[GMAT math practice question]

(statistics) The average of the numbers 1, 2, 3, … , 98, 99 and x is 100x. What is the value of x?

A. 1/10
B. 2/11
C. 5/51
D. 50/101
E. 50/201

=>

( 1 + 2 + 3 + … + 98 + 99 + x ) / 100 = 100x.
So, 1 + 2 + 3 + … + 98 + 99 + x = 10000x, and 1 + 2 + 3 + … + 99 = 9999x.
It follows that 9999x = 99*100/2 = 4950, and x = 4950 / 9999 = 550 / 1111 = 50 / 101.

Therefore, the answer is D.
Answer: D
_________________

[GMAT math practice question]

(number properties) x, y are two different positive integers greater than 6. What are x and y?

1) x * y =1008
2) the greatest common divisor of x and y is 6

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Assume x = 6a and y = 6b where a and b are relatively prime.
We have x*y = (6a)*(6b)= 1008 or ab = 28.
Then we have four pairs of (a,b) which are (1,28), (4,7), (7,4) and (28,1).
Thus we have four pairs of (x,y) which are (6, 168), (24, 42), (42, 24) AND (6,168) and we don’t have a unique answer.

Therefore, E is the answer.
Answer: E

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E. 
_________________

The squares of 1, 2, 3, 4, … are 1, 4, 9, 16, …, respectively and they have remainders of 1, 0, 1, 2,
if squares are divided by 4 series comes 1,0,1,0, Can you please explain how 2 came ?

[GMAT math practice question]

(number properties) Let f(n) be the number of positive factors of n. What is minimum x satisfying f(420)*f(x)=96?

A. 4
B. 6
C. 8
D. 10
E. 12

=>

Since 420 = 22*3*5*7, we have f(420) = (2+1)(1+1)(1+1)(1+1) = 24 and f(x) = 96 / f(420) = 96 / 24 = 4.
The smallest integer with 4 factors is 6, which has 4 factors, 1, 2, 3 and 6.

Therefore, B is the answer.
Answer: B
_________________

[GMAT math practice question]

(geometry) What is d?



1) b=√3
2) c= 2

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Remind that a 30°-60°-90° right triangle has the ration among sides 1:2: √3.

Condition 1)
We know all angles of the triangles, ABC and CDE. Since b = √3, we can find CE =1, c = 2 and BC = √3. It follows that d = BC + CE = √3 + 1.
Thus, condition 1) is sufficient.

Condition 2)
We know all angles of the triangles, ABC and CDE. Since c = 2, we can find CE = 1 and BC = √3. It follows that d = BC + CE = √3 + 1.
Thus, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D
_________________

[GMAT math practice question]

(number properties) a and b are positive integers. What is the value of b-a?

1) a/b = 2/7
2) a+b is a two-digit integer greater than 20

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (a and b) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since a/b= 2/7, we have 7a = 2b.
If a = 4 and b = 14, then a/b = 4/14 = 2/7, a+ b = 21 > 20, and b – a = 10.
If a = 6 and b = 21, then a/b = 6/21 = 2/7, a + b = 27 > 20, and b – a = 15.
Since both conditions together don’t yield a unique solution, they are not sufficient.

Therefore, E is the answer.
Answer: E

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.


Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
_________________

[GMAT math practice question]

(ratio) Tom travels the same distance in 4 steps as Alice travels in 5 steps. If the length of one of Alice’s steps is 36 cm, what distance, in cm, does Tom cover in 18 steps?

A. 5184
B. 3688
C. 1800
D. 1243
E. 810

=>

Let t and a be stride-lengths of Tom and Alice, respectively.
Then 4t = 5a and a = 36.
So, t = (5/4)a = (5/4)*36 = 45 cm.
It follows that 18t = 18*45 = 810 cm.

Therefore, E is the answer.
Answer: E
_________________

[GMAT math practice question]

(geometry) l is parallel to m, what is the value of x+y?

1) A=100
2) B=50



=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

<x = <A + <B and <y = 180 - <A.
Therefore, x + y = <A + <B + 180 - <A = <B + 180.

Thus, condition 2) is sufficient on its own.
Therefore, B is the answer.
Answer: B
_________________

[GMAT math practice question]

(algebra) The time is between 8 and 9 o’clock. The angle between the hour hand and the minute hand is 180o. What time is it?

A. 8 hours 9 and (9/10) minutes
B. 8 hours 10 and (10/11) minutes
C. 8 hours 11 and (11/12) minutes
D. 8 hours 12 and (12/13) minutes
E. 8 hours 13 and (13/14) minutes

=>

The minute hand moves 6° and the hour hand moves 0.5° for every minute.
The angle between the hour hand and the minute hand at 8:00 is 120°.
After x minutes, the angle between the hands is 120° + x * 6° - x * 0.5° = 120° + x * 5.5° = 180. So, x * 5.5° = 60° and x = 600 / 55 = 120 / 11 = 10 + (10/11).

Therefore, the answer is B.
Answer: B
_________________

[GMAT math practice question]

(geometry) In the figure, AB is parallel to EF. What is the value of z?



1) x =78°

2) y = 70°

=>


Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 3 variables (x, y and z) and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.



Conditions 1) & 2)
We draw lines through C and D, parallel to AB, as shown in the diagram.
Since <a and <ABC=105° are supplementary, a=75°.
<d = 180° - <X - <a = 180° - 78° - 75° = 27°, and <c = <d = 27° as they are alternate interior angles.
Since y = 70°, b = y – c = 70°-27° = 43°
Since <z and <b are alternate interior angles, <z = <b =43°.
Both conditions together are sufficient.

Therefore, C is the answer.
Answer: C
_________________