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On six seating chairs including one red chair around a big round table , A, B, C, D, E, F each sit on a chair. A must sit on the red chair and B must be beside A. How many seating arrangement are? A. 36 B. 40 C. 44 D. 48 E. 64

==> If you set A, and B sits on the left side of A, the rest can be seated with the number of cases of 4*3*2*1=24. If B sits on the right side of A, there can be 24 number of cases that they can be seated. Hence, 24+24=48. The answer is D. Answer: D
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On the coordinate planes system there are line L and line M, y=3x+2 and y=kx, respectively. For which of the following value of k did the line L intersect with the line M?

I. -3 II. -1 III. 3

A. none B. I only C. III only D. I & II E.II & III

==>For the two straight lines to intersect with each other, the slope has to be different. Hence, from y=kx, it has to be k≠3, so I & II is the answer. The answer is D.

It’s well known that a company's profit=20,000x-25x^2, where x is the number of workers. If the profit of the company is the greatest, what is the value of x?

A. 50 B. 100 C. 150 D. 250 E. 400

ANS: In general, from y=ax2+bx+c, the value of y from symmetric axis, x=-b/2a is the least or the greatest. For the profit has to be the greatest, x=-20,000/2(-25) = 400, hence the greatest profit. The answer is E.
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It is well known that the ratio of hydrogen’s mass to oxygen’s mass is 1 to 16. If the moving speed rates of hydrogen atoms and oxygen atoms are inversely proportional to square root of its mass, what is the ratio of the moving speed rate of hydrogen to that of oxygen?

A. 1:4 B. 4:1 C. 1:16 D. 16:1 E. 1:2

==>From v=k/√m, (v: moving speed and m: mass), if hydrogen=a, oxygen=16a If you substitute this, the rate of the speed of hydrogen atoms to that of oxygen atoms is k/√a : k/√16a =1:1/4=4:1. Hence the answer is B.

The revenue increased by 10 percent from March to April and then the revenue decreased by 10 percent from April to May. By what percent is increased (or decreased) the revenue from March to May?

A. 10% less B. 10% more C. 1% less D. 1% more E. 0.9% less

==>If the revenue increases by 10% from March to April, R -->R(1+10%)=1.1R, and if this decreases by 10% from April to May, 1.1R 1.1R(1-10%)=1.1R(0.9)=0.99R. If so, the revenue from March to May is R 0.99R, hence 1% decrease. The answer is C.

For a positive integer n, if 5^n is a factor of 20!, but 5^n+1 is not a factor of 20!, what is the value of n? A. 4 B. 6 C. 7 D. 8 E. 9

==> As 20!=5^n(integer), the question eventually asks the number of 5 in 20!. There are 20, 15, 10, 5 in 20! and therefore the number of 5 is 4. Hence, the answer is A. Answer: A
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There are 5 integers and a≤b≤c≤d≤e. If the median of them is 20 and the average (arithmetic mean) of them is 30, what is the smallest possible value of e? A. 35 B. 40 C. 45 D. 50 E. 55

==> In order to be the smallest value of e, the other values should be maximum, which makes a=b=c=20. Then, d+e=5*30-20*3=90 is derived. Since we are looking for the smallest value of e, which means d should be the maximum value, d=e=45 is derived. Hence, C is the answer. Answer: C
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In a certain theater, the first row has 8 seats and the next row has 2 more seats than pre row. If the last row has 56 seats, what is the total number of seats? A. 640 B. 720 C. 800 D. 1,000 E. 1,240

==> There are 8 seats in the first row. Since 2 more seats increase as you go farther, n=25 is derived from 8+(n-1)2=56. That is, there are 25 rows in total and the sum of all seats is 25(8+56)/2=800. Hence, the answer is C. Answer: C
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It’s well known that a company's profit=20,000x-25x^2, where x is the number of workers. If the profit of the company is the greatest, what is the value of x? A. 50 B. 100 C. 150 D. 250 E. 400

==>In general, from y=ax^2+bx+c, the value of y from symmetric axis, x=-b/2a is the least or the greatest. For the profit has to be the greatest, x=-20,000/2(-25) = 400, hence the greatest profit. The answer is E. Answer: E
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It is well known that the ratio of hydrogen’s mass to oxygen’s mass is 1 to 16. If the moving speed rates of hydrogen atoms and oxygen atoms are inversely proportional to square root of its mass, what is the ratio of the moving speed rate of hydrogen to that of oxygen? A. 1:4 B. 4:1 C. 1:16 D. 16:1 E. 1:2

==>From v=k/√m, (v: moving speed and m: mass), if hydrogen=a, oxygen=16a If you substitute this, the rate of the speed of hydrogen atoms to that of oxygen atoms is k/√a : k/√16a =1:1/4=4:1. Hence the answer is B. Answer: B
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The revenue increased by 10 percent from March to April and then the revenue decreased by 10 percent from April to May. By what percent is increased (or decreased) the revenue from March to May? A. 10% less B. 10% more C. 1% less D. 1% more E. 0.9% less

==>If the revenue increases by 10% from March to April, R -->R(1+10%)=1.1R, and if this decreases by 10% from April to May, 1.1R 1.1R(1-10%)=1.1R(0.9)=0.99R. If so, the revenue from March to May is R 0.99R, hence 1% decrease. The answer is C. Answer: C
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If n is the product of all the integers from 5 to 20, inclusive, what is the greatest integer k for which 2k is a factor of n ?

(A) 11

(B) 12

(C) 13

(D) 15

(E) 16

The questions asks you about the number of 2, from n=(5)(6)….(19)(20)=2k(int). Then, you only have to find even numbers from (5)(6)..(19)(20), and only 6,8,10,12,14,16,18,20 are even numbers. The numbers of 2 for each number are 6=(2)(3), 8=23, 10=(2)(5), 12=(22)(3), 14=(2)(7), 16=24, 18=(2)(32), 20=(22)(5), and 2 becomes 2, 23, 2, 22, 2, 24, 2, 22. Hence the number of 2 is 1+3+1+2+1+4+2+1=15.

If x-1, x+6, x+7 are 3 side lengths of a right triangle, what is the value of x?

==> (x-1)^2+(x+6)^2=(x+7)^2, x^2-2x+1+x2+12x+36=x^2+14x+49. From x^2-4x-12=0 to (x-6)(x+4)=0, x=6, since -4 does not work, you get x=6. Therefore, the answer is C. Answer: C
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