[GMAT math practice question]

What is the reminder when n^2 is divided by 4?

1) When n is divided by 2, the reminder is 1.
2) When n is divided by 3, the reminder is 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.

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)
Plugging-in numbers is suggested for remainder questions.
The integers which have a remainder of 1 when divided by 2 are odd.
So,
n: 1, 3, 5, 7, 9, …
and
n^2: 1, 9, 25, 49, 81, …
Each value of n2 has a remainder of 1 when it is divided by 4.
So, condition 1) is sufficient.

Condition 2)
When n = 1, n^2= 1 has a remainder of 1 when it is divided by 4.
When n = 4, n^2=16 has a remainder of 0 when it is divided by 4.
Since we don’t have a unique answer, condition 2) is not sufficient.

Therefore, A is the answer.

Answer: A

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]

X is proportional to Y. If Y is increased by 30%, by approximately what percent is X^2 increased?

A. 30%
B. 40%
C. 50%
D. 60%
E. 70%

=>

X = kY for some k.
Since (1.3kY)^2 = 1.69k^2Y^2 = 1.69(kY)^2 = 1.69X^2, we have (1.69X^2 – X^2) / X^2 = 0.69, which is approximately 70%.

Therefore, the answer is E.

Answer: E
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[GMAT math practice question]

If |x+1/2|=|y+1/2|, what is the value of x+y?

1) xy<0
2) x>0 and y<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.

|x+1/2|=|y+1/2|
=> |x+1/2|^2=|y+1/2|^2
=> (x+1/2)^2=(y+1/2)^2
=> x^2 + x + 1/4 = y^2 + y + 1/4
=> x^2 - y^2 + x - y = 0
=> (x-y)(x+y)+(x-y) = 0
=> (x-y)(x+y+1) = 0
=> x=y or x+y=-1

Condition 1)
Since xy < 0, we have x≠y and x+y = -1 from the original condition.

Condition 2)
Since x > 0 and y < 0, we have x≠y and x+y = -1 from the original condition.

Therefore, D is the answer.

By Tip 1), D is most likely to be the answer.

Answer: D
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[GMAT math practice question]

Alice studies for a hours on each of Monday, Tuesday, and Thursday. She also studies for b hours on Wednesday, and for c hours on Friday. What is her median study time for the school week?

1) a=10
2) b+c=30

=>

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.

Depending on their values, it is possible to rearrange a, a, a, b, c into ascending order in the following ways:
a, a, a, b, c
a, a, a, c, b
b, a, a, a, c
c, a, a, a, b
b, c, a, a, a
c, b, a, a, a
In every order, the median (middle value) is a. So, the question asks for the value of a.

Thus, only condition 1) is sufficient.


Therefore, A is the answer.

Answer: A
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[GMAT math practice question]

Which of the following is the greatest?

A. (1+2)^5
B. 10^3
C. (1^2+2^2)^4
D. (2^2+2^2+2^2)^3
E. (2^2+2^2+2^2+2^2)^2

=>

A. (1+2)^5 = 3^5 = 243
B. 10^3 = 1000
C. (1^2+2^2)^4 = 5^4 = 625
D. (2^2+2^2+2^2)^3 = 12^3 = 1728
E. (2^2+2^2+2^2+2^2)^2 = 16^2 = 256

Therefore, D is the answer.

Answer: D
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[GMAT math practice question]

If 50-√7<x<50+√7, then x=?

1) x is an odd 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.

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

Since 2< √7< 3, we have 47< x < 53 from the original condition ‘50-√7<x<50+√7’.

Condition 1)
As x is an odd integer, it could be 49 or 51.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
If √x is an integer, then x is the square of an integer.
49 is the only perfect square of an integer between 47 and 53.
Thus, x=49 and condition 2) is sufficient.

Therefore, B is the answer.

Answer: B
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[GMAT math practice question]

The number 123,k50 is a 6-digit integer, and k is a positive 1 digit integer. Which of the following cannot be a factor of 123,k50?

A. 2
B. 3
C. 4
D. 5
E. 6

=>

The last two digits tell us whether the number is divisible by 4.
Since 50 is not a multiple of 4, the number cannot be a multiple of 4.

Therefore, the answer is C.

Answer: C

Let’s see why the number could be divisible by each of the other options:

A: Since the units digit is an even number, the whole number is a multiple of 2.
B: A number is divisible by 3 if the sum of its digits is divisible by 3. If k = 4, then the sum of the digits is 1 + 2 + 3 + 4 + 5 + 0 = 15, which is a multiple of 3, and so the number is a multiple of 3.
D: Since the units digit is a multiple of 5, the number is a multiple of 5.
E: If k = 4, the number is divisible by 3 as seen in part B. Since it is also divisible by 2 (see part A), the number is divisible by 6.
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[GMAT math practice question]

If the average (arithmetic mean) price of apples, bananas and oranges is $3.00 per pound, what is their median price?

1) The price of apples is $3.00 per pound.
2) The price of bananas is $2.97 per pound.

=>

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 (a for apples, b for bananas and o for oranges) and 1 equation ( ( a + b + o ) / 3 = 3), 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.

This question is a new type of GMAT question. The average is provided in the original condition and the question asks for the value of the median. The same average applies for each of the conditions.

Conditions 1) & 2)
If a = 3.00 and b = 2.97, then ( a + b + o ) / 3 = 3.
So,
3.00 + 2.97 + o = 9
and
o = 3.03. Therefore, the median is 3.00.
Thus, both conditions together are sufficient.

Condition 1)
We consider three cases.
Case 1: a = b = c = 3.00
Since all prices are the same, the median is 3.00.

Case 2: b < 3.00
If b < 3.00, then we must have c > 3.00.
Therefore, the median is 3 since b < a < c.

Case 3: b > 3.00
If b > 3.00, then we must have c < 3.00.
Therefore, the median is 3 since c < a < b.

Thus, condition 1) is sufficient on its own.

Condition 2)
If a = 3.00, b = 2.97 and c = 3.03, then the median is 3.00.
If a = 2.00, b = 2.97 and c = 3.00, then the median is 2.97.
Since we don’t have a unique solution, condition 2) is not sufficient on its own.

Therefore, A is the answer.

Answer: A

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]

Is √x+x>-√y？

1) √x+√y = 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.

Since we have 2 variables (x and y) 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.

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

√x+x>-√y
=> √x+√y+ x>0
Since √x+√y≥0 is always true, we just need to check if x > 0.
Condition 2) is sufficient and since condition 1) is hard to check and condition 2) is easy to check, the answer is D by CMT (Common Mistake Type) 4B.

Condition 1)

We have x ≥ 0 from √x,
Then √x+√y + x ≥ 1 > 0.
Thus condition 1) is sufficient.


Condition 2)

Since √x+√y ≥ 0 and x > 1, we have √x+√y + x > 1 > 0.
Thus condition 2) is sufficient.

Therefore, D is the answer.

Answer: D

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]

A jar contains 4 balls, labeled 1,2,3, and 4. A ball is selected from the jar, and its number is recorded before it is returned to the jar. If a second ball is then selected from the jar, what is the probability that the difference between the numbers on the two balls selected is 1?

A. 1/3
B. 3/4
C. 1/2
D. 5/8
E. 3/8

=>

The total number of ways the two balls may be selected is 4C2 = ( 4 * 3 ) / ( 1 * 2 ) = 6.
There are three ways in which the numbers on the two balls selected can have a difference of 1: ( 1, 2 ), ( 2, 3 ), and ( 3, 4 ).

Thus, the probability is 3/6 or 1/2

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

In the x-y plane, what is the slope of the line segment joining the y-intercept and the positive x-intercept of the curve y=x2-5x-6?

A. 1
B. 2
C. 3
D. 4
E. 5

=>

First, we find the x-intercepts of the curve. These occur when y = 0. Now,
y=x^2-5x-6
=> y = (x+1)(x-6).

Thus, the x-intercepts are the points (-1,0) and (6,0), and the positive x-intercept is (6,0).
As the y-intercept occurs when x = 0, it is the point (0,-6).

Thus, the required slope is (6 – 0) / ( 0 – (-6) ) = 1.

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

Which of the following relationships must be inversely proportional to each other?

I. Work done and working time
II. Speed and Traveling Time
III. Profit and Cost

A. I only
B. II only
C. III only
D. II and III
E. I and II

=>

Statement I
Let R be the work rate, W be the work done and T be the working time. Then R = W/T and T = W*R. Therefore, T and W are directly proportional.

Statement II
Let V be the speed, D be the distance and T be the traveling time. Then
V = D/T, and V and T are inversely proportional.

Statement III
Let P be the profit, R be the revenue and C be the cost. Then
P = R – C, and P and C are neither directly nor inversely proportional.

Therefore, B is the answer.

Answer B
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[GMAT math practice question]

When a and b are positive integers, what is the greatest common divisor of a + b and a + 100?

1) a = 100
2) b = 99

=>
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.


Condition 1)
If a = 100, and b = 98, the greatest common divisor of a + b and a + 100 is 2.
If a = 100, and b = 99, the greatest common divisor of a + b and a + 100 is 1.
Thus, condition 1) is not sufficient as it does not yield a unique solution.

Condition 2)
a + b = a + 99 and a + 100 are consecutive integers. Therefore, their greatest common divisor is 1.
Thus, condition 2) is sufficient.


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

If the sum of the squares of two positive integers is 106, what is the sum of the two integers?

A. 13
B. 14
C. 15
D. 16
E. 17

=>
x^2 + y^2 = 106.
We can plug-in the numbers from 1 to 10 to find the values of x and y that work.
We find that 9^2 + 5^2 = 81 + 25 = 106. So, the two numbers are 5 and 9, and their sum is 9 + 5 = 14.

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

What is the median of m, n and 5?

1) m+n=10
2) m=5

=>

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 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) together:
Since m + n = 10 and m = 5, we have m = n = 5.
Thus, the median of m, n and 5 is 5.

Both conditions together are sufficient.

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): m + n = 10
If one of m and n is less than 5, the other one must be greater than 5. For example, if m < 5, then n > 5.
If one of m and n is 5, the other must also equal 5. For example, if m = 5, then n = 5.
In both cases, the median is 5.
Condition 1) is sufficient.

Condition 2) : m = 5.
If n > 5, the numbers are 5, 5, n in ascending order.
If n < 5, the numbers are n, 5, 5 in descending order.
If n = 5, the numbers are 5, 5, 5.
In each of the possible cases, the median is 5.
Condition 2) is sufficient.

Therefore, D is the answer.

Answer: D

Note: The VA approach tells us that the answer is most likely to be D, since this is a CMT(Common Mistake Type) 4B question.
Condition 1) is easy to check, but condition 2) is more difficult to work with. If you can’t figure out condition 2), you should choose D as the answer.

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]

Alice works from Monday to Friday only. How many days does she work in April?

1) April 1 is Saturday or Sunday
2) April 28 is Saturday or Sunday

=>

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.
When the day on which a specific date, for example, the first of April 2018 is fixed, we can determine the days on which other dates in April fall.
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 April 1st is Saturday, Alice works on the 3rd - 7th, 10th - 14th, 17th - 21st, 24th - 28th. This means she works on 20 days.
If April 1st is Sunday, Alice works on the 2nd - 6th, 9th - 13th, 16th - 20th, 23rd - 27th and 30th. This means she works on 21 days.
Since we don't have a unique solution, condition 1) is not sufficient.

Condition 2)
If April 28th is Saturday, Alice works on the 2nd - 6th, 9th - 13th, 16th - 20th, 23rd - 27th and 30th. This means she works on 21 days.
If April 28th is Sunday, Alice works on the 1st - 5th, 8th - 12th, 15th - 19th, 22nd - 26th, 29th and 30th. This means she works on 22 days.

Since we don't have a unique solution, condition 2) is not sufficient.
Conditions 1) & 2) together:
If the 1st of April is a Saturday or Sunday and the 28th of April is a Saturday or Sunday, the 1st of April must be a Saturday.
So, Alice works on the 2nd - 6th, 9th - 13th, 16th - 20th, 23rd - 27th and 30th. Therefore, she works on 21 days.
Since we have a unique solution, 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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the answer can also be E ?


Sent from my iPhone using GMAT Club Forum mobile app
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[GMAT math practice question] 4/30 trend 5.10

If the profit on a product is 20 percent of its sales price, what percent is the profit of the cost price?

A. 20%
B. 25%
C. 30%
D. 33+(1/3)%
E. 35%

=>

Let P, R and C be the profit, the sales price (revenue) and the cost price of the product, respectively.
Then P = R – C.
Since P is 20 percent of the sales price, the IVY method tells us that P = 20*(1/100)*R.
So, P = (1/5)R or R = 5P.
Plugging this into P = R – C yields P = 5P – C or C = 4P.
It follows that P = (1/4)C, and P is 25 percent of C.

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

x and y are positive integers. If p and q are different prime numbers, what is the number of factors of pxqy?

1) x=2 and y=3
2) p=2 and q=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 first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Since p and q are different prime numbers, the number of factors of pxqy is (x+1)(y+1).
Thus, the question asks for the value of (x+1)(y+1) and only condition 1) is sufficient.

Therefore, A is the answer.

Answer: A

When we count the number of factors, the prime numbers must be different in prime number factorizations.
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