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[GMAT math practice question]

(functions) \(f(x)○f(y)\) is defined as \(f(x+y+2xy)\). What is the value of \(1○(-1)\)?

\(1) f(-1)=-1\) and \(f(0)=1\)

\(2) f(x)=2x+1\)

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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.

So, we will first simply the original condition as follows:

\(a(\frac{1}{b} + \frac{1}{c}) +b(\frac{1}{c} + \frac{1}{a}) + c(\frac{1}{a} + \frac{1}{b})\)
\(= \frac{a}{b} + \frac{a}{c} + \frac{b}{c} + \frac{b}{a} + \frac{c}{a} + \frac{c}{b}\)
\(= \frac{(b+c)}{a} + \frac{(a+c)}{b} + \frac{(a+b)}{c}\)
\(= \frac{(a+b+c-a)}{a} + \frac{(a+b+c-b)}{b} + \frac{(a+b+c-c)}{c}\)
\(= \frac{(a+b+c)}{a} – \frac{a}{a} + \frac{(a+b+c)}{b} – \frac{b}{b} + \frac{(a+b+c)}{c} – \frac{c}{c}\)
\(= (a+b+c)[(\frac{1}{a})+(\frac{1}{b})+(\frac{1}{c})] – 3\)
If \(a+b+c= 0,\) then we have \((a+b+c)[(\frac{1}{a})+(\frac{1}{b})+(\frac{1}{c})] – 3 = -3.\)

Condition 1)
\(|a+b+c|≤0\) means \(a+b+c=0\) since \(|a+b+c|≥0\)
So, we have a\((\frac{1}{b} + \frac{1}{c}) +b(\frac{1}{c} + \frac{1}{a}) + c(\frac{1}{a} + \frac{1}{b}) = -3.\)
Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
Since we have \(a+b+c=0\), we have a\((\frac{1}{b} + \frac{1}{c}) +b(\frac{1}{c} + \frac{1}{a}) + c(\frac{1}{a} + \frac{1}{b}) = -3.\)
Since condition 2) yields a unique solution, it is also sufficient.

Therefore, D is the answer.
Answer: D

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

[GMAT math practice question]

(functions) What is the value of \(\frac{f(2)}{f(1)} + \frac{f(3)}{f(2)} + \frac{f(4)}{f(3)} + … +\frac{f(2006)}{f(2005)}\)?

\(1) f(1)=1\)

\(2) f(a+b)=f(a)f(b)\)

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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 many variables to determine a function and \(0\) equations, E is most likely to be the answer. However, by Tip 1) of the VA method, D is most likely to be the answer if condition 1) gives the same information as condition 2).

Condition 1)
\(1○(-1) = f(0)○f(-1)=f(0+(-1)+0)=f(-1)=-1.\)

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

Condition 2)
Since we have \(f(0)=1\) and \(f(-1)=-1,\) we have \(1○(-1) = f(0)○f(-1)=f(0+(-1)+0)=f(-1)=-1.\)

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

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(number properties) \(f(x)\) denotes the maximum prime factor of \(x\), where \(x\) is a positive integer. For example, \(f(30)=f(2*3*5)=5.\) What is the value of \(f(abc)\)?

\(1) f(a) = 2\)

\(2) f(b)+f(c)=14\)

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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 many variables to determine a function 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 we have \(f(2)=f(1+1)=f(1)f(1)\), we have \(\frac{f(2)}{f(1)} = \frac{f(1)f(1)}{f(1)} = f(1)\)

Since we have \(f(3)=f(2+1)=f(2)f(1)\), we have \(\frac{f(3)}{f(2)} = \frac{f(2)f(1)}{f(2)} = f(1)\)

Since we have \(f(4)=f(3+1)=f(3)f(1)\), we have \(\frac{f(4)}{f(3)} = \frac{f(3)f(1)}{f(3)} = f(1)\)
…
Since we have \(f(2006)=f(2005+1)=f(2005)f(1)\), we have \(\frac{f(2006)}{f(2005)} = f(1)\)

So, we have \(\frac{f(2)}{f(1)} + \frac{f(3)}{f(2)} + \frac{f(4)}{f(3)} + … +\frac{f(2006)}{f(2005)} = f(1) + f(1) + … + f(1) = 2005*f(1) = 2005.\)

Both conditions together yield a unique solution, so they are sufficient.

Therefore, C is the answer.
Answer: C

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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 \(3\) variables 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)

If \(a = 2, b = 7\) and \(c = 7\), then \(f(abc) = f(2*7^2) = 7\).
If \(a = 2, b = 3\) and \(c = 11\), then\(f(abc) = f(2*3*11) = 11.\)
Both conditions together do not yield a unique solution, so they are not sufficient.

Therefore, E is the answer.
Answer: E

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

[GMAT math practice question]

(function) What is the value of \(f(3)+f(2)\)?

\(1) f(a)f(b) = 3f(a+b)+f(a-b)\)

\(2) f(1) = 5\)

[GMAT math practice question]

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

\(1) f(a)+ 4f(\frac{1}{a})=15a\)

\(2) f(a)=f(-a)\)

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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 many variables to determine a function 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)
If \(a = 1\), and \(b = 0\), then we have \(f(1)f(0) = 3f(1+0) + f(1-0) = 4f(1) = 4(5) = 20\) or \(5f(0) = 20.\) So, we have \(f(0) = 4.\)

If \(a = 1\), and \(b = 1\), then we have \(f(1)f(1) = 3f(1+1) + f(1-1) = 3f(2) + f(0) = 3f(2) + 4.\) Therefore \(3f(2) + 4 = 25, 3f(2) = 21\), or \(f(2) = 7.\)

If \(a = 2, b = 1,\) then we have \(f(2)f(1) = 3f(2+1) + f(2-1) = 3f(3) + f(1) = 3f(3) + 5.\)
Therefore \(3f(3) + 5 = 5*7, 3f(3) + 5 = 35, 3f(3) = 30,\) or \(f(3) = 10.\)

Then, \(f(3)+f(2) = 10 + 7 = 17.\)

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 when the answer is A, B, C, or D.

[GMAT math practice question]

(statistics) In a basketball game, Watson, a new player, substitutes in for James. What is the height of James?

1) The height of Watson is 192 cm.
2) After the substitution, the average height of the 5 players increased by 1.8 cm.

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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 many variables to determine a function 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)
When we substitute a by \(\frac{1}{a}\) in condition 1), we have \(f(\frac{1}{a}) + 4f(a) = 15(\frac{1}{a})\) or \(4f(\frac{1}{a})+16f(a) = 60(\frac{1}{a}).\)

Then, since we have \(4f(\frac{1}{a}) = 15a – f(a)\) from condition 1), we have \(4f(\frac{1}{a}) + 16f(a) = (15a – f(a)) + 16f(a) = 15f(a) + 15a = \frac{60}{a}\) or \(f(a) = -a + \frac{4}{a}.\)

When we substitute a by \(–a\), we have \(f(-a) = a – \frac{4}{a}\) and \(–a + \frac{4}{a} = a – \frac{4}{a}.\)

Then, we have \(a = \frac{4}{a}\) or \(a^2 = 4\) and we have solutions \(a = 2\) or \(-2\).

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

Therefore, E is the answer.
Answer: E

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

[GMAT math practice question]

(geometry) \(AB\) is a straight line. What is the measure of the angle \(∠AOD\)?

\(1) ∠EOB = (\frac{4}{5})∠AOD\)

\(2) ∠DOE = \frac{∠AOB}{2}\)

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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.
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.

Assume j is James’ height, and w is Watson’s height.

Since we have \(2\) variables (\(j\) and \(w\)) 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 \(w = 192\) and \(\frac{( w – j )}{5} = 1.8\) which is the increased height after the substitution.

We have \(w – j =9\) and \(j = w – 9, j = 192 – 9\), and \(j = 183.\)

Since both conditions together yield a sufficient condition, C appears to be the solution.

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)
We have \(w = 192\), but condition 1) does not give us any information about \(w\).
Condition 1) is obviously not sufficient.

Condition 2)
We have \(j = w – 9.\)
Since condition 2) does not yield a unique solution, it 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]

(algebra) What is the length of \(BE\)?

1) \(AB = 24cm\) and \(C\) is the midpoint of \(AB\)

2)\(AD+CE = 5, AD = \frac{CD}{3}\)

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

Assume \(<AOD = x, <DOE = y\) and \(<EOB = z.\)

Then we have \(x + y + z = 180.\)

Since we have \(3\) variables (\(x, y\) and \(z\)) and \(1\) equation (\(x + y + z = 180\)), 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 \(z = (\frac{4}{5})x\) from condition 1) and \(y = 90\) from condition 2).
Then we have \(x + z = 90\) and \(x + (\frac{4}{5})x = 90\), or \((\frac{9}{5})x = 90.\)
Then, \(x = 50.\)

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 \(4\) variables (\(AD, DC, CE\), and \(EB\)) and \(0\) equations and each condition has \(2\) 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)
We have \(AD + DC + CE + EB = 24\) and \(CE + EB = 12\) from condition 1).

Since \(AD + CD = 12\) and \(AD = \frac{CD}{3}\), we have \(\frac{CD}{3} + CD = 12, \frac{4CD}{3} = 12, 4CD = 36\), or \(CD = 9\). Then \(AD + 9 = 12\), or \(AD = 3.\)

Then we have \(CE = 5 – AD, CE = 5 - 3\), or \(CD = 2.\)

So, \(EB = 12 – CE, EB = 12 – 2,\) or \(EB = 10.\)

Since both conditions 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 when the answer is A, B, C, or D.