Math Revolution Approach (DS)

Which of the following is equivalent to √3+√2?

A. \(√5\)
B. \(\sqrt{5+2√6}\)
C. \(\sqrt{6+2√5}\)
D. \(\sqrt{5-2√6}\)
E. \(\sqrt{5+4√6}\)

==> You get \(√a+√b=\sqrt{a+b+2√ab}\) because if you square both sides, you get \(a+b+2\sqrt{ab}\) on both sides.
Thus, you get \(√3+√2=\sqrt{5+2√6}\).

The answer is B. Answer: B

\(\frac{4.53}{0.146}\) is closest to which of the following?

A. 0.3
B. 3
C. 30
D. 300
E. 3,000

==> You get \(\frac{4.53}{0.146}≒\frac{4,500}{150}=30\). The answer is C.

Answer: C

If a and b are integers, is a an odd number?
1) a+b is an even
2) ab is an odd

==> In the original condition, there are 2 variables(a,b). In order to match with the number of equations, 2 equations are needed as well. For 1) 1 equation, for 2) 1 equation, which is likely to make C the answer.
Through 1) & 2), a=b=odd is derived, which is yes and sufficient. However, this is an integer question, one of the key questions, and apply the mistake type 4(A).
In case of 1), (a,b)=(1,1) yes, (2,2) no, which is not sufficient.
In case of 2), (a,b)=(odd,odd), which is yes and sufficient.
Hence, the answer is B, not C.

Answer: B

If x and y are positive integers, is x divisible by y?

1) x is divisible by 6
2) y is divisible by 6

==> In the original condition, there are 2 variables (x,y) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), (x,y)=(6,6) yes, but (x,y)=(6,12) no, hence it is not sufficient.

Therefore, the answer is E.
Answer: E

If \(ab>0\), is \(a^2\)\(b^3\) \(>0\) ?

1) \(a>0\)

2) \(b>0\)

==> If you modify the original condition and the question, is \(a^2b^3>0\)? becomes is b>0?, and If ab>0 becomes a>0?. Then, you get con 1) = con 2), and , the answer is D.

Answer: D

In the xy-plane, there is a rectangle ABCO shown as the above figure. What is the perimeter of the rectangle?

1) OB=5
2) B(3,4)

==> If you modify the original condition and the question, the quadrilateral ABCD is a rectangle, so in order to find the perimeter, you only need to know B, hence con 2) is sufficient.

Therefore, the answer is B.
Answer: B

\((x-y)^2=?\)
1) \(x^2+y^2=16\)
2) \((x+y)^2=4xy\)


==> If you modify the original condition and the question, you get \((x-y)^2=x^2+y^2-2xy=x^2+y^2+2xy-4xy=(x+y)^2-4xy\). The answer is B.
Answer: B

2x+3y=?

1) x=6
2) 4x+6y=18

==> In the original condition, you get 2x+3y=?. For con 2), if you divide both sides by 2, you get 2x+3y=9, hence it is unique and sufficient.

Therefore, the answer is B.
Answer: B

If b≠0, |a/b|=a/b, is b>0?

1) a>0
2) a+b>0

==> If you modify the original condition and the question, in order to satisfy |a/b|=a/b, you must get a/b≥0, and if you multiply b^2 on both sides, from (b^2)a/b≥(b^2)0, you get ab≥0. Since the question is b>0? becomes a≥0?, con 1) is yes and sufficient. For con 2), using CMT 4 (B: if you get A or B too easily, consider D), you only get ab≥0 and a+b>0 when a>0 and b>0, hence yes, it is sufficient.

Therefore, the answer is D.
Answer: D

What is the median of the 9 numbers?

1) The smallest 5 numbers of the 9 numbers are less than or equal to 10.
2) The largest 5 numbers of the 9 numbers are more than or equal to 10.

==> In the original condition, there are 9 variables, and in order to match the number of variables to the number of equations, there must be 9 equations. Since there is 1 for con 1) and 1 for con 2), you need 7 more equations, so E is most likely to be the answer. By solving con 1) and con 2),
From S1, S2, S3, S4, M, L1, L2, L3. L4, (Sn=smallest, M=median, Lm=largest)
S1, S2, S3, S4, M: 10 or less than 10
M, L1, L2, L3. L4: 10 or more than 10
Thus, M overlaps and 10 also overlaps, so M=10 and it is sufficient.

The answer is C.
Answer: C

When n and k are positive integers, what is the greatest common divisor of n+k and n?

1) n=2
2) k=1

==> In the original condition, there are 2 variables, and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), you get n+k=2+1=3 and n=2, and GCD(3,2)=1, hence it is unique and sufficient. Therefore, the answer is C. However, this is an integer question, one of the key questions, so you apply CMT 4 (A: if you get C too easily, consider A or B).
For con 1), k is unknown hence it is not sufficient.
For con 2), if k=1, n+k(=n+1) and n becomes 2 consecutive integers, so always GCD=1, hence it is unique and sufficient.

Therefore, the answer is B, not C.
Answer: B

If x+z=y, is x>y?

1) y>0
2) z<0

==> If you modify the original condition and the question, you get x>y?, x-y>0? Or –z>0? Or z<0?. Thus, from con 2), it is always yes and sufficient.

The answer is B.
Answer: B

If x and y are positive, is (y/x)+(x/y)>2?

1) x>y
2) x>1>y

==> If you modify the original condition and the question, even if you multiply xy on both sides, you get xy>0, hence the inequality sign doesn’t change. Thus, you get is (y/x)+(x/y)>2?, or \(y^2+x^2>2xy\)?, or \(y^2+x^2-2xy>0\)?, or \((x-y)^2>0?\), or x≠y?. From con 1) = con 2), it is always yes and sufficient.

The answer is D.
Answer: D

Is \(x^2>y^2\)?

1) x+y=2
2) x>y


==> If you modify the original condition and the question, you get\(x^2>y^2\)??, or\(x^2-y^2\)?0?, or (x-y)(x+y)>0?. There are 2 variables (x,y) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), you get x+y>2 and x-y>0, hence yes, it is always sufficient.

Therefore, the answer is C.
Answer: C

Which of the \(\frac{1}{x}\), \(x\), \(x^2\) is the greatest value?

1) x<1
2) x>0

==> If you modify the original condition and the question,
When x>1, you get \(\frac{1}{x}<1<\sqrt{x}<x<x^2\).
When 0<x<1, you get \(x^2<x<\sqrt{x}<1<\frac{1}{x}\). In other words, when
0<x<1, \(\frac{1}{x}\) is always the greatest and unique.

Therefore, the answer is C.
Answer: C

If x is a prime number, what is the number of factors of 75x?

1) \(x^2\) has 3 factors
2) x>5

==> When the words “factors” and “prime factors” are included in the question, you must consider the case when they are the same and when they are different. In other words, from \(75x=3^15^2x\), if x≠3,5, the number of factors of 75x becomes (1+1)(2+1)(1+1)=12. Then, if you look at con 2), it is x>5, hence you always get x≠3,5. Thus, the number of factors of 75x is 12, hence it is sufficient.

Therefore, the answer is B.

Answer: B

There are a lot of products. Is the standard deviation of the prices of the products less than $60?

1) The median value of their price is $100
2) The range of their price is $110

==> If you modify the original condition and the question, you get standard deviation(d)≤range/2. Then, from d≤range/2=$110/2=$55<$60, it is always yes and sufficient.

Therefore, the answer is B.
Answer: B