard deviations on each side of the mean.
Assignment 3
D
RILL1. If a certain concrete mixture contains gravel, water, and sand in a 1 to 3 to 5 ratio by weight, how many pounds of gravel would be needed to produce 72 pounds of the mixture?
72 40 24 9 8
2. A disc jockey plays only hip-hop and country-western records. If the disc jockey plays four country-western records for every seven hip-hop records he plays, what fraction of the records he plays are country-western?
3. It’s rumored that in 1976, Elvis consumed three times his body weight in peanut butter and banana sandwiches. If Elvis’s body weight in 1976 was 250 pounds, and if a peanut butter and banana sandwich weighs four ounces, then how many such sandwiches did Elvis consume in 1976? (1 pound = 16 ounces)
750
5. The average (arithmetic mean) of eight numbers is 7. If two numbers are discarded, the average of the remaining numbers is 6
1
2. What is the average of the two discarded numbers?
6. The average (arithmetic mean) of 6, 21, x, and y is 13, where x and y are integers with a product of 100. Which of the following could be x?
50
integers in increasing order, what is the average of the five numbers?
(1) a = 13 (2) a + e = 30
8. For the set of positive, distinct integers {v, w, x, y, z}, the median is 10. What is the minimum value of v + w + x + y + z?
9. The sum of the numbers in the set {5, 28, 10, 50, 24}, let x equal the median, v equal the mean, w equal the mode, and y equal the range. Which of the following is true?
v < w < x < y v < x < w < y y < v < w < x y < v < x < w w < y < v <x
11. The mean score on a certain test is 80. If the test scores have a normal distribution and the stan-dard deviation is 7, what percent of examinees score between 66 and 94?
50%
68%
82%
96%
98%
12. A courier van delivers a package from office A to office B. If the van travels the 20 miles between the two offices at 50 miles per hour, how fast must the van travel on the return trip if the round-trip travel time is to be 40 minutes?
55 mph 60 mph 65 mph 70 mph 75 mph
13. Runners X and Y started an 18-mile race at the same time. Runner X completed the course in 6 hours, and Runner Y finished 2 hours earlier.
Runner Y ran an average of how many miles per hour faster than Runner X?
1
14. The Lewiston Road Race is how many kilometers long?
(1) It takes 2 more hours to complete the race at an average speed of 40 kilome-ters per hour than at an average speed of 50 kilometers per hour.
(2) At an average speed of 20 kilometers per hour, it took Brett 10 hours to complete half the race.
15. Three typists worked to type a single manuscript.
What percent of the total number of words was typed by the slowest of the three typists?
(1) All three typists typed steadily and independently for 4 hours at the rates of 40, 50, and 60 words per minute to complete the manuscript.
(2) The fastest typist typed 150% as fast as the slowest.
Assignment 3
A
NSWERSANDE
XPLANATIONS 1. E Set up a Ratio Box:Gravel Water Sand Total Concrete
Ratio 1 3 5 9 multiplier for gravel, and 1 × 8 = 8. Note: you could also ballpark out some of the larger answer choices, like (A) and (B).
4. A Set up AD/BCE. Fact (1) gives you 3x = 4y. It seems you can’t solve for x or y, but then again, you don’t have to! You are just trying to get the ratio, so rearrange your equation to get
x y= 4
3 , and that’s sufficient! Cross off BCE. Looking at Fact (2), x = 4 alone is insufficient without any information about y. Eliminate (D).
5. D Draw some average pies! The first pie is for all 8 numbers, which average 7. This pie is going to be split into two groups, so you need two more pies, one for the remaining 6 numbers, and one for the discarded two. The average for the remaining 6 numbers is 6.5, which means the total is 39. Subtract that from the previous total of 56 from the first pie, 8 × 7, to get the remaining total 17. 17 is the total for the two discarded number pie, so the average is
81 2 .
6. B If the average of the four elements is 13, their total must be 13 × 4 = 52, so 6 + 21 + x + y = 52, and thus x + y = 52 – 21 – 6 = 25. So if x and y add up to 25 and their product is 100, then they could be 20 and 5. Since 5 is not an option, the answer is 20.
2. D The ratio of country-western to hip-hop records is 4 to 7, but the question asks, “what fraction of the records” are country-western, so you want the
3. D There are a couple of different proportions to set up. First, convert ounces to pounds,
4lb, or 0.25 lb per sandwich. Elvis ate a total of three times his body weight of 250 lbs, or 750 lbs, worth of sandwiches. The next proportion is:
. = . Cross-multiply
to get 750 = 0.25x, so x =
750
0 25. = 3,000 lbs.
7. D What is the missing piece of the puzzle? To get an average, you need a total, and number of things. We have 5 things, so you just need the total. Fact (1) tells you a = 13, and the other numbers are consecutive integers, so b = 14, c = 15, d = 16, e = 17, and we know we can add those up. Get rid of BCE. Fact (2) tells you a + e = 13, which looks like not enough information, but because you know the numbers are consecutive, e must be 4 numbers bigger than a and you can figure out both, and all the numbers in between.
If you are doubtful, plug in values for a and e, and the only ones that work are 14 and 17.
8. C The key here is that we’re trying to find the minimum value of five distinct, positive integers where the median is 10. If the middle number is 10, the smaller two numbers must be 1 and 2, and the larger two numbers must be 11 and 12, and 1 + 2 + 10 + 11 + 12 = 36.
9. B They tell us that the numbers in the set {5, 28, 10, 50, x, 7} have the sum of 120, so we can figure out that x = 20, which also happens to be the mean. But we’re looking for the median.
Rearranged, the set now looks like {5, 7, 10, 20, 28, 50}. Since 10 and 20 are the numbers in the middle, we’ll split the difference and get 15.
10. D It looks like we’re going to have to calculate the mean, median, mode, and range for {20, 14, 19, 12, 17, 20, 24}. Do the easy calculations first: The median x = 19, and the only recurring number, 20, is mode w. The range y = 24 – 12 = 12. The only answer choice in which y < x < w is (D), so we don’t have to calculate the mean after all.
11. D The range from 66 to 94 represents 14 points above and below the average. If the standard deviation is 7, then 14 points represents two standard deviations. Two standard deviations in each direction is 14 + 34 + 34 + 14 = 96% of all examinees.
12. E Make a Rate Pie! Plug in 20 miles for distance on top, and 50 miles per hour for the rate. The time it takes is then minutes. The round-trip time needs to be 40 minutes, so the time we have left for the return
trip is 40 24 16− = minutes, which is hour. So now we can make another Rate Pie, for the distance of 20 miles, and time of
13. B Make a rate pie for both Runner X and Runner Y. Runner X goes the 18-mile distance in 6 hours, so the speed is
18
6 =3 miles per hour. Runner Y finished 2 hours earlier, in 4 hours, so the rate is
14. D Set up AD/BCE. What is the missing piece of the puzzle? In order to get a distance, you need both rate and time. Fact (1) gives you both rates and times. Prove it to yourself by making 2 Rate Pies, one for a rate of 40 kilometers per hour, and one at a rate of 50. The times are Time + 2 and Time, respectively. Since the two distances are the same, you could say 40(T + 2) = 50T. One equation, one variable, so it’s solvable! Now check out Fact (2). We have Brett’s speed, and also the time for half the race, but you can double it and get the full time.
15. A Set up AD/BCE. Then figure out the missing piece of the puzzle. To get the percent of total number of words, you’ll need the total number, plus some information on how much of the total the slowest typist typed, which you could also get from the rates. Looking at Fact (1), which gives each individual’s rate and time, we could figure out each typist’s total number of words, and therefore the percentage of the slow one, so eliminate BCE. The second fact tells us the relative rates of two of the typists, but we have no information about the third person, so this is insufficient, and you’re left with (A).