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Overview of GMAT Math Question Types and Patterns on the GMAT

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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   07 Dec 2020, 04:17
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MathRevolution wrote:
[GMAT math practice question]

(Number Properties) \(a\) and \(b\) are positive integers. What is the value of \(2^a + 2^b\)?

1) \(a\) is the units digit of \(7^{1020}\) and \(b\) is the units digit of \(3^{224}.\)
2) \(a\) and \(b\) are neither prime numbers nor composite numbers.


=>

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 and each condition has 2 equations, C is most likely to be the answer. Let’s look at both conditions together. However, since the value of condition (1) is equal to the value of condition (2), by Tip 1, we get D as the most likely answer. Let’s look at each condition separately.

Let’s look at the condition 1). It tells us that \(a = 1\) and \(b = 1\).

Units of powers of \(7\) are \(7^1\)~\(7\), \(7^2\)~\(9\), \(7^3\)~\(3\), \(7^4\)~\(1\), \(7^5\)~\(7\), …

So, the units digits of \(7^n\) have a period of \(4\):

They form the cycle \(7 -> 9 -> 3 -> 1\).

Thus, \(7^n\) has a units digit of \(1\) if \(n\) has a remainder of \(0\) when it is divided by \(4\).

The remainder is \(0\) when \(224\) is divided by \(4\), so the units digit of \(7^{1020}\) is \(1\).

Units of powers of \(3\) are \(3^1\)~\(3\), \(3^2\)~\(9\), \(3^3\)~\(7\), \(3^4\)~\(1\), \(3^5\)~\(3\), …

So, the units digits of \(3^n\) have a period of \(4:\)

They form the cycle \(3 -> 9 -> 7 -> 1.\)

Thus, \(3^n\) has a units digit of \(1\) if \(n\) has a remainder of \(0\) when it is divided by \(4\).

The remainder is \(0\) when \(224\) is divided by \(4\), so the units digit of \(3^{224}\) is 1.

\(2^1 + 2^1 = 2 + 2 = 4.\)

The answer is unique, so the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at the condition 2), it tells us that \(a = 1\) and \(b = 1.\)

\(2^1 + 2^1 = 2 + 2 = 4.\)

The answer is unique, so the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Also, the original condition needs 2 equations.
Condition 1) has 2 equations.
Condition 2) has 2 equations.


Each condition ALONE is sufficient.

Therefore, D is the correct answer.
Answer: D
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   09 Dec 2020, 03:51
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MathRevolution wrote:
(Number Properties) p, q, and r are positive integers. What is the value of p + q + r?

1) p, q, and r are prime numbers.
2) The product of p, q, and r is 5 times the sum of p, q, and r.



Solution:

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.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to determine the value of p + q + r.

Follow the second and the third step: From the original condition, we have 3 variables (p, q, and r). To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer.

Let’s look at both conditions 1) & 2) together.

Since we have 3 variables (p, q, and r) 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) together give us that:

Since we have pqr = 5(p + q + r), we assume p = 5.

Then we have 5qr = 5(5 + q + r) or qr = q + r + 5.

Since we have qr – q – r = 5, we have qr – q – r + 1 = 6 (by adding 1 to both sides so that it can easily be factored). Factoring gives us (q - 1)(r - 1) = 6.

Then we have the possible pairs of q - 1 and r - 1 are (1, 6), (2, 3), (3, 2), and (6, 1).

If we have q – 1 = 1 and r – 1 = 6, we have q = 2 and r = 7.
If we have q – 1 = 2 and r – 1 = 3, we have q = 3 and r = 4.
If we have q – 1 = 3 and r – 1 = 2, we have q = 4 and r = 3.
If we have q – 1 = 6 and r – 1 = 1, we have q = 7 and r = 2.

However, p, q, and r are prime numbers from condition 1), and their possible numbers are 2, 5, and 7.

Thus, we have p + q + r = 2 + 5 + 7 = 14.

The answer is unique, so both conditions together are sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one. So, C seems to be the answer.

However, since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately.

Condition 1) tells us that we don’t have a unique solution, obviously.

If p = 2, q = 3, and r = 5, we have p + q + r = 2 + 3 + 5 = 10.
If p = 2, q = 5, and r = 7, we have p + q + r = 2 + 5 + 7 = 14.

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

Let’s look at the condition 2). It tells us that we don’t have a unique solution.

Assume p = 5 and since we have qr – q – r = 5, we have qr – q – r + 1 = 6 or (q - 1)(r - 1) = 6.

Then the possible pairs of q - 1 and r - 1 are (1, 6), (2, 3), (3, 2), and (6, 1).

If we have q – 1 = 1 and r – 1 = 6, we have q = 2, r = 7 and p + q + r = 5 + 2 + 7 = 14.
If we have q – 1 = 2 and r – 1 = 3, we have q = 3, r = 4 and p + q + r = 5 + 3 + 4 = 12.

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

Thus, really, both conditions 1) & 2) together are sufficient.

Therefore, C is the correct answer.
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   12 Dec 2020, 03:56
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MathRevolution wrote:
[GMAT math practice question]

(Number Properties) \(a\) and \(b\) are integers. If \(\frac{a}{504}\) is a terminating decimal, what is the value of \(a - b\)?

1) \(\frac{3}{b}\) is the simplest fraction of \(\frac{a}{504}\).

2) \(150 ≤ a ≤ 200\).


=>

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 \(\frac{a}{504}=\frac{a}{2^3∙3^2∙7}\) is a terminating decimal, \(a\) is a multiple of \(3^2 ·7 = 63.\)

Since we have \(2\) variables (\(a\) and \(b\)) and \(1\) equation, D 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) together tell us that we have \(ab = 3·504 = 2^3 · 3^3 · 7\) and \(a = 189.\)

Thus, we have \(b = \frac{(3·504)}{a} = \frac{(3·504)}{189} = \frac{504}{63} = 8.\)

Then we have \(a – b = 189 – 8 = 181\)

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

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.

Let’s look at condition 1). It tells us that \(ab = 3·504 = 2^3 · 3^3 · 7.\)

If \(a = 189\) and \(b = 8\), then we have \(a – b = 189 – 8 = 181.\)

If \(a = 63\) and \(b = 24\), then we have \(a – b = 39.\)

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

Let’s look at the condition 2). It tells us that \(a = 189.\)

If \(a = 189\) and \(b = 8,\) then we have \(a – b = 189 – 8 = 181.\)

If \(a = 189, b = 1\), then we have \(a – b = 189 – 1 = 188.\)

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

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   14 Dec 2020, 04:44
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MathRevolution wrote:
[GMAT math practice question]

(number properties) \(p\) and \(q\) are different positive integers. What is the remainder when \(p^2 + q^2\) is divided by \(4\)?

1) \(p\) and \(q\) are prime numbers.

2) \(p\) and \(q\) are not consecutive integers.


=>

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 (\(p\) and \(q\)) 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 \(p\) and \(q\) are prime numbers, which are not consecutive integers, \(p\) and \(q\) are odd integers.
So, both \(p^2\) and \(q^2\) have remainder \(1\) when they are divided by \(4\).
Thus, \(p^2 + q^2\) has remainder \(2\) when it is divided by \(4\).
Since conditions 1) & 2) yield a unique solution, when they are applied together, they 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 \(p = 2\) and \(q = 3\), then \(p^2 + q^2 = 4 + 9 = 13\), which has remainder \(1\) when it is divided by \(4\).
If \(p = 3\) and \(q = 5\), then \(p^2 + q^2 = 9 + 25 = 34\), which has remainder \(2\) when it is divided by \(4\).

Condition 1) is not sufficient since it doesn’t yield a unique solution.

Condition 2)
If \(p = 3\) and \(q = 5\), then \(p^2 + q^2 = 9 + 25 = 34\), which has remainder \(2\) when it is divided by \(4\).
If \(p = 3\) and \(q = 6\), then \(p^2 + q^2 = 9 + 36 = 45\), which has remainder \(1\) when it is divided by \(4\).

Condition 2) is not sufficient since it doesn’t yield 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.
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   17 Dec 2020, 03:09
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MathRevolution wrote:
[GMAT math practice question]

(Fractions) \(M\) is a simplified fraction. How many possible values of \(M\) exist?

1) \(M\) is between \(\frac{5}{13}\) and \(\frac{2}{3}.\)

2) The numerator of \(M\) is \(10.\)


=>

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

We assume \(M = \frac{a}{b}\) where \(a\) and \(b\) are relatively prime integers where \(b ≠ 0.\)

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) together give us:

The fractions with a numerator of \(10\) between \(\frac{5}{13}\) and \(\frac{2}{3}\) are \(\frac{10}{25}, \frac{10}{24}, \frac{10}{23}, … , \frac{10}{16}.\)

Simplified fractions among them are \(\frac{10}{23}, \frac{10}{21}, \frac{10}{19},\) and \(\frac{10}{17}.\)

Thus, we have \(4\) possible simplified fractions between \(\frac{5}{13}\) and \(\frac{2}{3}.\\
\)

The answer is unique, and conditions 1) and 2) together are 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 sufficient.

Therefore, C is the correct answer.
Answer: C

Normally, in problems that 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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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   20 Dec 2020, 03:37
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MathRevolution wrote:
(Algebra) What is the value of \(\frac{(3mr - nt)}{(4nt - 7mr)}\)?

1) \(\frac{m}{n}\)= \(\frac{4}{3}\) and \(\frac{r}{t}\) = \(\frac{9}{14}\)

2) m, n, r, and t are positive integers.



Solution:

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.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

Modify the original condition and question: \(\frac{(3mr - nt)}{(4nt - 7mr)}\).

=> Dividing the numerator and denominator by 'nt' gives us:

=> \(\frac{\frac{3mr - nt}{nt}}{\frac{4nt - 7mr}{nt}}\)

OR

=> \(\frac{\frac{3mr}{nt} -1}{4- \frac{7mr}{nt}}\)

OR

=> \(\frac{3 * \frac{m}{n} *\frac{r}{t} -1}{4- 7 * \frac{m}{n} *\frac{r}{t}}\)


So, we are looking for the value of \(\frac{m}{n}\) and \(\frac{r}{t}\).

Thus, let’s look at the condition (1). It tells us that \(\frac{m}{n} = \frac{4}{3}\) and \(\frac{r}{t} = \frac{9}{14}\), which is exactly what we are looking for.

=> Substituting the values, we get:

=> \(\frac{3 * \frac{4}{3} *\frac{9}{14} -1}{4- 7 * \frac{4}{3} *\frac{9}{14}}\)

OR

=> \(\frac{4 * \frac{9}{14} -1}{4- \frac{4}{3} *\frac{9}{2}}\)


OR

=> \(\frac{\frac{36}{14} -1}{4- \frac{36}{6}\)

OR

=> \(\frac{\frac{36}{14} -\frac{14}{14}}{4 - 6}\)

OR

=> \(\frac{\frac{22}{14}}{- 2}\)

OR

=> \(\frac{\frac{11}{7}}{- 2}\)

OR

=> \(\frac{-11}{14}\)

Since the answer is unique, and the condition is sufficient, according to CMT 2, which states that the number of answers must be one.

NOTE: We ideally don't have to solve for the value once we know that required ratios are given in the condition.

Condition (2) tells us that m, n, r, and t are natural numbers.

However, we cannot determine the unique values of m, n, r, and t to get the value of \(\frac{m}{n}\) and \(\frac{r}{t}\).

So, the condition is not sufficient, according to CMT 2, which states that the number of answers must be one.

Also, remember that if one condition has a statement with a ratio and another condition has a statement with numbers, then the condition with a ratio is more likely to be the answer. A is the correct answer because the condition with a ratio is the answer.

Condition (1) ALONE is sufficient.

Therefore, A is the correct answer.

Answer: A
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   23 Dec 2020, 06:25
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MathRevolution wrote:
[GMAT math practice question]

(Statistics) 100 students take a test. 20 students are in class A, 30 students in class B, and 50 students in class C. What is the average of the 100 students?

1) The average of class B is 10 points higher than that of class A.
2) The average of class C is 20 points higher than that of class B.


=>

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.

Assume \(a, b\) and \(c\) are the averages of classes \(A, B\), and \(C\), respectively.

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.

Conditions 1) & 2) together give us that \(b = a + 10\) and \(c = b + 20:\)

If \(a = 60, b = 70\) and \(c = 90\), then the average is
\(\frac{60·20 + 70·30 + 90·50}{100} = \frac{1200 + 2100 + 4500}{100 }= \frac{7800}{100} = 78\)

If \(a = 50, b = 60\) and \(c = 80\), then the average is
\(\frac{50·20 + 60·30 + 80·50}{100} = \frac{1000 + 1800 + 4000}{100} = \frac{6800}{100} = 68\)

The answer is not unique, and both conditions 1) and 2) together are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are not sufficient.

Therefore, E is the correct 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.
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   27 Dec 2020, 03:39
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MathRevolution wrote:
[GMAT math practice question]

(Fractions) \(M\) is a simplified fraction. How many possible values of \(M\) exist?

1) \(M\) is between \(\frac{5}{13}\) and \(\frac{2}{3}.\)

2) The numerator of \(M\) is \(10.\)


=>

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

We assume \(M = \frac{a}{b}\) where \(a\) and \(b\) are relatively prime integers where \(b ≠ 0.\)

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) together give us:

The fractions with a numerator of \(10\) between \(\frac{5}{13}\) and \(\frac{2}{3}\) are \(\frac{10}{25}, \frac{10}{24}, \frac{10}{23}, … , \frac{10}{16}.\)

Simplified fractions among them are \(\frac{10}{23}, \frac{10}{21}, \frac{10}{19},\) and \(\frac{10}{17}.\)

Thus, we have \(4\) possible simplified fractions between \(\frac{5}{13}\) and \(\frac{2}{3}.\\
\)

The answer is unique, and conditions 1) and 2) together are 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 sufficient.

Therefore, C is the correct answer.
Answer: C

Normally, in problems that 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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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   30 Dec 2020, 04:08
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MathRevolution wrote:
[GMAT math practice question]

(Number Property) \(x\) and \(y\) are positive integers. What is the difference between \(x\) and \(y\)?

1) \((x - 8)^2 = -|y - 36|\)

2) \((x + y)^2 + 3x + y = 1996\)


=>

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.

Thus, look at condition (1). It tells us that \(x = 8\) and \(y = 36\) for the following reason.

\((x - 8)^2 = -|y - 36|\)

⇔ \((x - 8)^2 + |y - 36| = 0\)

⇔ \(x = 8\) and \(y = 36\) since \((x - 8)^2 ≥ 0, |y - 36| ≥ 0\)

Then we have the difference \(y – x = 36 – 8 = 28.\)

It is exactly what we are looking for. The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2) tells us that \(x = 8\) and \(y = 36\) for the following reason.

\((x + y)^2 < (x + y)^2 + 3x + y = 1996 < 45^2\), since \(x\) and \(y\) are positive

⇔ \(x + y ≤ 44\)

Case 1: \(x + y = 44\)

⇔ \((x + y)^2 + 3(x + y) – 2y = 1996\)

⇔ \((44)^2 +3(44) – 2y = 1996 \)

⇔ \(1936 + 132 – 2y = 1996\)

⇔ \(2068 – 2y = 1996\)

⇔ \(-2y = -72\)

⇔ \(y = 36\)

Substituting \(y = 36\) into \(x + y = 44\) gives us \(x + 36 = 44\) and \(x = 8. \)

Thus, we have \(y = 36\) and \(x = 8.\)

Case 2: \(x + y ≤ 43\)

\(2y = (x + y)^2 + 3(x + y) – 1996 ≤ 43^2 + 129 – 1996 = -18.\)

We don’t have a solution in this case, since \(y\) is a positive integer.

Thus, we have a unique solution for \(x\) and \(y\), which is \(x = 8\) and \(y = 36.\)

The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Each condition ALONE is sufficient
Also, according to Tip 1, it is about 95% likely that D is the answer when condition (1) = condition (2).
This question is a CMT 4(B) question: condition 1) is easy to work with, and condition 2) is hard to work with. For CMT 4(B) questions, D is most likely the answer.
Therefore, D is the correct answer.
Answer: D
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   01 Jan 2021, 04:06
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MathRevolution wrote:
[GMAT math practice question]

(Function) \(f(x)\) is a function. What is the value of \(f(2020)\)?

1) \(f(10)=11\)

2) \(f(x+3)=\frac{f(x) - 1}{f(x) + 1}\)


=>

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Since we have many variables to determine a function, 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)

\(f(13)=\frac{f(10) - 1}{f(10) + 1}=\frac{11 - 1}{11 + 1}=\frac{5}{6}.\)

\(f(16)=\frac{f(13) - 1}{f(13) + 1}=\frac{5}{6} - 1/\frac{5}{6} + 1=\frac{-1}{11}.\)

\(f(19)=\frac{f(16)-1}{f(16)+1}=\frac{-1}{11}-1/\frac{-1}{11}+1=\frac{-6}{5}\)

\(f(22)=\frac{f(16) - 1}{f(16) + 1}=\frac{-6}{5} - 1/\frac{-6}{5} + 1=11.\)

Since we have \(2008 = 4*5002 + 0\) has a remainder \(0\) when it is divided by \(4\),\( f(2008) = f(16) = \frac{-1}{11}.\)
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 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.
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   03 Jan 2021, 04:00
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MathRevolution wrote:
[GMAT math practice question]

(Inequality) \(P = \sqrt{n+1}-\sqrt{n}\), and \(Q = \sqrt{m+1}-\sqrt{m}\) for positive integers \(m\) and \(n\). Which one is greater than the other?

1) \(n > m\).

2) \(n\) and \(m\) are consecutive integers.


=>

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.

\(\frac{1}{P} = \frac{1 }{ (\sqrt{n+1}-\sqrt{n})} = \sqrt{n+1}+\sqrt{n}\)

\(\frac{1}{Q} = \frac{1 }{ (\sqrt{m+1}-\sqrt{m})} = \sqrt{m+1}+\sqrt{m}\)

Since \(n > m \)from condition 1), we have \(\frac{1}{P} – \frac{1}{Q} = (\sqrt{n+1}+\sqrt{n}) – (\sqrt{m+1}+\sqrt{m}) > 0\) or \(\frac{1}{P} > \frac{1}{Q}.\)

Since \(P\) and \(Q\) are positive, we have \(P < Q.\)
Thus, condition 1) is sufficient.
Condition 2) is not sufficient, since we don’t know which one of m or n is greater.
Therefore, A is the answer.
Answer: A
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   05 Jan 2021, 04:45
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MathRevolution wrote:
[GMAT math practice question]

(Number Properties) \(a\) and \(b\) are integers. If \(\frac{a}{504}\) is a terminating decimal, what is the value of \(a - b\)?

1) \(\frac{3}{b}\) is the simplest fraction of \(\frac{a}{504}\).

2) \(150 ≤ a ≤ 200\).


=>

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 \(\frac{a}{504}=\frac{a}{2^3∙3^2∙7}\) is a terminating decimal, \(a\) is a multiple of \(3^2 ·7 = 63.\)

Since we have \(2\) variables (\(a\) and \(b\)) and \(1\) equation, D 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) together tell us that we have \(ab = 3·504 = 2^3 · 3^3 · 7\) and \(a = 189.\)

Thus, we have \(b = \frac{(3·504)}{a} = \frac{(3·504)}{189} = \frac{504}{63} = 8.\)

Then we have \(a – b = 189 – 8 = 181\)

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

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.

Let’s look at condition 1). It tells us that \(ab = 3·504 = 2^3 · 3^3 · 7.\)

If \(a = 189\) and \(b = 8\), then we have \(a – b = 189 – 8 = 181.\)

If \(a = 63\) and \(b = 24\), then we have \(a – b = 39.\)

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

Let’s look at the condition 2). It tells us that \(a = 189.\)

If \(a = 189\) and \(b = 8,\) then we have \(a – b = 189 – 8 = 181.\)

If \(a = 189, b = 1\), then we have \(a – b = 189 – 1 = 188.\)

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

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   07 Jan 2021, 03:22
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MathRevolution wrote:
[GMAT math practice question]

(statistics) If the median and average (arithmetic mean) of a set of 4 different numbers are both 10, what is the smallest number?

1) The range of the 4 numbers is 10
2) The sum of the smallest and the largest numbers is 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.

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

Let \(a, b, c\) and \(d\) be the \(4\) numbers, and suppose \(a < b < c < d.\)

Then \(\frac{( a + b + c + d )}{4} = 10\) and \(\frac{( b + c )}{2} = 10.\)

Since \(b + c = 20\) and \(a + b + c + d = 40\), we must have \(a + d = 20.\)

Condition 1)

Since \(d – a = 10\) by condition 1), we can figure out the values of \(a\) and \(d\). Thus, condition 1) is sufficient.

Condition 2)

\(a + d = 20\) can be deduced from the original condition as shown above.

So, condition 2) provides no additional information.

If \(a = 1, b = 9, c = 11\) and \(d = 19\), then the smallest number is \(1\).

If \(a = 2, b = 9, c =11\) and \(d = 18\), then the smallest number is \(2\).

Condition 2) is not sufficient since it does not yield a unique answer.

Therefore, A is the answer.
Answer: A
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   10 Jan 2021, 04:40
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MathRevolution wrote:
[GMAT math practice question]

If \(xyz≠0\), is \(x^3y^4z^5>0?\)

\(1) xz>0\)
\(2) xyz>0\)


=>

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.

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

Asking if \(x^3y^4z^5>0\) is equivalent to asking if \(xz > 0\) since we can ignore even exponents in inequalities.
Thus, condition 1) is sufficient.

Condition 2)
If \(x = 1, y = 1\) and \(z = 1\), then \(x^3y^4z^5>0\), and the answer is “yes”.
If \(x = 1, y = -1\) and \(z = -1\), then \(x^3y^4z^5<0\), and the answer is “no”.
Condition 2) is not sufficient since it does not yield a unique answer.

Therefore, A is the answer.
Answer: A
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   12 Jan 2021, 03:11
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MathRevolution wrote:
[GMAT math practice question]

(Function) What is the integer closest to \(r=\frac{1}{f(1)}+\frac{1}{f(2)}+…+\frac{1}{f(50)}\)?

1) \(f(a)=\sqrt{a}+\sqrt{a+1}\)

2) \(r\) is an irrational number.


=>

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. If we determine the value of \(f(x)\), then we can get a solution.

Condition 1)
Since \(f(a)= \sqrt{a}+\sqrt{a+1}, \)

we have
\(\frac{ 1}{f(a)}=\frac{1}{\sqrt{a}+\sqrt{a+1}}\)

\(= \frac{(\sqrt{a} - \sqrt{a+1})}{(\sqrt{a} + \sqrt{a+1})(\sqrt{a} - \sqrt{a+1})} \)(multiplying both the denominator and numerator by the conjugate)

\(=\frac{(\sqrt{a} - \sqrt{a+1}}{a - (a+1)}\) (multiplying the denominator)

\(=\frac{(\sqrt{a} - \sqrt{a+1})}{a - a - 1}\) (multiplying -1 through the bracket)

\(= \frac{(\sqrt{a} - \sqrt{a+1})}{-1}\) (adding like terms)

\(= -\sqrt{a} + \sqrt{a+1}\) (dividing by -1)

Then \(\frac{1}{f(1)}+\frac{1}{f(2)}+⋯+\frac{1}{f(50)}=(-\sqrt{1}+\sqrt{2})+(-\sqrt{2}+\sqrt{3})+⋯+(-\sqrt{50}+\sqrt{51})\\
= -1+\sqrt{51}.\)

Since \(7 < \sqrt{51} < 7.5\), we have \(6 < \sqrt{51}-1 < 6.5\) and the integer closest to \(r\) is \(6\).

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

Condition 2)

Since we don’t have any specific definition of f, condition 2) does not yield a unique solution, and it is not sufficient.

Therefore, A is the answer.
Answer: A
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   15 Jan 2021, 09:07
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MathRevolution wrote:
[GMAT math practice question]

(Algebra) What is the value of \(a^2 + b^2 + c^2 – ab – bc - ca\)?

1) \(a-b=-1\)

2) \(b-c=\sqrt{2}\)


=>

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.
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.

The question \(a^2 + b^2 + c^2 – ab – bc - ca\) is equivalent to \(\frac{1}{2}{(a-b)^2+(b-c)^2+(c-a)^2}\) for the following reason
\(a^2 + b^2 + c^2 – ab – bc - ca\)
\(= \frac{1}{2}{2a^2 + 2b^2 + 2c^2 – 2ab – 2bc - 2ca}\)
\(= \frac{1}{2}{(a^2-2ab + b^2)+(b^2-2bc + c^2)+(c^2 - 2ca+a^2)}\)
\(=\frac{1}{2}{(a-b)^2+(b-c)^2+(c-a)^2}\)

Conditions 1) & 2)
Since we have \(a-b=-1\) and \(b-c=\sqrt{2}\), we have \(c - a = -( c – b + b – a ) = -( √2 + (-1)) = 1 - √2.\)

\(a^2 + b^2 + c^2 – ab – bc - ca\)

\(= \frac{1}{2}{(a-b)^2+(b-c)^2+(c-a)^2} \)

\(= \frac{1}{2}{(-1)^2+\sqrt{2}^2+(1-\sqrt{2})^2}\)

\(= \frac{1}{2}(1+2+3-2\sqrt{2})\)

\(= 3 - \sqrt{2}\)

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 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.
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   18 Jan 2021, 03:43
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MathRevolution wrote:
[GMAT math practice question]

(Number Properties) \(x\) is an integer. What is the value of \(x\)?

1) \(x^2+4x+9\) is a perfect square.

2) \(x\) is a non-zero integer.


=>

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)
We have \(x^2 + 4x + 9 = k^2\) for some integer \(k.\)

Then we have
\(k^2 – (x^2 + 4x + 4) = 5\)

\(k^2 – (x + 2)^2 = 5\)

or \((k + x + 2)(k – x - 2) = 5.\)

We have four cases
Case 1) \(k + x + 2 = 1, k – x – 2 = 5\)

Adding both equations together gives us:
\(k + x + 2 + k – x – 2 = 1 + 5\)

\(2k = 6\)

\(k = 3\)

Then \(k + x + 2 = 1\) becomes

\(3 + x + 2 = 1\)

\(x = -4\)

Then we have \(k = 3, x = -4.\)

Case 2) \(k + x + 2 = 5, k – x – 2 = 1\)

Adding both equations together gives us:
\(k + x + 2 + k – x – 2 = 5 + 1\)

\(2k = 6\)

\(k = 3\)

Then \(k + x + 2 = 5 \)becomes

\(3 + x + 2 = 5\)

\(x = 0\)

Then we have \(k = 3, x = 0.\)

Case 3) \(k + x + 2 = -1, k – x – 2 = -5\)

Adding both equations together gives us:
\(k + x + 2 + k – x – 2 = -1 + -5\)

\(2k = -6\)

\(k = -3\)

Then \(k + x + 2 = -1\) becomes

\(-3 + x + 2 = -1\)

\(x = 0\)

Then we have \(k = -3, x = 0.\)

Case 4) \(k + x + 2 = -5, k – x – 2 = -1\)

Adding both equations together gives us:
\(k + x + 2 + k - x - 2 = -5 + -1\)

\(2k = -6\)

\(k = -3\)

Then \(k + x + 2 = -5\) becomes

\(-3 + x + 2 = -5\)

\(x = -4\)

Then we have \(k = -3, x = -4.\)

Thus, we have two solutions for \(x\), which are \(0\) and \(-4\).

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

Condition 2)

Since condition 2) does not provide enough information to yield a unique solution, it is not sufficient.

Conditions 1) & 2)
We have a unique solution \(-4.\)
Since both conditions together yield a unique solution, they are 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.
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   20 Jan 2021, 04:32
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MathRevolution wrote:
[GMAT math practice question]

(Algebra) \(p, q, r,\) and \(s\) are integers. \(s\) and \(t\) are two roots of \(x^2 + px + q = 0\) where \(q\) is a prime number. What the value of \(p + q\)?

1) \(s\) and \(t\) are consecutive integers.

2) \(p\) is positive.


=>

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 \(2\) variables (\(p\) and \(q\)) 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)

Assume \(t = s + 1\) from condition 1).
Then we have \(x^2 + px + q = (x - s)(x - t) = (x - s)(x - (s + 1)) = x^2 – (2s + 1)x + s(s + 1) = 0\) and \(p = -(2s + 1), q = s(s + 1).\)

Since \(q = s(s + 1)\) is a product of two consecutive integers, \(q\) is an even number, and the unique even prime is \(2\). So, we have \(q = 2.\)

Then we have \(s(s + 1) = 2, s^2 + s = 2\) or \(s^2 + s - 2 = (s + 2)(s - 1) = 0\). We have \(s = -2\) or \(s = 1.\)

Then we have \(p = -(2s + 1) = -(-3) = 3\) for \(s = -2\) or \(p = -(2s + 1) = -3\) for \(s = 1.\)

Since \(p\) is positive, we have \(p = 3\) and \(p + q = 5.\)

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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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   22 Jan 2021, 03:54
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MathRevolution wrote:
[GMAT math practice question]

(Algebra) What is the value of \(y^{x^2}/ y^{2\sqrt{3}x-5}\)?

1) \(x = \sqrt{3} + \sqrt{2}\)

2) \(y = \sqrt{2} - 1\)


=>

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 \(y^{{x^2}-2\sqrt{3}x+5}\), since we have \(y^{x^2}/ y^{2\sqrt{3}x-5} = y^{{x^2}-2\sqrt{3}x+5}.\)

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)
Since we have \(x = √3 + √2\), we have \(x - √3 = √2.\)
Then we have \(x^2 - 2√3x + 3 = 2\) or \(x^2 - 2√3x = -1\) by squaring.
Thus we have \(x^2 - 2√3x + 5 = 4\) by adding \(5\) to both sides.

Then we have \(y^{{x^2}-2\sqrt{3}x+5}= (√2 - 1)^4\), which a unique solution.

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

Therefore, C is the answer.
Answer: C

Normally, in problems that 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 a probability of 70%), and E is the answer (with a probability of 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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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   24 Jan 2021, 04:55
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MathRevolution wrote:
[GMAT math practice question]

(Probability) \(A = {1, 2, 3, …., 12}\). A1, A2, A3, …., An are all the subsets of \(A\) with m elements. If ak is the summation of Ak, what is a1 + a2 +….+ an?

1) \(m = 3\)

2) \(n = 220\)


=>

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.

Condition 1)

A1, A2, …, An are subsets of set \(A = {1, 2, …, 12}\) with three elements.

Then the number of subsets Ai’s containing \(1\) is 11C2 = \(11*\frac{10}{1}*2 = 55,\) which is the number of cases to choose \(2\) elements out of \(11\) elements.

Thus, when we calculate a1 + a2 + … + ak, each element is added \(55\) times.

Then we have a1 + a2 + … + ak = \(55(1 + 2 +…+ 12) = 4290.\)

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

Condition 2)
12C1 = \(\frac{12}{1} = 12\), 12C2 = \(12*\frac{11}{1}*2 = 66\) and 12C3 = \(12*11*\frac{10}{}1*2*3 = 220.\)

Then 12Cm = 12C3 and \(m = 3\).

Thus, condition 2) is equivalent to condition 1), and it is sufficient.

Therefore, D is the answer.
Answer: D


Note: Tip 1) of the VA method states that D is most likely the answer if condition 1) gives the same information as condition 2).
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink]
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