[GMAT math practice question]


In a certain theater, the first row has 12 seats, and each row has 1 more seat than the previous row. If the last row has 50 seats, what is the total number of seats in the theater?

A. 1003
B. 1029
C. 1129
D. 1209
E. 1,339

=>

The question asks for the value of 12 + 13 + ... + 50. This is the sum of an arithmetic sequence with first term a = 12, and last term l = 50.
The sum of n terms of an arithmetic sequence may be found using the formula n/2 (a + l).
The number of rows is n = 50 – 12 + 1 = 39.
So, the number of seats in the theater is 39 * ( 12 + 50 ) / 2 = 39 * 62 / 2 = 39 * 31 = 1209.

Therefore, the answer is D.

Answer: D
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[GMAT math practice question]

What is the remainder when a positive integer n is divided by 36?

1) The remainder when n is divided by 12 is 5.
2) The remainder when n is divided by 18 is 11.

=>

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 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.
Condition 1)
Since the divisor (12) of condition 1) is not a multiple of the divisor (36) of the question, condition 1) is not sufficient.
For example, if n = 5, the remainder when n is divided by 36 is 5.
If n = 17, the remainder when n is divided by 36 is 17.
Since we don’t have a unique solution, condition 1) is not sufficient.
Condition 2)
Since the divisor (18) of condition 2) is not a multiple of the divisor (36) of the question, condition 2) is not sufficient.
For example, if n = 11, the remainder when n is divided by 36 is 11.
If n = 29, the remainder when n is divided by 36 is 29.
Since we don’t have a unique solution, condition 2) is not sufficient.
Conditions 1) & 2)
The first integer satisfying both conditions is 29, and the lcm of 12 and 18 is 36. Therefore, the integers satisfying both conditions are 29, 65, 101, ….
Each of these integers has a remainder of 29 when it is divided by 36.
Both conditions together 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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[GMAT math practice question]

How many 4-digit integers have the form abcd, where b is even and d >= 2b?

A. 900
B. 1200
C. 1520
D. 2400
E. 2700

=>

Suppose abcd is a 4-digit number.
There are 9 possible values for a: a = 1, 2, …, 9.
There are 10 possible values of c: c = 0,1,2,…,9.
Since b is even, b can take on the values 0,2,4,6 and 8.
However, the condition d >= 2b limits the possible values of b to 0, 2 and 4.

Case 1 : b = 0 => d = 0, 1, … , 9
The number of possible values of d is 10.
There are 10 * 9 * 10 = 900 4-digit integers with b = 0.

Case 2: b = 2 => d = 4, 5, …, 9
The number of possible values of d is 6.
There are 6 * 9 * 10 = 540 4-digit integers with b = 2.

Case 3: b = 4 => d = 8, 9
The number of possible values of d is 2.
There are 2 * 9 * 10 = 180 4-digit integers with b = 4.
In total, there are 900 + 540 + 180 = 1520 possible 4-digit integers of this form.

Therefore, the answer is C.

Answer: C
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[GMAT math practice question]

The median of 5 numbers is 50, and their range is 40. If the median of the 3 smallest numbers is 40, which of the following could not be the range of the 3 largest numbers?

I. 0
II. 20
III. 40

A. I only
B. II only
C. I & II only
D.I & III only
E. I, II & III

=>

Suppose a, b, c, and d satisfy a ≤ b ≤ 50 ≤ c ≤ d. Since the median of the 3 smallest numbers is 40, we must have b = 40.

The range of the 3 largest numbers is d – 50.

Since we are told that d – a = 40, the maximum range of the 3 largest numbers occurs when a = b = 40.
Then d = 80, and the maximum range is d – 50 = 80 – 50 = 30.

The minimum range of the 3 largest numbers occurs when a = 10 and b = 40.
Then we have c = 50 and d = 50, and the range is d – 50 = 50 – 50 = 0.

The range of the 3 largest numbers lies between 0 and 30, inclusive.
Thus, 0 and 20 are the only possible values.

Therefore, the answer is C.
Answer: C
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[GMAT math practice question]

p, q, x, and y are positive integers. If p^mq^n=72, what is the value of p+q?

1) m>n
2) mn=6

=>

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 4 variables (p, q, m and n) and 1 equation, 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):
We can write 72 = 2^33^2=1^672^1. (find the “hidden 1”)
If p = 2, q = 3, m = 3 and n = 2, then p + q = 5.
If p = 1, q = 72, m = 6 and n = 1, then p + q = 73.
Since we don’t have a unique solution, both conditions, taken together, are not sufficient.

Therefore, the answer is E.

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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Hi chetan2u,

I was wondering how did you get the remainder to be 6 when you divided 443 by 7? I am getting 2 instead. 7*63=441, which leaves us a remainder of 2. For 48 however, I am getting a remainder of 6. I am getting two different values 2 or 6. Could you please explain where I am going wrong? Would greatly appreciate it!

[GMAT math practice question]

When x and n are positive integers, if x^{2n}>(3x)^n, which of the following must be true?

A. x>3
B. n>1
C. x=3
D. n=3
E. x=n

=>

x^{2n}>(3x)^n
=> x^{2n} > 3^nx^n
=> x^n > 3^n
=> x > 3

Therefore, A is the answer.
Answer: A
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[GMAT math practice question]

In the x-y plane, does the parabola y=ax^2+bx+c have x-intercepts?

1) b^2-4ac<0
2) 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.

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

For y=ax^2+bx+c to have an x-intercept, we must have b^2-4ac ≥ 0.
Thus, condition 1) gives the answer of ‘no’. Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, condition 1) is sufficient.

Condition 2)
If a = -1, b = 0, c = 0, we have y = -x, which has an x-intercept of zero, so the answer is ‘yes’.
If a = -1, b = 0, c = -1, we have y = -x2 – 1, which has no x-intercept. The answer is ‘no’ in this case.
Since we don’t have a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A
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[GMAT math practice question]

If the sum of the minimum and the maximum of 7 consecutive odd integers is 114, what is the average (arithmetic mean) of the seven integers?

A. 19
B. 38
C. 57
D. 76
E. 114

=>

Let the 7 consecutive odd integers be 2n-5, 2n-3, 2n-1, 2n+1, 2n+3, 2n+5, and 2n+7. Then their minimum is 2n-5 and their maximum is 2n+7. The sum of the maximum and minimum is ( 2n – 5 ) + ( 2n + 7 ) = 4n + 2 = 114 and so n = 28.
Since the average and the median of consecutive odd integers are equal, the average is 2n + 1 = 57.

Therefore, the answer is C.

Answer: C

Another way to solve this problem is to note that the average of the consecutive odd integers is equal to the average of the two end points. That is, 114 / 2 or 57.
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[GMAT math practice question]

Is |x|+x <2?

1) x>-1
2) x<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.

The definition of the absolute value gives us two cases to consider when examining the question.

Case 1: x ≥ 0
|x| + x < 2
=> x + x < 2
=> 2x < 2
=> x < 1
The question asks if 0 ≤ x < 1 in this case.

Case 2: x < 0
|x| + x < 2
=> (-x) + x < 2
=> 0 < 2
As this is always true, the answer is always “yes” if x < 0.

Combining these two cases shows that the question asks if x < 1.

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient

Thus condition 1) is not sufficient, but condition 2) is sufficient since the solution set of the question includes the solution set of condition 2), but it doesn’t include that of condition 1).

Therefore, B is the answer.

Answer: B
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[GMAT math practice question]

If x and y are prime numbers, and n is a positive integer, what is the number of factors of xnyn?

1) xy=6
2) n=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.

Since we have 3 variables (x, y and n) 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 x and y are prime numbers and xy = 6, we must have x = 2 and y = 3, or x = 3 and y = 2.
If x = 2 and y = 3, then x^ny^n = 2^23^2 has (2+1)(2+1) = 9 factors, since x and y are different prime numbers and n = 2.
If x = 3 and y = 2, then x^ny^n = 3^22^2 has (2+1)(2+1) = 9 factors, since x and y are different prime numbers and n = 2.
Since we have a unique answer, both conditions together 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)
Since it doesn’t give us any information about the variable n, condition 1) is not sufficient.

Condition 2)
Since it doesn’t give us any information about the variables x and y, condition 2) is not 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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[GMAT math practice question]

N is the 36th prime number. N has remainder 1 when it is divided by 3 and it has remainder 1 when it is divided by 5. What is the remainder when N is divided by 2?

A. -1
B. 0
C. 1
D. 2
E. 3

=>

All prime numbers but 2 are odd integers.
Thus, the remainder when N is divided by 2 is 1.

Therefore, the answer is C.
Answer: C
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[GMAT math practice question]

n is an integer strictly between 10 and 20. What is the value of n?

1) The tens digit of n^2 is 2.
2) The hundreds digit of n^2 is 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.

The squares of the integers that are strictly between 10 and 20 are
11^2 = 121, 12^2 = 144, 13^2 = 169, 14^2 = 196, 15^2 = 225, 16^2 = 256, 17^2 = 289, 18^2 = 324, 19^2 = 361.

Condition 1):
The values for which the tens digit of n^2 is 2 are n = 11, n = 15 and n = 18.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
The values for which the hundreds digit of n^2 is 3 are n = 18 and n = 19.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
n = 18 is the unique solution that satisfies both conditions 1) and 2).
Conditions 1) and 2) are sufficient, when considered together.

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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[GMAT math practice question]

A law firm charges 1500 dollars in total for jobs that take at most 10 hours to complete, and 100 dollars/hour for jobs that take more than 10 hours to complete. If the firm completed 2 jobs, how many hours did it take to complete the two jobs?

1) The cost of one of the jobs was 1500 dollars
2) The total cost of the two jobs was 3500 dollars

=>

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.
Let T1 and T2 be the numbers of hours it took to complete the first and second jobs, respectively.
Since we have 2 variables 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 the cost of the first job is $1500, the cost of the second job is $3500 - $1500 = $2000. Therefore, it took T2 = 2000/100 = 20 hours to complete the second job. However, we cannot determine how long the first job took. For example, we could have T1 = 5 < 10, or T1 = 1500/100 = 15 hours.
Since we don’t have a unique solution, both conditions together are not sufficient.

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.
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[GMAT math practice question]

What is the sum of the remainders when the first 30 positive integers are divided by 5?

A. 50
B. 55
C. 60
D. 65
E. 70

=>

1, 6, 11, 16, 21 and 26 have remainder 1 when they are divided by 5.
2, 7, 12, 17, 22 and 27 have remainder 2 when they are divided by 5.
3, 8, 13, 18, 23 and 28 have remainder 3 when they are divided by 5.
4, 9, 14, 19, 24 and 29 have remainder 4 when they are divided by 5.
5, 10, 15, 20, 25 and 30 have remainder 0 when they are divided by 5.

The sum of the remainders is
1*6 + 2*6 + 3*6 + 4*6 + 0*6 = ( 1 + 2 + 3 + 4 + 0 ) * 6 = 60.

Therefore, the answer is C.

Answer: C
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[GMAT math practice question]

At a fruit store, apples are sold for $4 per pound and oranges are sold for $6 per pound. Is the total weight of apples sold greater than the total weight of oranges sold?

1) The average (arithmetic mean) price of fruit sold is less than $5.
2) The total weight of fruit sold is 10 pounds.

=>

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 x and y be the weights of apples and oranges sold, respectively.
The question asks if x > y.

Condition 1)
The average price of fruit sold is given by (4x + 6y)/(x + y). Now,
( 4x + 6y ) / ( x + y ) < 5
=> 4x + 6y < 5(x+y)=5x+5y
=> 6y - 5y < 5x – 4x
=> y < x
and the answer is ‘yes’.
Thus, condition 1) is sufficient.

Condition 2)
Condition 2) tells us that x + y = 10, but we cannot determine whether x >y.
Condition 2) is not sufficient.

Therefore, A is the answer.

Answer: A

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[GMAT math practice question]

693 and n have the same prime factors and n is a multiple of 693 that is greater than 693. What is the smallest possible value of n/693?

A. 2
B. 3
C. 5
D. 7
E. 11

=>

Since 693 = 3^2*7*11, the smallest integer multiple of 693 that is greater than 693 with prime factors 3, 7 and 11 only is n = 693*3.
Thus, n / 693 = ( 693 * 3 ) / 693 = 3.

Therefore, the answer is B.
Answer : B
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[GMAT math practice question]

If An+1=120 + (1/4)An for each positive integer n, and A1=150, which of these ranges of values includes A12?

A. 110-120
B. 120-130
C.130-140
D. 140-150
E. 150-160

=>

A1 = 150, A2 = 120 + (1/4)150 = 120 + 32.5 = 152.5. As An is increasing,
A12 must lie between 150 and 160.

Therefore, the answer is E.

Answer: E
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[GMAT math practice question]

If n is positive integer, is 21 a factor of n?

1) 21 is a factor of 3n
2) 21 is a factor of n^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.

Since we have 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
If n = 21, then 21 is a factor of n and the answer is “yes”.
If n = 7, then 21 is not a factor of n and the answer is “no”.
Since the solution is not unique, condition 1) is not sufficient.

Condition 2)
Since 21 = 3*7 is a factor of n^2, each of 3 and 7 is a factor of n^2.
If 3 is a factor of n^2, then, since 3 is a prime number, 3 is a factor of n.
If 7 is a factor of n^2, then, since 7 is a prime number, 7 is a factor of n.
Thus, 21 is a factor of n.
Condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

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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