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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If n is the sum of the first 50 positive integers, what is the greatest prime factor of n?

A. 3
B. 5
C. 7
D. 17
E. 51

=>

1 + 2 + 3 + … + 50 = 50*(50 + 1)/2 = 25*51 = 5^2*3*17
17 is the greatest prime factor of n.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

x and y are integers such that 4x^2 – y^2 +4x + 4y – 3 = 0. Which of following is true?

A. y is an even number.
B. y is an odd number.
C. y is positive.
D. y is negative.
E. y is a prime number

=>

4x^2 – y^2 +4x + 4y – 3 = 0
=> 4x^2 + 4x + 1 – y^2 + 4y – 4 = 0
=> 4x^2 + 4x + 1 = y^2 - 4y + 4
=> (2x+1)^2 = (y-2)^2
Since 2x + 1 is an odd integer, (y-2)^2 is an odd integer and y-2 is an odd integer.
Thus, y is also an odd integer.

_________________
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Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

The total price of books A, B, and C is \$306. What is the median price of books A, B, and C?

1) The price of book A is \$102.
2) The price of book B is \$20 more than that of book C.

=>

Let a, b and c be the prices of books A, B and C, respectively.
Since we have 3 variables (a, b and c) and 1 equation, 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.

The equation given by the original condition is a + b + c = 306.

Conditions 1) & 2):
Since a = 102 and b = c+ 20, we have a + b + c = 102 + c + 20 + c = 2c + 122 = 306 or 2c = 184.
So, c = 92, b = 112 and a = 102.
Thus, the median is 102.

Since this question is a statistics question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
a = 102 is the average price of the books since the total price is 306.
If b < 102, then c > 102, and the median is a = 102.
If b = 102, then c = 102, and the median is 102.
If b > 102, then c < 102, and the median is a = 102.
Thus, the median is 102 in all cases.
Condition 1) is sufficient.

When we have 3 data values, if one value is equal to the average value, then it is the median.

Condition 2)
If a = 102, b = 92 and c = 112, then the median price is 102.
If a = 86, b = 100 and c = 120, then the median price is 100.
Since it doesn’t give us a unique solution, condition 2) is not sufficient.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If m and n are integers greater than 1, is m^n>500?

1) n>8
2) n>4m

=>

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 (m and n) 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)
As the question asks if m^n > 500, we need to find the minimum possible value of m^n.
Since m ≥ 2 and n ≥ 9, the minimum possible value of m^n is 2^9 = 512 > 500.
Both conditions together are sufficient.

Since this question is an inequality question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since m ≥ 2 and n ≥ 9, the minimum possible value of m^n is 2^9 = 512 > 500.
Condition 1) is sufficient.

Condition 2)
Since m ≥ 2 and n > 4m ≥ 4*2 = 8, it follows that n ≥ 9 and m^n ≥ 2^9 = 512 > 500.
Condition 2) is sufficient, too.

Note: Since condition 1) is the same as condition 2), D is most likely to be the answer by Tip 1).

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

78% of all students at a school enrolled in an English class and 79% enrolled in a Math class. What percent of the students enrolled in neither English nor Math classes?

1) 13% of students enrolled in a Math class only.
2) 12% of students enrolled in an English class only.

=>
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.

For questions related to 2x2 matrices and percentages, we need 3 values or percentages for sufficiency. Since the original condition gives us 2 percentages, we need 1 additional percentage to solve the problem.

Each condition gives us one percentage. Thus, each condition is sufficient.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If the average of five positive integers is 16, and the largest of the integers is 40, then the median of the five integers could be which of the following?

I 10
II 15
III 20

A. I only
B. II only
C. III only
D. I & II only
E. I, II & III only

=>

Since the average of the five positive integers is 16, their sum is 5*16 = 80. So, the four smallest numbers must sum to 80 – 40 = 40.

As the integers are positive, the smallest possible median occurs if the integers are 1,1,1,37 and 40. The largest possible median occurs if the third and fourth integers are as large as possible. For this to occur, the two smallest integers must be as small as possible, that is, 1 and 1. In this case, the two remaining integers add to 40 – (1 + 1) = 38. The largest possible median occurs if the two remaining integers are equal to 38/2 = 19. Therefore, the median lies between 1 and 19, inclusive. So, 20 is not the median.

If the numbers are 1,1,10,28 and 40, the median is 10, and if the numbers are 1,1,15, 23 and 40, the median is 15.

_________________
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Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If the product of 5 different positive integers is less than 200, what is the greatest possible value for the largest of these 5 integers?

A. 3
B. 6
C. 7
D. 8
E. 9

=>
The greatest possible value of the largest of the integers occurs when the other integers are as small as possible. In other words, the other four integers should be 1,2,3, and 4.

We try different values for the fifth integer:
1*2*3*4*5 = 120
1*2*3*4*6 = 144
1*2*3*4*7 = 168
1*2*3*4*8 = 192
1*2*3*4*9 = 216

The largest product that is less than 200 occurs when the fifth integer is 8.
Thus, the greatest possible value of the largest integer is 8.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

If m and n are positive integers, m+n=?

1) m-n=-1
2) n^2<5

=>

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 (m and n) 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 0 < n^2<5, we must have n = 1 or n = 2.

Case 1: n = 1
Since m – n = m – 1 = -1, we have m = 0, which is not positive.
So, n is not equal to 1.

Case 2: n = 2
Since m – n = m – 2 = -1, we have m = 1.
Thus m + n = 1 + 2 = 3.
Because we have a unique solution, both conditions together are sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
If m = 1 and n = 2, then m + n = 3.
If m = 2 and n = 3, then m + n = 5.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
Since we don’t have any information about m, condition 2) is not sufficient.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

The relationship between the cost (c) and time (t) is given by the equation c=mt+b. If the time is increased by 10, by how many dollars is the cost increased?

1) m=20
2) b=30

=>
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.

The increase in the cost when the time is increased by 10 is given by
m(t + 10) + b – (mt + b) = 10m. As this depends on the variable m only,
condition 1) is sufficient.

_________________
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Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If m and n are positive integers, m=?

1) 7^m11^n=847
2) n=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.
Since we have 2 variables (m and n) 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 847 = 7^1*11^2, we must have m = 1 and n = 2.
Both conditions together are sufficient.
Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.
Condition 1)
Since 847 = 7^1*11^2, we must have m = 1 and n = 2.
Thus, condition 1) is sufficient.
Condition 2)
Since condition 2) gives us no information about m, it is not sufficient.
For condition 2), we should consider CMT3. We should forget everything about condition 1), since conditions 1) and 2) are independent.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

Alice and Bob drive the same car. Alice drives the car for 1/4 of one trip, and Bob drives the car for the remainder of the same trip. The difference between the distance driven by Bob and the distance driven by Alice is 40 km. What is the total distance travelled on the trip?

A. 60 miles
B. 80 miles
C. 100 miles
D. 120 miles
E. 125 miles

=>

Let d be the total distance traveled on the trip.
Alice drives (1/4)d and Bob drives (3/4)d.
The difference between the distances they drive is (3/4)d – (1/4)d = (1/2)d = 40.
Thus, d = 80.

_________________
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Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If Alice traveled on a 10 km trip at a constant speed of 25 km/h and a 48 km trip at a constant speed of 30 km/h, what was her average speed, in kilometers per hour, for the two trips?

A. 26 km/h
B. 28 km/h
C. 29 km/h
D. 30 km/h
E. 32 km/h

=>

The total time taken for the two trips is 10 / 25 + 48 / 30 = 0.4 + 1.6 = 2 hours.
The total distance traveled on the two trips is 10 + 48 = 58 km.
Thus, the average speed for the two trips is the total distance over the total time, or 58 / 2 = 29 km / h.
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GPA: 3.89
The Ultimate Q51 Guide [Expert Level]  [#permalink]

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Hi All,

I have many hardships in RC. Even I have no patiance to read the paragraphs, I am getting exhausted/bored very quickly. What should I do? What would you advise to cope with this Paramount problem?

Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If the positive integer n has 4 different factors, n=?

1) n has 1 prime factor
2) n<10

=>

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 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
2^3 and 3^3 are positive integers with 4 different factors.
Since the solution is not unique, condition 1) is not sufficient.

Condition 2)
6 = 2*3 and 8 = 2^3 are positive integers with 4 different factors.
Since the solution is not unique, condition 2) is not sufficient.

Condition 1) & 2)
8 = 2^3 is the unique positive integer less than 10 with 4 different factors and one prime factor.
Since the solution is unique, both conditions 1) & 2) are sufficient, when considered together.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

If f(x) = 4x^2 + px, what is the minimum value of f(x)?

1) f(1) = -4
2) f(2) = 0

=>
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 1 variable (p) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
Since f(1) = -4, we have 4 + p = -4 and p = -8.
f(x) = 4x^2 -8x = 4( x^2 - 2x + 1 - 1 ) = 4(x-1)^2 – 4.
The minimum of f(x) is -4.
Condition 1) is sufficient.

Condition 2)
Since f(2) = 0, we have 16 + 2p = 0 and p = -8.
f(x) = 4x^2 -8x = 4( x^2 - 2x + 1 - 1 ) = 4(x-1)^2 – 4.
The minimum of f(x) is -4.
Condition 2) is sufficient.

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

If the operation % is defined by x % y = px^2 + qy^2 (p and q are constants), what is the value of 2 % 4?

1) 1 % 2 = 5
2) 1 % 1 = 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.
2 % 4 = 4p + 16q = 4 ( p + 4q ) = 4( p* 1^2 + q*2^2 ) = 4(1%2)
Thus, condition 1) is sufficient.

Condition 2)
Now, 1%1 = p + q = 2.
If p = 1 and q = 1, then 2 % 4 = 1*2^2 + 1*4^2 = 20.
If p = 3 and q = -1, then 2 % 4 = 3*2^2 + (-1)*4^2 = 12 – 16 = -4.
Since we don’t have a unique solution, condition 2) is not sufficient.

_________________
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Thanks!
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8141
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

Water enters a cylindrical barrel at a constant speed of 500 cm3/min, and the height of the barrel increases at a constant speed of 10 cm per minute. What is the approximate radius of the barrel, in centimeters?

A. 1
B. 2
C. 3
D. 4
E. 5

=>

Let r be the radius of the barrel.
The area of the water surface 3.14*r^2.
The volume of water poured in 1 minute is 10*3.14*r^2.
Then, 10*3.14*r^2 = 500 or 31.4*r^2 = 500.
r^2 = 500 / 31.4 ≒ 16.

Thus, the radius is approximately 4 cm.

_________________
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Joined: 16 Aug 2015
Posts: 8141
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[GMAT math practice question]

Which of the following equations has a graph that can pass through only one coordinate (x,y) in which both x and y are integers?

A. y=2x
B. y=√2*x+1
C. y= 1/x^2 - 1
D. y=1/x^3
E. y=(-1/x)+1

=>

A. x = 1, y = 2 and x = 2, y = 4 lie on this graph.
B. Since √2 is irrational, the only point on this graph with integer coordinates is x = 0, y = 1.
C. x = 1, y = 0 and x = -1, y = 0 lie on this graph.
D. x = 1, y = 1 and x = -1, y = -1 lie on this graph.
E. x = 1, y = 0 and x = -1, y = 2 lie on this graph.

_________________
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Joined: 16 Aug 2015
Posts: 8141
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[GMAT math practice question]

When the members of group A are divided into groups of 13 people, m subgroups are formed. When the members of group B are divided into groups of 11 people, n subgroups are formed, and 8 people are left over. What is the number of members of group B?

1) m = n
2) The numbers of members of groups A and B are the same.

=>

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 (m and n) 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)
The conditions give the equations m = n and 13m = 11n + 8.
Plugging the first equation into the second yields 2n = 8 or n = 4.
Thus, group B has 11*4 + 8 = 52 members.
Both conditions together are sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
The equation m = n is not sufficient for determining the value of n.
Thus, condition 1) alone is not sufficient.

Condition 2)
13m = 11n + 8 does not give enough information to find the value of n.
Thus, condition 2) is not sufficient.

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.

_________________ Re: The Ultimate Q51 Guide [Expert Level]   [#permalink] 15 Jul 2018, 18:21

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# The Ultimate Q51 Guide [Expert Level]

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