Math Revolution Approach (DS)

[GMAT math practice question]

Is x<y?

1) 3^x<2^y
2) x>0 and y>0

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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):
Since 3^x < 2^y and x, y are positive, we have 2^x < 3^x < 2^y or 2^x < 2^y. It follows that x < y.
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)
If x = 1 and y = 2, then the answer is “yes”.
If x = -1 and y = -1, then the answer is “no”.
Thus, condition 1) is not sufficient on its own since it does not give a unique solution.

Condition 2)
If x = 1 and y = 2, then the answer is “yes”.
If x = 2 and y = 1, then the answer is “no”
Thus, condition 2) is not sufficient on its own since it does not give a unique solution.

Therefore, C is the answer.

Answer: C

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]

Is |x|+x <2?

1) x>-1
2) x<0

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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 definition of the absolute value gives us two cases to consider when examining the question.

Case 1: x ≥ 0
|x| + x < 2
=> x + x < 2
=> 2x < 2
=> x < 1
The question asks if 0 ≤ x < 1 in this case.

Case 2: x < 0
|x| + x < 2
=> (-x) + x < 2
=> 0 < 2
As this is always true, the answer is always “yes” if x < 0.

Combining these two cases shows that the question asks if x < 1.

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient

Thus condition 1) is not sufficient, but condition 2) is sufficient since the solution set of the question includes the solution set of condition 2), but it doesn’t include that of condition 1).

Therefore, B is the answer.

Answer: B

[GMAT math practice question]

If n is a positive integer, is 3^4+3^{n+4} divisible by 5?

1) n is an even integer.
2) 3^8 +3^{n+8} is divisible by 5.

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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)
If n = 2, then 3^4 + 3^{n+4} = 3^4 + 3^6 = 3^4(1+3^2) = 81*10 = 810, and the answer is “yes”.
If n = 0, then 3^4 + 3^{n+4} = 3^4 + 3^4 = 3^4(1+1) = 81*2 = 162, and the answer is “no”.
Condition 1) is not sufficient.

Condition 2)
Now, 3^8 + 3^{n+8} = 3^8(1+3^n) is divisible by, but 3^8 is not divisible by 5. Since 5 is a prime number, 1+3^n must be divisible by 5.
Thus, 3^4 + 3^{n+4} = 3^4(1+3^n) is also divisible by 5.
Condition 2) is sufficient.

Therefore, B is the answer.
Answer: B

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.

[GMAT math practice question]

If x and y are prime numbers, and n is a positive integer, what is the number of factors of xnyn?

1) xy=6
2) n=2

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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 n) 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)
Since x and y are prime numbers and xy = 6, we must have x = 2 and y = 3, or x = 3 and y = 2.
If x = 2 and y = 3, then x^ny^n = 2^23^2 has (2+1)(2+1) = 9 factors, since x and y are different prime numbers and n = 2.
If x = 3 and y = 2, then x^ny^n = 3^22^2 has (2+1)(2+1) = 9 factors, since x and y are different prime numbers and n = 2.
Since we have a unique answer, 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 it doesn’t give us any information about the variable n, condition 1) is not sufficient.

Condition 2)
Since it doesn’t give us any information about the variables x and y, condition 2) is not sufficient.

Therefore, C is the answer.

Answer: C

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.

[GMAT math practice question]

△ is one of operations addition, subtraction, multiplication and division. What is the value of 2△2?

1) 4△2=2
2) 3△1=3

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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 operation △ can be considered to be one variable. Since we have 1 variable 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 4△2=2, △ is subtraction or division.
If △ is subtraction, then 2△2 = 0.
If △ is division, then 2△2 = 1.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
Since 3△1=3, △ is multiplication or division.
If △ is multiplication, then 2△2 = 4.
If △ is division, then 2△2 = 1.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
The two conditions show that △ must be division. Thus, 2△2 = 1.
Since we have a unique solution, condition 2) is sufficient.


Therefore, C is the answer.
Answer: C

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.

[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.

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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.

Therefore, A is the answer.

Answer: A

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]

n is an integer strictly between 10 and 20. What is the value of n?

1) The tens digit of n^2 is 2.
2) The hundreds digit of n^2 is 3.

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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 squares of the integers that are strictly between 10 and 20 are
11^2 = 121, 12^2 = 144, 13^2 = 169, 14^2 = 196, 15^2 = 225, 16^2 = 256, 17^2 = 289, 18^2 = 324, 19^2 = 361.

Condition 1):
The values for which the tens digit of n^2 is 2 are n = 11, n = 15 and n = 18.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
The values for which the hundreds digit of n^2 is 3 are n = 18 and n = 19.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
n = 18 is the unique solution that satisfies both conditions 1) and 2).
Conditions 1) and 2) are sufficient, when considered together.

Therefore, C is the answer.

Answer: C

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.

[GMAT math practice question]

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

1) n>8
2) n>4m

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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 mn ≥ 29 = 512 > 500.
Condition 2) is sufficient, too.

Therefore, D is the answer.

Answer: D

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.


[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.

Therefore, the answer is D.

Answer: D

[GMAT math practice question]

A law firm charges 1500 dollars in total for jobs that take at most 10 hours to complete, and 100 dollars/hour for jobs that take more than 10 hours to complete. If the firm completed 2 jobs, how many hours did it take to complete the two jobs?

1) The cost of one of the jobs was 1500 dollars
2) The total cost of the two jobs was 3500 dollars

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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.
Let T1 and T2 be the numbers of hours it took to complete the first and second jobs, respectively.
Since we have 2 variables 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 the cost of the first job is $1500, the cost of the second job is $3500 - $1500 = $2000. Therefore, it took T2 = 2000/100 = 20 hours to complete the second job. However, we cannot determine how long the first job took. For example, we could have T1 = 5 < 10, or T1 = 1500/100 = 15 hours.
Since we don’t have a unique solution, both conditions together are not sufficient.

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.

[GMAT math practice question]

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

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

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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.


Therefore, C is the answer.

Answer: C

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]

At a fruit store, apples are sold for $4 per pound and oranges are sold for $6 per pound. Is the total weight of apples sold greater than the total weight of oranges sold?

1) The average (arithmetic mean) price of fruit sold is less than $5.
2) The total weight of fruit sold is 10 pounds.

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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.

Let x and y be the weights of apples and oranges sold, respectively.
The question asks if x > y.

Condition 1)
The average price of fruit sold is given by (4x + 6y)/(x + y). Now,
( 4x + 6y ) / ( x + y ) < 5
=> 4x + 6y < 5(x+y)=5x+5y
=> 6y - 5y < 5x – 4x
=> y < x
and the answer is ‘yes’.
Thus, condition 1) is sufficient.

Condition 2)
Condition 2) tells us that x + y = 10, but we cannot determine whether x >y.
Condition 2) is not sufficient.

Therefore, A is the answer.

Answer: A

[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

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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.

Therefore, the answer is A.

Answer: A

Hello,thanks so much for this helpful explanation. But why doesn't the equation 2x+y not equal to zero included in the original condition not count as an equation making it a 2variable and one equation case?

Posted from my mobile device

[GMAT math practice question]

If m and n are positive integers, m=?

1) 7^m11^n=847
2) n=2

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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.

Therefore, A is the answer.
Answer: A

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]

In the xy-plane, the line K passes through a point (a,b), where a>0 and b>0. Is the slope of line K greater than 0?

1) Line K passes through the point (-a,-b).
2) Line K passes through the point (c,d), where a>c and b>d

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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.

Condition 1):
The slope of line K is ( b – ( -b ) ) / ( a – ( -a ) ) = 2b / 2a = b/a. Moreover, b/a > 0 since a>0 and b>0.
Thus, condition 1) is sufficient.

Condition 2)
The slope of line K is ( b – d ) / ( a – c ). This is greater than zero since b > d and a > c.
Thus, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

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.

[GMAT math practice question]

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

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

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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.

Therefore, the answer is C.
Answer: C

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.