Math Revolution Approach (DS)

What is the perimeter of a certain right triangle?

1) The hypotenuse’s length is 10
2) The triangle’s area is 24

==> In the original condition, for a right triangle, there are 2 variables (2 legs) 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 6:8:10 and the perimeter of the right triangle becomes 6+8+10=24, hence unique and sufficient.

Therefore, the answer is C.
Answer: C

Is a positive integer n greater than 1 divisible by 2?

1) 24/n is an integer
2) 32/n is an integer

==> In the original condition, there is 1 variable (n) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer. For con 1), n=2 yes, but n=3 no, hence it is not sufficient.
For con 2), from \(32=2^5\), n always has 2 as its factor, hence yes, it is sufficient.

The answer is B.
Answer: B

If the average (arithmetic mean) of set A is 10,000 and the average (arithmetic mean) of set B is 10,000, what is the range of set A and set B combined?

1) The range of set A is 6,000
2) The range of set B is 3,000

==> If you modify the original condition and the question, from range= Max-min, you get set A: Ra =Ma-ma and set B: Rb=Mb-mb. Then, there are 6 variables and 2 equations, and in order to match the number of variables to the number of equations, there must be 4 more equations. Since there is 1 for con 1) and 1 for con 2), E is most likely to be the answer. By solving con 1) and con 2), the answer becomes E as well. Also, there is no relationship between the average and the range, and thus the answer is definitely E.

Answer: E

Is a positive integer x a factor of 12?

1) 3x is a factor of 48
2) 2x is a factor of 12

==> In the original condition, there is 1 variable (x) and in order to match the number of variables to the number of equations, there must be 1 more equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer.
For con 1), “3x is a factor of 48” becomes “x is a factor of 16”, and x=2 yes but x=8 no, hence it is not sufficient.
For con 2), “2x is a factor of 12” becomes “x is a factor of 6”, hence it is always yes and sufficient.

The answer is B.
Answer: B

There are 30 consecutive integers. What is the sum of the integers?

1) The sum of the smallest integer and the greatest integer is -1
2) The greatest integer is 14.

==> In the original condition, there is 1 variable (n, n+1, n+2,……n+29), and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer.
For con 1), you get -15~14, so the sum=-15, hence unique and sufficient.
For con 2), you also get -15~14, so the sum=-15, hence unique and sufficient.
Thus, con 1) = con 2).

The answer is D.
Answer: D

Is a positive integer n a multiple of 12

1) n is a multiple of 6
2) n is a multiple of 24

==> In the original condition, there is 1 variable (n) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer. For remainder questions, you always use direct substitution.
For con 1), n=6 no, n=12 yes, hence not sufficient.
For con 2), n=24,48,…, hence it is always yes and sufficient.

Therefore, the answer is B.
Answer: B

Is x>y>z?

1) x>y
2) y>z

==> In the original condition, there are 3 variables (x,y,z) and in order to match the number of variables to the number of equations, there must be 3 equations. Since there is 1 for con 1) and 1 for con 2), E is most likely to be the answer. By solving con 1) and con 2), from x>y>z, it is always yes and sufficient.

Therefore, the answer is C.
Answer: C

n=?

1) twice n equals to n+1
2) n times n equals to n

==> In the original condition, there is 1 variable (n) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer.
For con 1), from 2n=n+1, you get n=1, hence sufficient.
For con 2), from \(n^2=n\) and \(n^2-n=0\), n(n-1)=0, you get n=0,1, hence it is not unique and not sufficient.

The answer is A.
Answer: A

Is a positive integer x a factor of 24?

1) 3x is a factor of 48
2) 2x is a factor of 24

==> In the original condition, there is 1 variable (x) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer.
For con 1), x is a factor of 16, so x=8 yes, x=16 no, hence not sufficient.
For con 2), x is a factor of 12, so it is always yes, hence sufficient.

Therefore, the answer is B.
Answer: B

If x and y are integers greater than 1 and x>y, what are the values of x and y?
1) x+y=13
2) xy=22

==> 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), you get x=11 and y=2, hence it is unique and sufficient. The answer is c. However, this is an integer question, one of the key questions, so you apply CMT 4(A).
For con 1), from (x,y)=(11,2),(10,3), it is not unique and not sufficient.
For con 2), you only get (x,y)=(2,11), hence it is unique.

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

In the x-y plane there is a line K, (x/a)+(y/b)=1. What is the x-intercept of line K?

1) a=b
2) a=1

==> If you modify the original condition and the question, the x-intercept is the value of x when y=0, hence from (x/a)=1, you get x=a, so you only need to find a.

Therefore, the answer is B.
Answer: B

Is wp+st>0?
1) ws+pt>0
2) wt+ps>0

==> In the original condition, there are 4 variables (w,p,s,t) and in order to match the number of variables to the number of equations, there must be 4 equations. Since there is 1 for con 1) and 1 for con 2), E is most likely to be the answer. By solving con 1) and con 2), you get (w,s,p,t)=(1,1,1,1) yes, but (w,s,p,t)=(3,-1,-4,-1) no, hence it is not sufficient.

Therefore, the answer is E.
Answer: E

How much is the total price of the 3 products?

1) The sum of any two prices of these 3 products is $8,000
2) At least one of them is 4,000

==> In the original condition, there are 3 variables, and in order to match the number of variables to the number of equations, there must be 3 equations. Since there is 1 for con 1) and 1 for con 2), E is most likely to be the answer. By solving con 1) and con 2), the price of each 3 products becomes $4,000, hence it is unique and sufficient. This is an inequality 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), the price of each 3 products always becomes $4,000, hence it is unique and sufficient. For con 2), it is unknown, hence it is not sufficient.

Therefore, the answer is A.
Answer: A

m=?

1) 5 is a factor of m
2) m is a prime number

==> In the original condition, there is 1 variable (m) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer.

For con 1), from m=5,10…, it is not unique and not sufficient.
For con 2), from m=5,7…, it is not unique and not sufficient.
By solving con 1) and con 2), you get m=5, hence it is unique and sufficient.

Therefore, the answer is C.
Answer: C

is an integer n even?

1) n(n+2)=even
2) n(n+3)=even

==> In the original condition, there is 1 variable (n) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer.

For con 1), in order to get n(n+2)=even, you get n=even, hence yes, it is sufficient.
For con 2), n=2 yes, but x=3 no, hence it is not sufficient.

Therefore, the answer is A.
Answer: A

When a positive integer \(m^2\) is divided by 4, what is the remainder?

1) When m is divided by 3, the remainder is 1
2) When m is divided by 2, the remainder is 1

==> In the original condition, there is 1 variable (m) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer. For remainder questions, it is best to use direct substitution.

For con 1), if you substitute from m=3p+1=1,4,7…, from m=1, 1^2=1=4(0)+1, the remainder=1, and if m=4, from 4^2=16=4(4)+0, the remainder=0, hence it is not unique and not sufficient.
For con 2), from m=2q+1=1,3,5,7,…, you get m2=1,9,25,49.., and the remainder divided by 4 always becomes 1, hence it is unique and sufficient.

Therefore, the answer is B.
Answer: B

Is x<1<y?

1) x<√x<y
2) x<√y<y

==> 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), from x<√x, you get 0<x<1, and from √y<y, you get y>1. Then, you get x<1<y, hence always yes.

The answer is C.
Answer: C