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==> In the original condition, there are 2 variables (x,y), so C is highly likely to be the answer. Through 1) & 2), 2) is true, and from 2) x<1, both sides of the equation can be multiplied by y, and you get xy<y. From 1<y, you multiply both sides of the equation by x, you get x<xy, then you get x<xy<y, hence yes, and sufficient. B is the answer Answer: B
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If x and y are positive integers, when x^4-y^4 is divided by 4, what is the remainder? 1) x-y is divisible by 4 2) x^2+y^2 is divisible by 4 ==> If you modify the original condition and the problem, you get x^4-y^4=(x-y)(x+y)(x^2+y^2). Then, from 1)=2), the remainder of what are both divided by 4 is 0, hence unique, and sufficient. The answer is D. Answer: D
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==> In the original condition, there are 2 variables (u,v), so C is highly likely to be the answer. In the case of 1), u=0 and in the case of 2), u=v=0, then u=0 all the time, hence unique and sufficient. The answer is D. Answer: D
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As always, there are more and more problems like CMT 3,4 types recently. Take a look at below, and you will exactly the type of a question which falls under CMT 4(B), a 5051 level problem. You should be prepared for this type of questions. You must know the relation between the variable approaches and the CMT.

If x and y are integers, is x+y an odd?

1) (x^2+y^2)(x-y)=odd 2) x^2+y^2 is not divisible by 2

==> C is the answer, and D is also the answer, hence a typical CMT. This is a common 5051-level problem. CMT 4(B) Answer: D
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When the average (arithmetic mean) of a, b, and c is 5, what is the median of them? 1) The average (arithmetic mean) of a and b is 5 2) c=5

==> In the original condition, there are 3 variables (a,b,c), and 1 equation (a+b+c=15), and you need 2 more of them, so C is highly likely to be the answer. If you do 1) & 2), you get 1)=2, so D is the answer.

The question below is a typical statistics problem that is 5051-level.

(statistics) What is the median number of cloths sold in the last 15 days? 1) 21 cloths are sold every day in the last 8 days 2) The total number of clothes sold was 600 in the last 15 days.

==>If you change the condition and the problem, the median of the number of all the clothes sold for 15 days is the 8th number when all figures are arranged in an ascending order. In the case of 1), 21 clothes were sold each day for the last 8 days, hence median=21, and sufficient. The answer is A. This kind of questions(integer and statistics) is the current 5051-question. Answer: A _________________

==>In the original condition, there are 2 variables (a,b), so C is the answer. However, if you modify the problem, and divide the numerator and denominator of √[(a+2b)/(a+b)] by a, you get sqrt[1+2(b/a)][1+(b/a)], so you only need to know b/a. B is the answer. In general, if one condition is a number, and the other condition is a ratio, the answer is highly likely to be the ratio.

Also, CMT 4(A) is still a common type of questions

If 23,ab2 is 5-digit positive integers, what is the remainder when 23,ab2 is divided by 25?

1) a=9 2) b=5

==>If you modify the condition and the problem, in general, the remainder of a certain integer,n, divided by 25 is the same as the value when divided by 25 to the nearest hundreds digit. That is because 100=25*4. 2,3,a do not have any influence when you divide 23,ab2 by 25 like b2, so you only need to know what b is. The answer is B. Answer: B _________________

==>If you look at the original condition, there are 2 variables (a,b) and 1 equation (|a|>|b|), so in order for them to match with the number of equations there has to be 1 more equation. Hence D is the answer. In the case of 1), you get a>|b|≥b, then a>b, hence always yes, and sufficient. In the case of 2), you get |a|>|b|=-b>b, then a>b, hence always yes, and sufficient. The answer is D. Answer: D _________________

If r and s are positive integers is r/s a terminating decimal?

1) r is a factor of 100 2) s is a factor of 200

==>If you modify the original condition and the problem, in order for r/s to become a terminating decimal, prime factors of s, the denominator, should be only 2 or 5. However, in the case of 2), in order for s to be a factor of 200, you always get 200=2352, hence yes, and sufficient. The answer is B. Answer: B _________________

If |m-n|=3, is m>n? 1) mn<0 2) The distance between m and 0 is 2

==>In the original condition, there are 2 variables(m,n) and 1 equation(|m-n|=3), so D is highly likely to be the answer. In the case of 1), (m,n)=(2,-1),(-2,1), so yes and no coexist, hence not sufficient. In the case of 2), you get |m-0|=|m|=2, m=±2, then (m,n)=(2,-1),(-2,1), hence yes/no, so not sufficient. Through 1) & 2), you get (m,n)=(2,-1),(-2,1), so yes/no, hence not sufficient. The answer is E.

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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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. IIIonly 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

==>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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When a positive integer n has 6 different factors, n=? 1) n has 2 prime factors 2) n<18

==> In the original condition, in case of standard deviation questions, there is 1(n) variable, and therefore D is most likely to become the answer. In case of con 1), you get n=2231, 2132, hence it is not unique and not sufficient. In case of con 2), you get only n=2231, hence it is unique and sufficient. Therefore, the answer is B.

==> In the original condition, there is 1 variable, and therefore D is most likely to be the answer. In case of con 1), from x^5<0, if you divide both sides by x^4, you get x<0, hence yes, it is sufficient. In case of con 2), it cannot be factored, and this is when you use CMT 4 (B). In other words, if you get A and B too easily, then consider D. Since x^5+x=-1, x(x^4+1)=-1, and x=-1/(x^4+1) always agree with x^4+1>0, and hence yes, it is sufficient. Therefore, the answer is D. Answer: D
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==> If you modify the original condition and the question, when you divide both sides of a^3*b^4*c^5<0 by a^2*b^4*c^4, you get ac<0, and the question abc<0? becomes b>0?. From con 2) b>0, it becomes yes, hence it is sufficient. Therefore, the answer is B.

As always, more developed questions that combine both CMT 3 and 4 are being released. As shown below, this question, which was recently released, is a 5051 level question that also corresponds to the CMT 4 (A). You must be aware of this type of questions. You need to know the relationship between variable approach and CMT.

Number of pets Number of People 0 4 1 5 2 6 A 7 5 B

In the above list, what is the median number of pets? 1) A=3 2) B=2

==> In the original condition, there are 2 variables (A, B), and therefore c is most likely to be the answer. By solving con 1) and con 2), C becomes the answer, but since it is a statistical question, one of the key questions, when you apply CMT 4 (A) and look at con 2), the median becomes (2+2) /2=2, and hence it is sufficient. Therefore, the answer is B. Answer: B
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