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14 Jun 2020, 18:55
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[GMAT math practice question]
(Algebra) x and y are real numbers. What is the value of x?
1) x^2 - 6xy + 9y^2 + x - 3y = 6
2) y = 1 + 2√3
=>
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) together give us:
x^2 - 6xy + 9y^2 + x - 3y = 6 (from Condition 1)
⇔ x^2 - 6xy + 9y^2 + x - 3y - 6 = 0 (subtracting 6 from both sides)
⇔ (x - 3y)^2 + (x - 3y) - 6 = 0 (factoring the first 3 terms)
⇔ m^2 + m – 6 = 0 (substituting m for (x - 3y))
⇔ (m – 2)(m + 3) = 0 (trinomial factoring)
⇔ (x - 3y - 2)(x - 3y + 3) = 0 (substituting (x – 3y) for m)
⇔ x – 3y – 2 = 0 or x – 3y + 3 = 0
⇔ x - 3y = 2 or x - 3y = -3
⇔ x = 3y + 2 or x = 3y – 3
⇔ x = 3(1 + 2√3) + 2 or x = 3(1 + 2√3) – 3 (substituting (1 + 2√3) from condition 2 for y)
⇔ x = 5 + 6√3 or x = 2√3 (simplifying)
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.
Therefore, E is the answer.
Answer: E
Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
(Algebra) x and y are real numbers. What is the value of x?
1) x^2 - 6xy + 9y^2 + x - 3y = 6
2) y = 1 + 2√3
=>
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) together give us:
x^2 - 6xy + 9y^2 + x - 3y = 6 (from Condition 1)
⇔ x^2 - 6xy + 9y^2 + x - 3y - 6 = 0 (subtracting 6 from both sides)
⇔ (x - 3y)^2 + (x - 3y) - 6 = 0 (factoring the first 3 terms)
⇔ m^2 + m – 6 = 0 (substituting m for (x - 3y))
⇔ (m – 2)(m + 3) = 0 (trinomial factoring)
⇔ (x - 3y - 2)(x - 3y + 3) = 0 (substituting (x – 3y) for m)
⇔ x – 3y – 2 = 0 or x – 3y + 3 = 0
⇔ x - 3y = 2 or x - 3y = -3
⇔ x = 3y + 2 or x = 3y – 3
⇔ x = 3(1 + 2√3) + 2 or x = 3(1 + 2√3) – 3 (substituting (1 + 2√3) from condition 2 for y)
⇔ x = 5 + 6√3 or x = 2√3 (simplifying)
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.
Therefore, E is the answer.
Answer: E
Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
15 Jun 2020, 18:04
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[GMAT math practice question]
(Number Property) f(x) denotes the remainder when x is divided by 10. For example, f(5) = 5, f(92) = 2 and f(271) = 1. What is the value f(2^2006) + f(3^2006) + f(5^2006) + f(7^2006) + f(9^2006)?
A. 24
B. 26
C. 28
D. 30
E. 32
=>
When we repeat multiplying a number every four times, we have the same unit digit.
2^2006 has the same unit digit of 2^2, which is 4, since 2006 = 4·501 + 2.
3^2006 has the same unit digit of 3^2, which is 9, since 2006 = 4·501 + 2.
5^2006 has the same unit digit of 5^2, which is 5, since 2006 = 4·501 + 2.
7^2006 has the same unit digit of 7^2, which is 9, since 2006 = 4·501 + 2.
9^2006 has the same unit digit of 9^2, which is 1, since 2006 = 4·501 + 2.
f(2^2006)+ f(3^2006)+ f(5^2006)+ f(7^2006)+ f(9^2006) = 4 + 9 + 5 + 9 + 1 = 28.
Therefore, C is the answer.
Answer: C
(Number Property) f(x) denotes the remainder when x is divided by 10. For example, f(5) = 5, f(92) = 2 and f(271) = 1. What is the value f(2^2006) + f(3^2006) + f(5^2006) + f(7^2006) + f(9^2006)?
A. 24
B. 26
C. 28
D. 30
E. 32
=>
When we repeat multiplying a number every four times, we have the same unit digit.
2^2006 has the same unit digit of 2^2, which is 4, since 2006 = 4·501 + 2.
3^2006 has the same unit digit of 3^2, which is 9, since 2006 = 4·501 + 2.
5^2006 has the same unit digit of 5^2, which is 5, since 2006 = 4·501 + 2.
7^2006 has the same unit digit of 7^2, which is 9, since 2006 = 4·501 + 2.
9^2006 has the same unit digit of 9^2, which is 1, since 2006 = 4·501 + 2.
f(2^2006)+ f(3^2006)+ f(5^2006)+ f(7^2006)+ f(9^2006) = 4 + 9 + 5 + 9 + 1 = 28.
Therefore, C is the answer.
Answer: C
18 Jun 2020, 18:25
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[GMAT math practice question]
(Statistics) What is the value of x?
1) The standard deviation of 1 and 3 is greater than the standard deviation of 1, 3, and x.
2) x is an 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.
Let’s look at the condition 1). It tells us that the average of 1 and 3 is 2. If x is between 1 and 3 inclusive, then the standard deviation of 1, 3, and x is less than that of 1 and 2.
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.
Condition 2) is not sufficient, obviously.
When we consider both conditions 1) & 2) together, 1, 2, and 3 are possible values of x.
The answer is not unique, and both conditions 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
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.
(Statistics) What is the value of x?
1) The standard deviation of 1 and 3 is greater than the standard deviation of 1, 3, and x.
2) x is an 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.
Let’s look at the condition 1). It tells us that the average of 1 and 3 is 2. If x is between 1 and 3 inclusive, then the standard deviation of 1, 3, and x is less than that of 1 and 2.
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.
Condition 2) is not sufficient, obviously.
When we consider both conditions 1) & 2) together, 1, 2, and 3 are possible values of x.
The answer is not unique, and both conditions 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
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.
