The Ultimate Q51 Guide [Expert Level]

Below is CMT4 quesiton
When p and t are positive integers and p is a factor of t, is t/p an even?
1) t is an even
2) p+t is an odd

There are 2 variables in the original condition, hence, there is a high chance that C is the correct answer. Using 1) & 2), C is the correct answer. However, since this is a key question, if we apply CMT 4(A), using condition 2), the answer becomes yes and the condition is sufficient. Thus, the correct answer is B.

Hi Senthil 7,

In fact, co-ordinate geometry questions given are not that difficult. The distance between two spots, areas of triangles, squares, parallelograms, questions to solve with the mistake type 4(A) of DS are given. It may seem difficult when you actually draw, but it is not that difficult.

Best regards,

Jin
Math Revolution

If 5-digit positive integer ab,aba is divided by 3, what is the remainder?
1) a=3
2) b=1
We can modify the original condition and the question. If we divide ab and abc by 3, the remainder is same as a+b+a+b+a=3a+2b divided 3. Hence, since 3a is same as 3, we only have to know 2b. Thus, the correct answer is B.

There are some 50-51 level geometry questions.
A triangle has 3 sides x, y, z such that x<y<z. Is this a right triangle?
1) x^2+y^2=4^2+5^2
2) y^2+z^2=5^2+(√41)^2
There are 3 variables (x,y,z) in the original condition. Hence, the correct answer is E. Using con 1) and 2), we get the answer as yes when (x,y,z)=(4,5,√41) and the answer as no when (x,y,z)=(3, √32, √34).

Below is 50-51 level question
The product of the units digit, tens digit, and hundreds digit of a positive 3-digit integer n less than 500 is 15, what is the value of n?
1) n>350
2) The sum of the units digit, tens digit, and hundreds digit of a positive 3-digit integer n is 8.
There are 3 variables in the original condition and 1 equation. Hence, there is a high chance that C is the correct answer. Since this is a key question, one of integer questions, we need to apply the CMT 4(A). Hence, using con 1), we get n=351. The condition is sufficient and the correct answer is A.

Is |x-1|>|x+3|?
1) x<-1
2) x<1

If we modifiy the original condition and the question, we should square both sides. Then we get x^2-2x+1>x^2+6x+9? ==> -2x+1>6x+9?, x<-1?. Hence the correct answer is A.

could you please solve this question?

There are five homes. If the median price of a home is $200,000, is the range of all prices greater than $80,000?
1) The average price of the five homes is $240,000.
2) Three of five homes have the same price.

and this one too please

If |r|>|s|, is r>s?
1) r>0
2) s>0

Answer is D, both statements are individually enough


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As always, questions like CMT 3,4 are often on the exam in a more developed form. Take a look at below and you’ll see the type of question that’s really common these days. This one, too, is a 5051-evel problem that is equivalent to CMT 4(A). You must get used to these. You need to know how variable approaches and CMT are related.

(ex 1) (integer) If x and y are postive integers, what is the remainder when 100x+y is divided by 11?
1) x=22
2) y=1
==> If you change the original condition and the prolem, you all ways get the remainder of 1 if you divide 100x by 11 regardless of the value of x. Thus, you only need to know y. Therefore, the answer is . This is a typical 5051 level problem. (CMT 4(A))
Answer: A

Your answer is wrong. If x = 0, the remainder in the division of 100x by 11 is 0, and if x = 3, the remainder is 3.

Not the first time you've posted a wrong solution in this thread though.

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2) The question below is a 5051-level problem, and an inequality problem that ignores squared numbers
(ex 2) (inequality) If x and y are positive, is y/x > x/y?
1) y>x
2) y=x+2
==> If you change the original condition and the problem, since x and y are positive numbers, the sign of inequality does not change even if both sides are multiplied by xy. If so, in y/x>x/y?, y2-x2>0?, (y-x)(y+x)>0?, y+x>0, thus, y-x>0?, y>x?. In the case of 1) the answer is yes, and in the case of 2) y-x=2>0 yes, thus the answer is D.

Answer: D

3) Let’s always be careful about the CMT 4(A) like the one below

(ex 3) (statistics) What is the standard deviation of 10 numbers?
1) The standard deviation of them is d, then d2-d=0
2) The product of all the 10 numbers is negative

==> In the original condition, the answer is highly likely to be E with 10 variables. If you go with 1) & 2) you get, from d(d-1)=0, you get d=0,1, and in the case of 2) in order for the mulplication of ten numbers to get negative number, there has to be at least one number that is different from each other. Thus, d≠0, and d=1, hence suffi. Therefore, the answer is C.
Answer: C

If a and b are integers, is ab+1 an odd?
1) a is an even
2) b is an odd
==> In the original condition, C is likely to be the answer since there are 2 variables (a,b). If you solve 1) & 2), you actually get C as an answer. However, this problem, too, is a integer problem(a key question), and if you apply CMT 4(A), if you do 1), a=even, it becomes ab=even ab+1=even+1=odd, hence yes, and suffi. Therefore, the answer is A.
Answer: A

Wait hold on. If the 3 digits are 1, 3, and 5, how is it possible that their sum is 8?

This is a terribly written question.

CMT 4(B) still often comes out these days.

If a and b are prime numbers, what is the value of a+b?

1) 30≤a<b≤40
2) ab=1,147

==>In the original condition, the answer is highly likely to be C since there are 2 variables (a,b). If you do 1)&2), in the case of 1), you get a=31, b=37, hence unique, and suffi. In the case of 2),too, it becomes 31*37=1,147, therefore suffi. The answer is D. You must not forget about CMT 4(B) like this one with 1) easy and 2) difficult. If you can do A, B easily, consider D. Hence the answer is D.

Answer: D

If x≠0, is x^2/|x|<1?
1) -1<x<1
2) 0<x

==>If you change the original condition and the problem, you get is x^2/|x|<1? --> is |x|^2/|x|<1? --> is |x|<1? --> -1<x<1?, so A is the answer.

Answer: A