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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   27 Oct 2019, 19:55
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[GMAT math practice question]

(algebra) What is the value of x + 1/y?

1) y + 1/z = 1
2) z + 1/x = 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.

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)
Since y + 1/z = 1, we have y = 1 – 1/z, y = (z-1)/z and 1/y = z/(z-1).
Since z + 1/x = 1, we have 1/x = 1 - z or x = 1/(1-z).
Then x + 1/y = 1/(1-z) + z/(z-1) = 1/(1-z) - z/(1-z) = (1-z)/(1-z) = 1.

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

Therefore, C is the answer.
Answer: C
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   11 Feb 2020, 19:11
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[GMAT math practice question]

(Algebra) What is the value of [x] + [-x]? ([x] means the greatest integer less than or equal to x.)

1) 0 ≤ x.
2) x is not 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.

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.

If x = n + h where n is an integer and 0 ≤ h < 1, then [x] = n. Here n is the integer part of n, and h is the decimal part of n.

If x is an integer, then we have x = n + h where h = 0, [x] = n, [-x] = -n and [x] + [-x] = n + (-n) = 0.

Assume x is not an integer.
Then we have x = n + h where 0 < h < 1, [x] = n.
We have -x = -n - h, -x = -n - 1 + 1 - h, -x = -(n + 1) + (1 - h) where 0 < 1 - h < 1.
Thus [-x] = -n - 1 and we have [x] + [-x] = n + (-n - 1) = -1.

Condition 2) tells us that x is not an integer. Therefore [x] + [-x] = -1 and condition 2) yields a unique solution.

Condition 2) is sufficient.

Condition 1)
If x = 0 which is an integer, then we have [x] + [-x] = 0.
If x = 1.5 which is not an integer, then we have [x] + [-x] = -1.

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

Therefore, B is the answer.
Answer: B
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   01 Mar 2020, 18:44
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[GMAT math practice question]

(Number Properties) A, B, and C are positive numbers. We have an equation A^2 + 2B^2 = C^2. What is the value of A + B + C?

1) A, B, and C are less than 11.
2) A, B, and C are 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.

Since A^2 + 2B^2 = C^2 , we have 2B^2 = C^2 - A^2 or 2B^2 = (C + A)(C - A).
C^2 - A^2 = (C + A)(C - A) is an even number, both A and C must either odd numbers or even numbers.
Then 2B^2 is a product of even numbers, and 2B^2 is a multiple of 4.
B is an even number.

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)
A, B, and C are positive integers less than or equal to 10.

Since we have 0 < C^2 - A^2 = 2B^2 < 10^2 or 0 < C^2 - A^2 = 2B^2 < 100, we have 0 < B < √50 and B = 2, 4 or 6.
We have C + A > C – A.

Case 1: B = 2
Since we have (C + A)(C - A) = 8, we have C + A = 4 and C – A = 2.
Then we have C = 3, A = 1 and A + B + C = 1 + 2 + 3 = 6.

Case 2: B = 4
Since we have (C + A)(C - A) = 32, we have C + A = 16, C - A = 2 or C + A = 8, C – A = 4.
If C + A = 16, C – A = 2, then we have A = 7, C = 9 and A + B + C = 7 + 4 + 9 = 20.

Since both conditions together 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 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 Mar 2020, 18:44
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[GMAT math practice question]

(Statistics) By adding n into {3, 6, 9, 10}, the average is equal to the median. What is the sum of possible values of n?

A. 24
B. 26
C. 28
D. 30
E. 32

=>

The possible medians are 6, n, and 9 only.

Case 1: Assume n ≤ 6.
The median of n, 3, 6, 9, 10 is 6 and we have (n + 28) / 5 = 6 or n + 28 = 30.
Thus, we have n = 2.

Case 2: Assume 6 < n ≤ 9.
The median of 3, 6, n, 9, 10 is n and we have (n + 28) / 5 = n or n + 28 = 5n.
Thus, we have 4n = 28 or n = 7.

Case 3: Assume 9 < n.
The median of 3, 6, 9, n, 10 is 9 and we have (n + 28) / 5 = 9 or n + 28 = 45.
Thus, we have n = 17.

Hence, the possible values of n are 2, 7, and 17.
Then we have 2 + 7 + 17 = 26.

Therefore, B is the answer.
Answer: B
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   05 Mar 2020, 19:21
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[GMAT math practice question]

(Algebra) What is the value of x?

1) x^2 + 4x + 9 is a perfect square of an integer.
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 to be the answer. So, we should consider each condition on its own first.

Condition 1)

x^2 + 4x + 9 = x^2 + 4x + 4 + 5 = (x + 2)^2 + 5 = k^2 for some integer k.

Then we have
5 = k^2 - (x+2)^2 (subtracting (x + 2)^2 from both sides)
5 = (k + x + 2)(k – x – 2) (factoring using difference of squares).

If k + x + 2 = 5 and k – x – 2 = 1, then we have 2x + 4 = ( k + x + 2 ) – ( k – x – 2 ) = 5 -1 = 4 and x = 0.

If k + x + 2 = 1 and k – x – 2 = 5, then we have 2x + 4 = ( k + x + 2 ) – ( k – x – 2 ) = 1 - 5 = -4 and x = -4.

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

Condition 2)

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

Conditions 1) & 2)

The reasoning in condition 1) can be applied to both conditions together too.

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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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   05 Mar 2020, 19:45
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Quote:
What is the value of x?

1) x^2 + 4x + 9 is a perfect square of an integer.
2) x is an integer.


St1) x^2 + 4x + 9 is a perfect square of an integer.
or, (x+2)^2 +5 is a perfect square of an integer
x can be equal to 0, -4, \sqrt{21}-2
NOT Sufficient

St2) x is an integer, hence it can be equal to -1, 2, 3, 4, 7 any integer.
NOT Sufficient

Even after combining St 1 & 2, x can be 0 or -4
NOT Sufficient

Answer E
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   08 Mar 2020, 20:09
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[GMAT math practice question]

(Algebra) What is (a-b)^2?

1) |a| = 4, |b| = 3
2) b/a < 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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)

Since |a| = 4, and |b| = 3, we have a = ±4, and b = ±3 from condition 1) and ab < 0 from condition 2).
Then we have a^2 = 16, b^2 = 9 and ab = -12.
Thus (a - b)^2 = a^2 - 2ab + b^2 = 16 – 2(-12) + 9 = 49.

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   13 Mar 2020, 03:04
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[GMAT math practice question]

(algebra ) If x = 1/2+√3, what is x^2 - 4x?

A. -2
B. -1
C. 0
D. 1
E. 2

=>

We have x = 1/2+√3
x = 2-√3/(2+√3)(2-√3) (multiplying by the conjugate)
x = 2-√3/4-3 (foiling the denominator)
x = 2-√3 (simplifying)
x – 2 = -√3 (subtracting 2 from both sides)
(x - 2)^2 = (-√3)^2 (squaring both sides)
x^2 – 4x + 4 = 3 (simplifying)
x^2 – 4x = -1 (subtracting 4 from both sides)

Therefore, the answer is B.
Answer: B
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   30 Mar 2020, 00:37
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[GMAT math practice question]

(Geometry) We have a square □ABCD, and △AEF is a triangle inscribed in □ABCD. What is the length of BE?

1) □ABCD is a square with AB = 10
2) △AEF is an equilateral triangle.

Attachment:
3.26(ds).png
3.26(ds).png [ 5.63 KiB | Viewed 2087 times ]


=>

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 a triangle and a square, we have 3 variables and 1 variable for the triangle and the square, respectively. Since we have 4 variables 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)

Attachment:
3.26ds(a).png
3.26ds(a).png [ 9.37 KiB | Viewed 2104 times ]


Assume BE = x (0 < x < 10).
Then we have BE = DF since the triangle AEF is equilateral.
AE^2 = 100 + x^2 = EF^2 = (10 - x)^2 + (10 - x)^2.
Then we have
100 + x^2 = (10 - x)^2 + (10 - x)^2
100 + x^2 = 2(10 – x)^2 (adding like terms)
100 + x^2 = 2(10 – x)(10 – x)
100 + x^2 = 2(100 – 10x – 10x + x^2) (foiling out the brackets)
100 + x^2 = 2x^2 – 40x + 200 (multiplying 2 through the bracket and combining like terms)
x^2 – 40x + 100 = 0 (bringing all terms to one side)
Thus, we have x = 20 ± 10√3. (using the quadratic formula)
But we have x = 20 - 10√3 since 0 < x < 10.

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   02 Apr 2020, 19:33
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[GMAT math practice question]

(Function) For a positive integer n, f(n) and g(n) are defined as follows

f(n)={0 ,(n is a multiple of 5) 1 , (n is not a multiple of 5)
g(n)={0 ,(n is a multiple of 7) 1 , (n is not a multiple of 7)

Moreover, h(n) is defined as (1 - f(n))(1 - g(n)).
What is the value of h(3) + h(6) + h(9) + … + h(2004) + h(2007)?

A. 13
B. 19
C. 38
D. 57
E. 152

=>

h(n) = 0 when f(n)=1 or g(n) = 1
h(n) = 1 when f(n)=0 and g(n)=0
Thus, we have h(n) = 1 when n is a multiple of both 5 and 7. It means we have h(n) = 1 when n is a multiple of 35.

3, 6, 9, … , 2016, 2019 are multiples of 3.
The function values of h for those numbers is 1 when they are multiples of 105 = 35*3.
2019 = 105*19 + 24
Thus, we have 19 multiples of 105 between 1 and 2007, inclusive.

Therefore, B is the answer.
Answer: B
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   06 Apr 2020, 04:01
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[GMAT math practice question]

(Function) What is the value of f(2019)?

1) f(3) = 5
2) f(x+2)=f(x - 1)/f(x + 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.

Since we have many variables to determine the function f(x), 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)
Since f(3) = 5, we have:
f(5) = (f(3) - 1) / (f(3) + 1)
f(5) = (5 - 1) / (5 + 1)
f(5) = 4/6 = 2/3.

Then we have:
f(7) = (f(5) - 1) / (f(5) + 1)
f(7) = ((2/3) - 1) / ((2/3) + 1)
f(7) = -(1/3) / (5/3) = -(1/5).
We have:
f(9) = (f(7) - 1) / (f(7) + 1)
f(9) = (-(1/5) - 1)/(-(1/5) + 1)
f(9) = -(6/5) / (4/5) = -(3/2).

We have:
f(11) = (f(9) - 1) / (f(9) + 1)
f(11) = (-(3/2) - 1)) / (-(3/2) + 1)
f(11) = -(5/2) / -(1/2) = 5.

Since f(3) = f(11), we have f(8k - 5) = 5.

Then we have:
f(8k-3) = 2/3, f(8k-1) = -(1/5) and f(8k+1) = -(3/2).
f(2007) = f(8*251-1) = -(1/5).

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

Therefore, C is the answer.
Answer: C
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   07 Apr 2020, 04:37
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[GMAT math practice question]

There are 1 card written ‘1’, 2 cards written ‘2’, …, and n cards written ‘n’. The average of all the numbers written on the cards is 17. How many cards are there? (Use the fact : 1 + 2 + … + n = n(n+1)/2, 1^2 + 2^2 + … + n^2 = n(n+1)(2n+1)/6)

A. 25 B. 125 C. 225 D. 325 E. 425

=>

( 1*1 + 2*2 + 3*3 + … +n*n ) / ( 1 + 2 + 3 + … + n ) = 17
[(1/6)n(n+1)(2n+1)] / [(1/2)n(n+1) ] = (2n+1)/3 = 17.
Thus, we have 2n+1 = 51 or n = 25.

Then the number of cards is
1 + 2 + 3 + … + 25 = (1/2)25*26 = 325

Therefore, D is the answer.
Answer: D
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   08 Apr 2020, 18:56
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[GMAT math practice question]

(Geometry) We have a right triangle ABC with ∠A = 90, AD = 6 and CD = 8. AD and BC are perpendicular to each other. If x = AC and y = BD, what is the value of xy?

Attachment:
4.6ps.png
4.6ps.png [ 8.7 KiB | Viewed 2013 times ]


A. 40
B. 42
C. 45
D. 50
E. 56

=>

We have x^2 = 6^2 + 8^2 = 36 + 64 = 100 and x = 10.
Since we have 6^2 = 8y or 8y = 36, we have y = 9/2.

Thus, we have xy = 10*(9/2) = 45.

Therefore, C is the answer.
Answer: C
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   03 May 2020, 20:44
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[GMAT math practice question]

(Algebra) abc ≠ 0. What is the value of a^2+b^2+c^2?

1) a+b+c=3.
2) a^3+b^3+c^3=27.

=>

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 (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)

If a = 1, b = -1, c = 3, then we have a^2 + b^2 + c^2 = 1 + 1 + 9 = 11.
If a = 2, b = -2, c = 3, then we have a^2 + b^2 + c^2 = 4 + 4 + 9 = 17.

Since both conditions together 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 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   11 May 2020, 19:55
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[GMAT math practice question]

(Geometry) The figure shows a square ABCD, and point E is on the extension with DE = 6 and CE = 10. What is the area of △BCE?

Attachment:
5.6PS.png
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A. 20
B. 24
C. 30
D. 32
E. 40

=>

Since CD^2 = 10^2 - 6^2 = 64, we have CD = 8.
Then the base BC of the triangle is 8, and its height is 8 as well.
Thus, the area of the triangle is (1/2)8*8 = 32.

Therefore, D is the answer.
Answer: D
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   26 May 2020, 19:58
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[GMAT math practice question]

(Inequality) P = √n+1-√n, and Q = √m+1-√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.

1/P = 1 / (√n+1-√n) = √n+1+√n
1/Q = 1 / (√m+1-√m) = √m+1+√m
Since n > m from condition 1), we have 1/P – 1/Q = (√n+1+√n) – (√m+1+√m) > 0 or 1/P > 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   28 May 2020, 18:37
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[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   31 May 2020, 21:22
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[GMAT math practice question]

(Equation) For a quadratic equation x^2 + px + q = 0, what is the value of p + q?

1) The roots of x^2 + px + q = 0 are consecutive positive integers.
2) The difference between the squares of the two roots of x^2 + px + q = 0 is 25.

=>

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.

Thus, look at condition 1).
Assume r and r+1 are roots of the equation x^2 + px + q = 0. It tells us that p = -25 and q = 156 for the following reason, which is exactly what we are looking for.
(r + 1)^2 – r^2 = r^2 + 2r + 1 – r^2 = 2r + 1 = 25 or r = 12.
Then we have x^2 + px + q = (x - r)(x -(r + 1)) = x^2 – (r + r + 1)x + r(r + 1) = x^2 – (2r+1)x + r(r+1) and we have p = -2r-1 = -25 and q = r(r+1) = 12*13 = 156.

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 1) ALONE is sufficient.

Condition 2)
If 1 and 2 are roots of the equation, then we have (x - 1)(x - 2) = x^2 - 3x + 2 = x^2 + px + q, p = -3 and q = 2.
If 2 and 3 are roots of the equation, then we have (x - 2)(x - 3) = x^2 - 5x + 6 = x^2 + px + q, p = -5 and q = 6.
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.

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   04 Jun 2020, 20:43
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[GMAT math practice question]

(Algebra) If x^2-5x+1=0, x^3+2(x+1/x)+1/x^3 = ?

A. 90
B. 100
C. 110
D. 120
E. 130

=>

When we divide both sides of the equation x^2 - 5x + 1 = 0 by x, we have x - 5 + 1/x = 0 or x + 1/x = 5.

x^3 + 1/x^3 = (x + 1/x)^3 – 3x(1/x)(x + 1/x) by factoring using sum of cubes.
Then x^3 + 1/x^3 = (x + 1/x)^3 – 3x(1/x)(x + 1/x) = 5^3 – 3·5 = 110.
Thus, x^3 + 2(x + 1/x) +1/x^3 = x^3 + 1/x^3 + 2(x + 1/x) = 110 + 2·5 = 120.

Therefore, D is the answer.
Answer: D
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink] New post   07 Jun 2020, 13:30
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There is a yin-yang symbol shown as above figure such that its radius is 2. What is the area of the region shaded?
1) Both arc MNO and arc OCD are the same semi-circles.
2) The area of region shaded is half of the area of the circle

ANSWER SHOULD BE D.

let us start with 2 which is more simple.
2. suff. you dont need to solve, since you are given the radius of the symbol. 1/2*pie*2 is the area of the shaded reason. thus-sufficient.
1. you are given that both semi circles are equal and they both have radius of 1. so in fact you can either look at it as the semi bigger circle or 1/2*pie*2^2. sufficient.
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT [#permalink]
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