РЕГИОНЫ РОССИЙСКОЙ ФЕДЕРАЦИИ 10 страницаColumn B: 5 by any power is always a number where the units’ digit is 5. Try some numbers: 52 = 25, 53 = 125 and so on. Column A = 1 and column B = 5, B is the answer.
A: The area of ABC. B: 10. AI (additional Information): AB = AC, AD = 8.
The best answer is D. In order to know if the area of the triangle is bigger or smaller than 10, you need to know the length of the base, which is not given and therefore the relationship cannot be determined.
A: The area of the inner circle. B: The area of the shaded region. AI: the Square ABCD is inscribed in the large circle. The small circle is inscribed in the square ABCD.
The best answer is A. Let’s define X as the length of the radius of the large circle. According to the Pythagoras principle, the ratio between the sides of the square to the diagonal is 1:1: The area of the smaller circle is equal to Since
3. AI: Line Q goes through (1, 2) and (9, 8). Line W is perpendicular to Q. A: The slope of line Q. B: The slope of line W.
The best answer is A. Without drawing the lines, we know that line Q has a positive slope and since line W is perpendicular to Q, it has a negative slope. And therefore the answer is A. If drawing a quick sketch helps, do so, but always remember that time is running.
4. A: 234 + 675 + 898. B: 324 + 756 + 989.
The best answer is B. This question can be solved quickly by adding the numbers, but a quicker way is to think. You can see that each of the numbers in column B is bigger than those in column A and therefore B is greater.
5. A: 100Z B: 100/Z AI: Z > 0.
The best answer is D. Take Z = 2, A = 200 and B = 50. Take Z = ½, A = 50 and B = 200. And therefore you cannot determine which column is greater.
6. A: (X + Y) 2 B: X2 + Y2 AI: XY is a positive integer.
The best answer is A. A = X2 + 2XY + Y2 and B = X2 + Y2. Subtract similar variables from each side and the question comes to: is XY > 0, if so than A is the answer. Since the additional information stated that XY is positive, the answer is A.
7. A: A2 + B2 B: (A – B)2 AI: A < 0, 0 < B < 1.
The best answer is B. A = A2 + B2 and B = A2 + B2 -2AB. If AB is positive then the answer is A, else the answer is B. According to the additional information, A is negative and B is positive and therefore AB is negative. The answer is B.
8. A: (X + Y)2 B: (X – Y)2 AI:
The best answer is D. The difference between the two columns is the presence of the variable XY. If it’s positive then A is greater and if it’s negative B is greater but if it’s equal to zero the columns are equal. Since the additional information is not distinct (A can be also zero), the answer cannot be determined.
9. A: 3X B: 3X AI: X < 0
The best answer is B. Since X is negative, 3X is negative, whereas 3X is positive. Try out a number: X = -2. A = -6 and B = 3-2 = 1/9 and therefore B is greater.
10. A: 4XY B: (XY)4 AI: X and Y are integers
The best answer is D. Pick up numbers: X = 2, Y=2. A = 16 and B = 256. Take X=1, Y=1: A = 4 and B = 1. Since picking up different numbers that fit the data resolves in a different answer, the answer cannot be determined.
11. A: X B: 5 AI: X + Y = 20, Y - X = 10.
The best answer is C. Take the given equations and add them, the result is Y = 15. Now find X = 5. The columns are equal.
12. A: X B: Y AI: 14X -16Y = 27, 81 – 42X = -48.
The best answer is D. In order to determine which column is greater we need two equations. If you multiply the first equation by 3 and you play with it a little, you’ll get the second equation and therefore we lack an equation to determine the answer.
13. A: B: The best answer is A. Column B can be written as
14. A: B:
The best answer is B. A = B = A is close to 220/340, which is 11/17 and B is close to 50/65, which is 10/13. 10 divided into 13 parts is a lot bigger than 11 divided into 17 parts and therefore B is greater. Remember, you don’t need to solve the numbers, just to compare.
15. A: B: The best answer is A. Don’t open the parenthesis, just compare. Both A and B are fractions, the one in A is powered by a smaller number and thus the result is bigger. The rule in fractions is so, the greater the power, the smaller the number.
16. A: 9P B: P2 AI: 2 < P < 4
The best answer is A. Find the highest and the lowest values of A and B. A is between 18 and 36 while B is between 4 and 16. We can see that for every P, the value of column A is greater than that of B.
17. A: P B: 27 AI: Q + P = 27, Q – P = 27.
The best answer is B. Adding the equations will give: Q = 27 and therefore P = 0.
18. A: Z B: M AI:
The best answer is D. Rewrite the expression: M = 2Z – 3. Now pick up numbers, take Z = 1 è M = -1. Now take, Z = 3 è M = 3. Since the answer is not distinct the answer is D.
19. A: X B: Y AI: The best answer is A. Rewrite the expression: And therefore X = Y + 10, thus X is greater than Y.
20. A: Y B: 2 AI: 8Y3 = 128Y
The best answer is D. IF 8Y3 = 128Y then Y2 = 16 è Y = 4 or Y = -4. Since 2 is bigger than -4 and smaller than 4 the answer is D.
21. A: Z B: 7 AI: 7Z3 = 175Z
The best answer is B. If 7Z3 = 175Z then Z2 = 25 and so Z = 5 or Z = -5. Since 7 is bigger than both, column B is greater.
22. A: The distance between X and Y. B: The distance between Y and Z. AI: X + Y = 4 + 2Y, Z = X + 10.
The best answer is D. This question has a little trick, no one said that the points are on a straight line. From the data we know that the distance between X and Y is 4 and that the distance between X and Z is 10. We cannot conclude that the distance between Y and Z is 6 because we were not told that X, Y and Z are on a straight line.
1. A: 3. B: The average (arithmetic mean) of X, Y and Z. AI: X + Y = 6 Y = 8 – Z Z = 4 – X
The best answer is C. If you add all three equations, you’ll get 2X + 2Y + 2Z = 18, divide both sides by 6 to get: (X + Y + Z)/3= 3, which is exactly the average.
2. A: X B: Y AI: 2X = 8 + 3Y Y = 2X
The best answer is A. Substitute 2X for Y in the first equation to get: 2X = 8 + 6X è X = -2, Y = -4 and therefore X is greater than Y.
3. A: X B: -1 AI: 2X = 10 - 3Y Y = -4X The best answer is C. Substitute -4X for Y in the first equation to get: 2X = 10 + 12X è X = -1 and therefore the answer is C.
4. A: X B: 6 AI: 5X + 3 < 4X + 10
The best answer is D. 5X + 3 < 4X + 10 can be written as X < 7 and therefore X can be equal to 6 or even smaller than 6 and therefore the answer cannot be determined and the answer is D.
5. A: 13 B: Z AI: 27 – 13Z > 40 – 14Z
The best answer is B. The additional information can also be written as Z > 13 and therefore Z can be 14 and up, thus column B is greater.
6. A: X B: Y AI:
The best answer is D. The expression can be simplified: Now, we can see that the answer cannot be determined because of the 10 supplement in the right wing of the inequality. X is only smaller if you add 10 to Y and therefore the answer is not clear from the data given.
7. A: 49/14 B: The average (arithmetic mean) of G and H. AI: G + 2H = 15 H – 2G = 10
The best answer is C. Multiply the first equation by two and add the second equation to get: 5H = 30 +10 à Y=8 and X= -1. The average of X and Y is 3.5, which is equal to 49/14 or 7/2.
8. A: The average (arithmetic mean) of a, b, c, d and e. B: c. AI: a, b, c, d and e are 5 consecutive numbers.
The best answer is C. By definition, the average of an odd amount of numbers is the median one and therefore c, which is the median among the 5, is the average of all the numbers.
9. A: 25. B: The average (arithmetic mean) of 5, 12, 14, 24 and 65.
The best answer is A. Find the average of the 5 given numbers. Average = (5+12+14+24+650/5 = 120/5 = 24. Column A is greater and therefore the answer is A.
10. A: X B: Y AI: X and Y are negative integers.
The best answer is B. Multiply both sides by XY to get: Y – 2XY = X è Y – X = 2XY, which is a positive number. We see that the difference between Y and X is a positive number and therefore Y is greater than X.
Word problems
11. A: Mike’s age in two years B: John’s age two years ago AI: Mike is twice as old as he was 14 years ago John is one third as old as he will be in 12 years
The best answer is A. Put the words into variables. Let M be the age of Mike today and J as the age of John today. We can write two equations: M = 2(M – 14) àM = 28. J = (J + 12)/3 à J = 6. A= 30 and B = 4 and therefore Column A is greater.
12. A: Kevin’s age today. B: Val’s age today. AI: Kevin is three times older than he was 16 years ago. Val is half as old as he will be 10 years from now.
The best answer is A. Replace the words with variables. Let K be Kevin’s age today and let V be Val’s age. We can write two equations: K = 3(K – 16) à K = 24. 2V = V + 10 à V = 10. A=24 and B=10 and therefore the answer is A.
13. A: The amount that Rachael saves every month. B: The amount that Rachael spends every month. AI: Rachael spends one fifth of her monthly allowance on new cloths and one third of the rest on driving lessons. The rest, Rachael saves in a savings account.
The best answer is B. Rachael spends 1/5 + (1/3)(4/5), which is (1/5 + 4/15 = 7/15) of her allowance each month. The rest, which is (1 – 1/5 - 8/15 = 4/15) she saves up. 7/15 > 4/15 and therefore the answer is B.
14. A: Money Travis spends on his Children’s education. B: Money Travis spends on basic necessities. AI: Travis deposits 40% of his salary in a trust fund and 20% of the rest he spends on basic necessities. The rest of the money goes to his children’s education.
The best answer is A. Travis deposits 2/5 of his salary and (1/5)(3/5), which is 3/25, goes to basic necessities. The rest of the money, which is (1 – 2/5 – 3/25 = 12/25), goes to his children’s education. A = 12/25 and B = 3/25 and therefore the answer is A.
15. A: 50 B: The number of years until Laura will be 5 times as old as she is now. AI: In 10 years, Laura will be twice as old as she is now.
The best answer is A. Define L as the age of Laura today. Write the equation: 2L = L + 10 à L = 10. If today she is 10, in 40 years she will be 50, which is 5 times as old as she is now. 40 < 50 and therefore the answer is A.
16. A: 22 B: The number of years until George will be twice as old as he is now AI: In 8 years from now, George will be one and a half as old as he was two years ago
The best answer is C. Let G be the age of George now. Write the equation: G + 8 = 1.5(G – 2) à 0.5G = 11 à G = 22. In 22 years, George will be twice as old as he is now. A=22 and B=22 and therefore the answer is C.
17. A: The number of 4$ coins that Owen used. B: The number of 7$ coins that Owen used. AI: Owen is playing a board game where there are only 4 and 7 dollar coins. Owen paid the cashier $100 with $4 and $7 coins only.
The best answer is D. Let X represent the number of $4 coins that he used and Y as the number of $7 coins. We can write the equation: 4X + 7Y = 100. There are only two solutions to this equation which are integers, X = 18, Y = 4 and X = 4, Y = 12. A distinct answer cannot be determined and therefore the answer is D.
18. A: The number of milk cartons that Erika used. B: The number of loafs of bread that Erika used. AI: In Weerky, people still trade goods. Two milk cartons are worth one fifth of a sheep and three loafs of bread are worth one eighth of a sheep. Erika traded milk cartons for half the value of the sheep and the other half with loafs of bread.
The answer is B. The question is quite easy although there is much data. One milk carton is worth 1/10 of a sheep. One loaf of bread is worth 1/24 of a sheep. Half a sheep was bought with milk, thus 5 milk cartons. The other half was bought with bread, thus 12 loaf of bread and therefore B is the answer.
19. A: The number of 3 eggs cartons that Alf used. B: The number of 4 eggs cartons that Alf used. AI: Alf made a cake with 14 eggs using 3 and 4 eggs cartons only.
The best answer is We need to find a combination of 4 and 3 that will resolve in 14. The only combination will be 3+3+4+4 = 14 and therefore the answer is C.
РЕГИОНЫ РОССИЙСКОЙ ФЕДЕРАЦИИ
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. If the diagonal is equal to 2R then the area of the square is equal to 2R2. The area of the shaded region is equal to 
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, the answer is B.
. Since both A and B are fractions, the power of the fraction will determine the answer. The higher the fraction à the smallest the number and therefore A is larger.
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. Since the numbers are not easy to work with, let’s be rough.
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è X-Y – 2 =8.
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