РЕГИОНЫ РОССИЙСКОЙ ФЕДЕРАЦИИ 5 страница
6. A: 163 x 322 x 8 B: 642 x 43 x 16
The best answer is A. Translate all the numbers to a base of 2. Column A = 212 x 210 x 23 = 225. Column B = 212 x 26 x 24 = 222. We can see that column A is greater.
7. A: XY B: YX AI: Y > X
The best answer is D. Take two cases: X=0, Y=1 and X=-2, Y=0: In the first case, column A = 0 and column B = 1. In the second case, column A = 1 and column B = 0. Since the answer changed, D is the right answer.
8. A: (X + Y)(X – Y) B: X2 + Y2 AI: Y < 0 and X > 0
The best answer is B. Simplify column A = X2 – Y2. Subtract X2 from both sides: A = -Y2 and B = Y2. Column A is negative and column B is positive and therefore B is the right answer.
9. A: X2 + 4 B: X + 8 AI: X > 1
The best answer is D. Replace X with two numbers: 2 and 10. For X = 2: column A = 8 and column B = 10. For X = 10: column A = 104 and column B = 18. Since the answer changed, D is the right answer.
10. A: B: AI: X – Y = Y – X
The best answer is C. The additional data tells us that 2X = 2Y à X = Y. Replace X with Y in both columns or the opposite to see that both columns are equal to: And therefore the answer is C.
11. A: C + 3 B: C – 3
The best answer is A. No matter what C is, adding 3 to its value will surely make it larger and so the answer is A.
12. A: XZ B: YZ AI: X > Y Z < 0
The best answer is B. Plug in easy numbers: X=2, Y=1 and Z= -1. Column A = -2 and column B = -1 and therefore column B is greater.
13. A: P/121 B: P/212 AI: P < 0
The best answer is B. Lets say P = -(121)(212). Plug it into both columns: Column A = -212 and column B = -121. The number in column B is greater than that in column A.
14. A: (-13)X B: (-3)X AI:
The best answer is D. If X is a negative number, column A is greater, otherwise column B is greater and therefore the answer is not distinct.
15. A: B:
The best answer is A. Multiply both columns by Column A = 2 and column B = 3/2 = 1.5. We can see that column A is greater.
16. A: B:
The best answer is A. Multiply both columns by Column A = 7. Column B = And therefore column A is greater.
17. A: B:
The best answer is C. Multiply both columns by Column A = 7. Column B = We can see that both columns are equal and thus the answer is C.
18. A: X + Z B: Y + Z AI: X > Y
The best answer is A. Subtract Z from both sides and then you compare between X and Y. We know that X > Y and therefore Column A is greater.
19. A: X – Y B: W – Z AI: X + Z > W + Y
The best answer is A. Take the expression from the extra data and manipulate it: X + Z > W + Y è X – Y > W – Z and therefore column A is greater than column B.
20. A: Z9 B: Z10 AI: 0 < Z < 1
The best answer is A. When dealing with a fraction, remember, the smallest the power the larger is the result. The power of the fraction (Z) in column A is smaller and thus the number there is greater.
21. A: B: The best answer is C. Simplify the expression in column A:
We can see that the expressions in both columns are equal.
22. A: B:
The best answer is A. Simplify the expression in column A:
The expression in the parenthesis in both columns is positive. Since the denominator in column B is greater, the answer is A.
23. A: B:
The best answer is B. Column A = Column B = The bigger the fraction in these cases, the larger the result and therefore column B is greater.
24. A: Y B: Y2 AI: Y < 1
The best answer is D. Take two different cases: Y = 0 and Y = -2. In the first case, both columns are equal to zero. In the second case, column A = -2 and column B = 4. Since we got a different answer each time, D is the right answer.
25. A: B + A – C B: C + A – B AI: A > B > C
The best answer is A. Subtract A from both sides. Column A = B – C. Column B = C – B. Since B > C, column A is greater.
26. A: B: AI: X – Y > 0 X – Y > Z Z < 0
The best answer is D. Let’s take two representative cases: (X – Y)=3, Z= -6 and (X – Y)= 4, Z = -1. In the first case, column A = -2 and column B = -1/2. In the second case, column A = -1/4 and column B = -4. Since we got different results each time, D is the right answer.
27. A: (333)(327) B: (329)(331)
The best answer is B. The best way to solve this question is by using the formula: (X - Y)(X + Y) = X2 – Y2. Column A = (330 + 3)(330 – 3) = 3302 – 32. Column B = (330 – 1)(330 + 1) = 3302 – 12. Now the answer is obvious, column B is greater.
28. A: (914)(980) B: (950)(944)
The best answer is B. The best way to solve this question is by using the formula: (X - Y)(X + Y) = X2 – Y2. Column A = (947 – 33)(947 + 33) = 9472 – 332. Column B = (947 + 3)(947 – 3) = 9472 – 32. Now the answer is obvious, column B is greater.
29. A: B: AI:
The best answer is A. First simplify both columns: Column A = S4 and column B = T4. Since the power is even, the numbers can even be even, it’s the absolute value that was interested in and therefore column A is greater.
30. A: B: AI:
The best answer is D. Simplify the expressions in both columns: Column A = Q5 and column B = P5. Since the power is odd, the absolute value is not enough to determine which is bigger. If for example one of the numbers is negative, we wouldn’t see it using the absolute value criteria. More data is required to determine the answer.
31. A: B: AI:
The best answer is B. Simplify both columns: Column A can be written as X4. Column B can be written as Y4. Since the power is even, the numbers can even be even, it’s the absolute value that was interested in and therefore column B is greater. If one of the numbers is negative, the even power will transform it into a positive number.
Relations between groups and overlapping
32. A: 25 B: The number of students that are excellent students and married AI: There are 75 students in the faculty. 35 are excellent students and 55 are married.
The best answer is D. No information was given to us about the overlapping between the groups and as far as we are concerned, all the excellent students can be married. We can also take the minimum case of overlapping: (55 + 35 – 75 = 15). The answer cannot be determined since not enough data was supplied.
33. A: The minimum number of customers that are satisfied with the service and that are not in a hurry B: 0 AI: In a supermarket there are 80 customers, 45 are in a hurry and 35 are satisfied of the service.
The best answer is C. The sum of the two groups is smaller than the whole and therefore the minimum overlapping can be zero and so the answer is C since both columns are equal.
34. A: The number of left handed dancers that don’t know how to dance the Waltz B: 5 AI: There are 120 dancers in the club, 80 are left handed and 50 don’t know how to dance Waltz.
The best answer is A. Let’s find the minimum and the maximum cases of overlapping. The maximum is when all the dancers who don’t know how to dance the Waltz are left handed, thus 50. The minimum overlapping is (50 + 80 – 120 = 10). Since both cases are larger than 5, column A is always greater than column B.
35. A: 10 B: The maximum number of horses that are not brownish yet domesticated AI: In a horse ranch 95 horses. 45 horses are domesticated and 55 are brownish.
The best answer is B. There are 55 brownish horses and therefore (95 – 55 = 40) are not brownish. This group can be contained inside the domesticated group and therefore the maximum number of horses that are not brownish yet domesticated is 40. Column B is greater than A.
36. A: 70% B: The percent of Americans that are proud citizens and that own a car AI: 35% of the Americans aren’t proud citizens and 80% own a car
The best answer is A. (100 – 35 = 65%) of the citizens are proud citizens. The largest overlapping between the proud citizens and the citizens that own a car is 65% and therefore 70% is larger. Column A is greater than column B.
Percent Comparison Questions
37. A: The money George spent B: The money Gloria spent AI: George spent 50% of his allowance. Gloria spent 30% of her allowance. The difference between the money they spent is 35 dollars.
The best answer is D. We can’t compare percentages of different wholes. We don’t know how much money each had prior to the spending process and therefore we cannot calculate anything.
38. A: The percent of Michael Jackson fans in Alabama B: The percent of Michael Jackson fans in North Carolina AI: 10% of the guys and 30% of the girls in Alabama are fans of Michael Jackson. In North Carolina 35% of the guys and 45% of the girls are Michael Jackson fans.
The best answer is B. The percent of Michael Jackson fans in Alabama is somewhere between 10% and 30% while in North Carolina, its between 35% and 45% and therefore column B must be greater than column A.
39. A: $34,000 B: The price of the car after the holidays AI: The price of a car is $34,000. During the holidays, there is a 12% decrease in the car’s price. After the holidays the price increases by 12%.
The best answer is A. This is an important principle to remember, an increase and then a decrease of a number (by the same percent) always results in a smaller number. Let’s work with easier numbers just to understand. Let’s say the price of the car is 30,000 and the decrease and increase was by 10%: During the holidays, the car costs: 27,000 After the holidays, the car costs: 29,700, which is lower than 30,000!
40. A: Jim’s income today B: $3,000 per month AI: Jim had an average paycheck of $3,000 per month. Last month Jim got promoted and his income grew by 15%. Today, his income decreased by 15%.
The best answer is B. This is an important principle to remember, an increase and then a decrease of a number (by the same percent) will always results in a smaller number and therefore his income decreased past $3,000 and so column B is greater.
41. A: The entire weight of all the lions in the zoo B: The entire weight of all the zebras in the zoo AI: In the local zoo, there are 20% more lions then zebras. The average weight of a lion is lower by 20% than that of a zebra.
The best answer is B. Let’s make up numbers, say that there are 10 zebras in the zoo and that each weighs 100 pounds è there are 12 lions and each weigh 80 pounds. The total weight of the lions is 12 x 80 = 960 pounds. The total weight of the zebras is 10 x 100 = 1000 pounds. Column B is greater than column A.
42. A: The cost of a sporting good in sport store A B: The cost of a sporting good in sport store B AI: In Somalia there is a 22% tax on all sporting goods. Sport store A sells sporting goods with a 25% discount on the price after tax. Sport store B sells sporting goods with a 25% discount on the price before tax.
The best answer is C. Let P be the price of a sporting good prior to the discount and the tax. The price in store A will be = P x 122% x 75%. The price in store B will be = P x 75% x 122%. The order of the multiplication doesn’t change the result and therefore both columns are equal.
Ratio Comparison Problems
43. A: The old ratio between the fans B: The new ratio between the fans AI: The ratio between “Bullets” fans and “Pirates” fans is 1:4. One fan joined each group of fans.
The best answer is A. If we want to keep the same ratio we need to add 4 fans to one group and 1 to the other. Since we added only one fan for each of the groups, the ratio between the groups got smaller. It was 1:4, now its 2:5.
Note: Zahi, I had a dilemma in this question- is 2:5 a smaller ratio than 1:4 or the opposite. If it’s a mistake, please fix it.
44. A: The ratio between the number of stamps that Jill has in her collection to the number of stamps that Joe has B: The ratio between the number of stamps that Joe has in her collection to the number of stamps that Jill has AI: Joe has 70 stamps in his collection more than Jill. Jill has 126 stamps in her collection.
The best answer is B. A ratio is actually a fraction. The ratio in column A is smaller than 1 since Jill has less stamps. The ratio in column B is greater than the ratio in column A since the ratio in column B is greater than 1.
45. A: The original ratio B: The new ratio AI: In the aviation camp, the ratio between the number of cadets to the number of instructors is 6:3. One cadet and two instructors joined the camp.
The best answer is C. The people who joined the camp are according to the ratio. If at first it was 3:6 or 1:2 now its (3 + 1):(6+2) = 1:2 and therefore the ratio didn’t change and the columns are equal.
46. A: 1:1 B: The new ratio AI: The ratio between the number of boys to the number of girls in the pool is 3:7. Seven boys and three girls joined the pool.
The best answer is D. At first, the 3 referred to the boys and 7 to the girls. The new kids that came to the pool came also in a ratio of 3:7, but in favor of the boys this time and thus we cannot compare between the ratios. Its also easy to look at it in a different way, there could have been 30 boys and 70 girls and so if kids joined the pool not in this ratio, it would have been impossible to find the new ratio.
47. A: The distance ratio between Mike and Spike B: The distance ratio between Slay and Mike AI: Slay, Spike and Mike are having a skateboard race. The ratio between their speeds is 6:3:2 respectively. The ratio between their traveling time is 4:6:12 respectively.
The best answer is A. The ratio between their distances is actually the ratio between their times multiplied by their ratio of speeds. The ratio of distances is: 24:18:24 or 4:3:4. And therefore the ratio in column A = 4/3 and the ratio in column B = 1. 4/3 > 1 and therefore column A is greater.
Digits and letters
48. A: X B: Y AI: X, Y and Z are different digits. Y + Z = XX.
The best answer is B. AA is a double digit number and since its made out a sum of two digits à A=1. Y can’t be zero because then Z would have to be larger than 9 and it cant be 1 because all the digits are different. The conclusion is that Y must be greater than 1 and so column B is greater.
49. A: B B: 7 AI: A, B and C are different digits. AB = 5C.
The best answer is B. AB is a double digit number that is a factor of 5 and therefore the units’ digit (B) is either 0 or 5. In either case B is smaller than 7 and so column B is greater.
50. A: 2 B: C AI: A, B and C are three different digits. AC = 3B
The best answer is B. AC is a double digit number that is a factor of 3 and therefore the units’ digit (B) is either 3 or 6 or 9. In either case C is larger than 2 and so column B is greater.
51. A: X B: Y AI: X and Y are different digits.
The best answer is A. There are only two possibilities for the digit A: 5 and 6. If X = 5 à Y = 2. If X = 6 à Y = 3. Either way, Y is smaller than X and therefore column A is greater.
Note: the following is the hardest question I’ve written in the GRE
52. A: A B: C AI: A, B, C, D and E are different digits. A + B < 10.
The best answer is B. The result has a hundreds digit and therefore A + B is bigger than 10, since the data says otherwise there’s only one option- that the 1 from the adding of B and C jumped to A + B and therefore B + C >10. If A + B < 10 and C + B > 10 è C must be bigger than A and therefore column B is greater than column A.
1. A: A B: The average (arithmetic mean) of A, B and C AI: B = A/2 C = 2A
The best answer is D. Calculate the average: (A + A/2 + 2A)/3 = 3.5A/3. Since we don’t know if X is positive, negative or even zero we can’t determine which column is greater.
2. A: X B: The average (arithmetic mean) of X, Y and Z AI: Y = X/3 Z = 3X X is a positive integer
The best answer is B. Calculate the average: (X + X/3 + 3X)/3 =
3. A: The average price per day that Josh spent on hotels during his first 5 days B: The average price per day that Josh spent on hotels during the whole trip AI: Josh went for a trip in Europe. In the first three days of his trip he stayed At a 85 Euro per night hotel. On the next two days he stayed at a 100 Euro per night hotel. On his last night he slept in a 120 Euro hotel.
The best answer is B. Since on his last day, Josh spent the largest amount of money on the hotel, this day is above average and so the average with out the last day is definitely smaller than the average with the last day. Therefore column B is greater than column A.
4. A: The average running time in the first 5 days B: The average running time from Sunday to Friday inclusive AI: On Sunday Ross ran for 12 minutes. In the next 4 days, Ross ran 2 minutes extra each day from the previous one. On Friday, Ross ran for 16 minutes.
The best answer is C. On Sunday, Ross ran 12 minutes. On Monday 14, Tuesday 16, Wednesday 18, Thursday 20 and Friday 16. Calculate the average for column A: (12+14+16+18+20)/5 = 16 minutes. Since on Friday he ran exactly 16 days the average doesn’t change and therefore both columns are equal.
5. A: The average (arithmetic mean) of the series B: Z – 1 AI: X < Y < Z < W are an arithmetic progression. The difference between the variables is 2.
The best answer is C. The average of the series is the median number. Since there is an even number of variables, the average is (Y + Z)/2 which is also equal to (Z – 2 + Z)/2 = Z – 1 and therefore both columns are equal.
6. A: The average level of water in the entire month B: The level of water on the 15’Th of March AI: There are 30 days in March. The level of water rises by 1.5 centimeters every day.
The best answer is A. There are 30 days in March, which is an even number of days. The average level of water in the entire month is the average of the median days, thus the 15’Th and the 16’Th. The average of those two days is greater than the level in the 15’Th only and therefore column A is greater than column B.
7. A: The average height of the plant over a period of an entire week B: The height of the plant on Wednesday AI: A creeper plant grows tall by 2 inches every day.
The best answer is C. The average height of the plant over a period of an entire week is actually the height of the plant in the median day, thus Wednesday. Therefore both columns are even.
8. A: The number of foxes in the wild group B: The number of foxes in the tamed group AI: The average weight of a fox in a wild group is 25 Kg’s. The average weight of a fox in a tamed group is 30 Kg’s. The average weight of a fox among both groups together is approximately 28.5 Kg’s.
The best answer is B. If there were the same number of foxes in both groups, the average would be 27.5. Since the average is higher, there are more foxes in the tamed group where the average weight is higher.
9. A: The number of children in the forth grade B: The number of children in the fifth grade AI: The average height of all the forth graders in the school is 1.35 meters. The average height of all the fifth graders in the school is 1.48 meters. The average weight of the two classes together is approximately 1.4 meters.
The best answer is A. If there were the same number of children in both classes, the average would be approximately 1.41. Since it’s lower, there are more children in the forth grade, where the average height is lower.
10. A: The concentration of chlorite in the final solution B: 32% AI: Two solutions are mixed together, one quantity of the first and two quantities of the second. The first with 40% chlorite and the second with 28% chlorite.
The best answer is C. The final solution contains: 1 dose of 40% chlorite, 1 dose of 28% chlorite and another dose of 28% chlorite. The percent of chlorite in the final solution is (28 + 28 + 40)/3 = 96/3 = 32% and therefore both columns are equal.
11. A: The concentration of the final solution B: AI: Two solutions are mixed together, one quantity of the first and two quantities of the second. The first with a concentration of A and the second with a concentration of B.
The best answer is B. The final solution contains: 1 dose of A, 1 dose of B and another dose of B. The concentration of the final solution is (A + B + B)/3 =
12. A: The average (arithmetic mean) of 372, 375, 378 and 375.
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, which is greater than X since X is a positive integer.
and therefore column B is greater since its denominator is smaller.
