Overview of GMAT Math Question Types and Patterns on the GMAT

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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.

Since we have \(3\) variables (\(a, b\), and \(c\)) and \(0\) equations, E is most likely 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 question --------- is equivalent to ----------- for the following reason

Conditions 1) & 2)

When we multiply both sides of the equation \(a + \frac{4}{b} = 1\) by \(b\), we have \(ab + 4 = b.\)

When we multiply both sides of the equation \(b + \frac{1}{c} = 4\) by \(c\), we have \(bc + 1 = 4c\) or \(bc = 4c – 1\).

When we multiply both sides of the equation \(ab + 4 = b\) by \(c\), we have \(abc + 4c = bc.\)

When we replace \(bc\) of the equation \(abc + 4c = bc\) by \(4c – 1\), we have \(abc + 4c = 4c – 1\) or \(abc = -1\).

Since both conditions together yield a unique solution, they are 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 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.

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have \(1\) variable \((N)\) and \(0\) equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
The possible values of \(N\) are \(75, 150,\) …. Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
\(2^{74}\) and \(2^4*3^4*5^2\) have \(75\) factors. Since condition 2) does not yield a unique solution, it is not sufficient.

Conditions 1) & 2)
\(N = 3^{24}*5^2\) and \(N=2^4*3^4*5^2\) have \(75\) factors. Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the answer.
Answer: E

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.

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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 is considered as a variable. Since we have \(1\) variable and \(0\) equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since \(2#1 = 2\), \(#\) is one of the operations, multiplication and division.
If \(#\) is the multiplication operation, then \(0#1 = 0\).
If \(#\) is the division operation, then \(0#1 = 0.\)
Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
Since \(4#2 = 2, #\) is one of the operations, subtraction and division.
If \(#\) is the subtraction operation, then \(0#1 = -1\).
If \(#\) is the division operation, then \(0#1 = 0\).
Since condition 2) doesn’t yield a unique solution, it is not sufficient.

Therefore, A is the answer.
Answer: A

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have \(1\) variable (\(x\)) and \(0\) equations, D is most likely the answer. So, we should consider each condition on its own first.

Condition 1)

Since we have \(2 < \sqrt{|x-2|} < 4\), we have \(4 < | x – 2 | < 16\), by squaring everything.

Two Cases
First Case – positive value
\((x – 2) = 4\)
\(x = 6\)

\((x – 2) = 16\)
\(x = 18\)
Then we have \(6 < x < 18\)

Second Case – negative value
\(-(x – 2) = 4\)
\(-x + 2 = 4\)
\(-x = 2\)
\(x = -2\)

\(-(x - 2) = 16\)
\(-x + 2 = 16\)
\(-x = 14\)
\(x = -18\)
Then we have \(-14 < x < -2\)

It means we have \(-14 < x < -2\) or \(6 < x < 18.\)
However, we don’t have either a maximum value of \(x\) or a minimum value of \(x\).

Since we can’t specify a unique solution, it is not sufficient.

Condition 2)

Since condition 2) does not yield a unique solution, it is not sufficient.


Conditions 1) & 2)
Using our solutions from earlier, namely \(-14 < x < -2\) or \(6 < x < 18\), we know that the possible values of \(x\) satisfying both conditions are \(-13, -12, … , -3\) and \(7, 8, … , 17.\)

The maximum value of \(x\) is \(17\) and the minimum value of \(x\) is \(-13.\)

The difference between the maximum and the minimum value of \(x\) is \(17 – (-13) = 30.\)

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.

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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. A standard deviation is the square root of an average of squares minus the square of the average. Thus, C is the answer.

The following is more detailed reasoning.

Assume \(m\) is the average of x1, x2, …, xn and p is the average of x1^2, x2^2, …, xn^2.
Then we have (x1 + x2 + … + xn) / n = m or (x1 + x2 + … + xn) = mn, and we have
(x1^2+x2^2+⋯+xn^2)/n= p or (x1^2+x2^2+⋯+xn^2)= pn.

Then we have
(x1-m)^2+(x2-m)^2+⋯+(xn-m)^2/n
= x1^2 + x2^2 +⋯+ xn^2- 2m(x1 + x2 + ⋯ + xn) + nm^2/n
= np - 2m∙nm + nm^2/n = p - m^2
The standard deviation is \(\sqrt{p-m^2}.\)
Since we have m = 1 and p = 5 from both conditions 1) and 2), we have the standard deviation \(\sqrt{5-1^2}=\sqrt{4}=2.\)
Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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.

We have \(x = {x} + [x]\) since \({x}\) is the integer part of \(x\) and \([x]\) is the decimal part of \(x\).

Since we have \(1\) variable (\(x\)) and \(0\) equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
\({x} = 2 \)means we have \(2 ≤ x < 3.\)

Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)

\(0.3 < [x] < 0.5\) means that we have \(0.3 < x < 0.5, 1.3 < x < 1.5, … .\)

Since condition 2) does not yield a unique solution, it is not sufficient.

Conditions 1) & 2)
Since \(0.3 < [x] < 0.5 \)and \({x} = 2,\) we have \(2.3 < {x} + [x] < 2.5. \)
Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the answer.
Answer: E

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.

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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.

\(a^3b + a^2b^2 + ab^3\)?

\(= ab(a^2 + ab + b^2)\) (taking out a common factor of \(ab\))

\(= ab((a^2 - 2ab + b^2 + 3ab))\) (because \(-2ab + 3ab = ab\) from the equation in the previous line)

\(= ab((a-b)^2+3ab)\) (factoring)

Thus, we need both conditions.
Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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.

Since we have \(3\) variables (\(x, y,\) and \(z\)) and 0 equations, E is most likely 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)
\((x^2+A)(x^2-B) = x4^+(A-B)x^2-AB = x^4+x^2-n\)
Then we have \(A – B = 1\) and \(AB = n\)
If \(A = 2\) and \(B = 1\), we have \(n = AB = 2.\)
If \(A = 3\) and \(B = 2\), we have \(n = AB = 6.\)

The answer is not unique, and the conditions combined are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions (1) and (2) together 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.

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Condition 2)
The minimum possible value of \(x\) is \(3\), and the minimum possible value of \(y\) is \(5\) since \(y\) is a prime number.

Then we have \(z = 2*3*5 = 30\), which is the possible maximum value of \(z\).

\(x = 3, y = 5\) and \(z = 30\) are the unique tuple of solutions.

Since condition 2) yields a unique solution, it is sufficient.

Condition 1)
When we take reciprocals, we have \(\frac{1}{30} < \frac{1}{z} < \frac{1}{y} < \frac{1}{x} ≤ \frac{1}{3}\)

We have \(\frac{1}{2} < \frac{1}{x} + \frac{1}{y} < \frac{1}{x} + \frac{1}{x} = \frac{2}{x}\) from \(\frac{1}{x} + \frac{1}{y} = \frac{1}{2} + \frac{1}{z}.\)

Then we have \(x = 3\) since \(x < 4\) and \(x ≥ 3\).

Since \(\frac{1}{x} + \frac{1}{y} = \frac{1}{2} + \frac{1}{z} > \frac{1}{2},\) we have \(\frac{1}{3} + \frac{1}{y} > \frac{1}{2} or \frac{1}{y} > \frac{1}{6}.\)

Therefore, we have \(y < 6.\)

Since we have \(3 = x < y < 6\) and \(y\) is a prime number, we have \(y = 5.\)

Then, we have \(\frac{1}{z} = (\frac{1}{x} + \frac{1}{y}) – \frac{1}{2} = (\frac{1}{3} + \frac{1}{5}) - \frac{1}{2} = \frac{8}{15} - \frac{30}{60} = \frac{1}{15}\) or \(z = 15.\) Since condition 1) yields a unique solution, it is sufficient.

Since each condition yields a unique solution, the answer is D.
Answer: D

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.

Note: Since this question is a CMT 4(B) question because condition 2) is easy to understand and condition 1) is hard. When one condition is easy to understand, and the other is hard, D is most likely the answer.

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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.

\(f(a+b) = c\)

=> \(-3(a+b) + 16 = -3c + 16\)

=> \(-3a - 3b + 16 = -3c + 16\)

=> \(-3a - 3b = -3c\) (by subtracting \(16\) on both sides)

=> \(a + b = c\) (by dividing by \(-3\) on both sides)

Since we have \(3\) variables (\(a, b\), and \(c\)) and \(1\) equation (\(a + b = c\)), C is most likely 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 have \(a = 8\) from condition 1) for the following reason.

\(f(a) = -a\)

=> \(-3a + 16 = -a\)

=> \(2a = 16\)

=> \(a = 8\)

We have \(b = 4\) from condition 2) for the following reason.

\(f(b) = b\)

=> \(-3b + 16 = b\)

=> \(4b = 16\)

=> \(b = 4.\)

Then we have \(c = a + b = 8 + 4 = 12.\)

\(f(|c|) = f(|12|) = f(12) = -3(12) + 16 = -36 + 16 = -20.\)

Since both conditions together yield a unique solution, they are 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.

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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.
The question asks the value of \(|x - y|.\)

Condition 1)
Since the first odd prime number is \(3,\) we have \(|x - y| = 3.\)

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
\(3x = 23 – 5y\)

If \(y = 1\), then we have \(3x = 23 – 5 = 18\) or \(x = 6\) and \(|x - y| = 5.\)

If \(y = 2\), then we have \(3x = 23 – 10 = 13\) and we don’t have an integer solution.

If \(y = 3,\) then we have \(3x = 23 – 15 = 8\) and we don’t have an integer solution.

If \(y = 4\), then we have \(3x = 23 – 20 = 3\) or \(x = 1\) and \(|x - y| = 3\)

If \(y = 5,\) then we have \(3x = 23 – 25 = -2\) and we start to have negative numbers and we can stop this substitution process.

Since condition 2) does not yield a unique solution, it is not sufficient.

Therefore, A is the answer.
Answer: A

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have \(2\) variables (\(x\) and \(y\)) and \(0\) equations, C is most likely 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)
In order to have a maximum value of \(ax – y, x\) must be the maximum value, and \(y\) is the minimum, which means \(x = 2\) and \(y = 1\).

The answer is \(ax - y = 2a - 1.\)

Since both conditions together yield a unique solution, they are 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.

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have \(1\) variable (\(x\)) and \(0\) equations, D is most likely the answer. So, we should consider each condition on its own first.

Condition 1)
Since \(2 ≤ x < 3\), we have \(\frac{[x]}{x - [x]} - \frac{x}{[-x + 1] + x} = \frac{2}{x - 2} - \frac{x}{-2+ x} = \frac{2 - x}{x - 2} = \frac{-1(x - 2)}{x - 2} = -1.\)

Since condition 1) yields a unique solution, it is sufficient.

Condition 2) If \(x = n + h\) where n is the integer part of \(x\) and \(h\) is the positive decimal part of \(x\), then we have:

\(\frac{[x]}{x - [x]} - \frac{x}{[-x + 1] + x} = \frac{n}{h} - \frac{n + h}{-n + n + h} = \frac{n}{h} - \frac{n + h}{h} = \frac{-h}{h} = -1.\)

Since condition 2) yields a unique solution, it is sufficient.

Therefore, D is the answer.
Answer: D

This question is a CMT 4(B) question: condition 1) is easy to work with, and condition 2) is difficult to work with. For CMT 4(B) questions, D is most likely the answer.

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.

=>

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 original condition \(x^3-y^3>x^2+xy+y^2\) is equivalent to \(x > y + 1\) as shown below:
\(x^3-y^3 > x^2+xy+y^2\)
\(=> (x-y)(x^2+xy+y^2)>x^2+xy+y^2\)
\(=> x – y > 1\) after dividing both sides by \(x^2+xy+y^2\), since \(x^2+xy+y^2 > 0.\)

Since the final inequality is equivalent to \(x > y + 1\), condition 1) is sufficient.

Condition 2)
If \(x = 3\) and \(y = 1\), then \(x – y = 2 > 1\), and the answer is ‘yes’.
If \(x = 1\) and \(y = \frac{1}{2},\) then \(x – y = \frac{1}{2} < 1,\) and the answer is ‘no’.
Since it doesn’t give a unique answer, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have \(2\) variables (\(a\) and \(b\)) and \(0\) equations, C is most likely 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 have \(\frac{a}{3} + \frac{b}{4} = 1\) since the point \(P(a, b)\) is on the line \(\frac{x}{3} + \frac{y}{4} = 1.\)

We have \(a = b\) since \(\frac{x}{3} + \frac{y}{4} = 1\) is parallel to the line \((\frac{a}{3})x + (\frac{b}{4})y = 1\) or \(\frac{1}{3} : \frac{a}{3} = \frac{1}{4} : \frac{b}{4}.\)

Then, we have \(\frac{a}{3} + \frac{a}{4} = 1, \frac{4a}{12} + \frac{3a}{12} = 1, (\frac{7}{12})a = 1\) or \(a = \frac{12}{7}.\)

Thus we have \(a + b = \frac{12}{7} + \frac{12}{7} = \frac{24}{7}.\)

Since both conditions together yield a unique solution, they are 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.

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have \(3\) variables (\(x, a\) and \(b\)) and \(1\) equation, \(x = ab(10a+b)\), 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) and 2)
Since \(a\) and \(b\) are prime numbers less than or equal to \(9\) and \(10a + b\) is a prime number, the possible values of \(a\) and \(b\) are \(a = 2\) and \(b = 3; a = 3\) and \(b = 7; a = 5\) and \(b = 3;\) and \(a = 7\) and \(b = 3.\)

If \(a = 2\) and \(b = 3\), then \(x = 2*3*23 = 138,\) which is not a \(4\)-digit number.
If \(a = 3\) and \(b = 7,\) then \(x = 3*7*37 = 777,\) which is not a \(4\)-digit number.
If \(a = 5\) and \(b = 3\), then \(x = 5*3*53 = 795\), which is not a \(4\)-digit number.
If \(a = 7\) and \(b = 3,\) then \(x = 7*3*73 = 1533,\) which is a \(4\)-digit number.
\(x = 1533\) is the unique solution, and both conditions together are sufficient.


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)
Since \(a\) and \(b\) are prime numbers less than or equal to \(9\) and \(10a + b\) is a prime number, the possible values of \(a\) and \(b\) are \(a = 2\) and \(b = 3; a = 3\) and \(b = 7; a = 5\) and \(b = 3;\) and \(a = 7\) and \(b = 3.\)
If \(a = 2\) and \(b = 3,\) then \(x = 2*3*23 = 138.\)
If \(a = 3\) and \(b = 7,\) then \(x = 3*7*37 = 777.\)
If \(a = 5\) and \(b = 3,\) then \(x = 5*3*53 = 795.\)
If \(a = 7\) and \(b = 3,\) then \(x = 7*3*73 = 1533.\)
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
Condition 2) is obviously not sufficient on its own.

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.

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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.
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Since we have \(1\) variable (\(x\)) and \(0\) equations, D is most likely the answer. So, we should consider each condition on its own first.

Condition 1)

Since we don’t have a unique value from condition 1), it does not yield a unique solution, and it is not sufficient.

Condition 2)

\(10\) and \(40\) are possible values of \(x\), since \(\sqrt{360*10} = \sqrt{3600} = 60\) and \(\sqrt{360*40} = \sqrt{14400} = 120.\)

Since condition 2) does not yield a unique solution, it is not sufficient.

Conditions 1) & 2)

\(10\) and \(40\) are possible values of x satisfying both conditions as well.

Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the answer.
Answer: E

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.