If n is an integer from 1 to 96, what is the probability for n*(n+1)*(n+2) being divisible by 8?
A) 25%
B) 50%
C) 62.5%
D) 72.5%
E) 75%
n is even - anytime n is even, it is divisible by 8
total nos using sequence theorm 96 = 2 + (#-1) 2,
hence # = 48
n is odd - again 48 no
but in 1-8, only one combination is divisible by 8, when n is 7,15, 22... hence 12 cases
probalility = possible outcomes / total outcomes = 48+12 / 96 = 60/96 = .625 = 62.5%
OR
we need to check for what values of n, n(n+1)(n+2) is divisible 8....
n(n+2) is divisible by 8 for all values of even numbers and there are 48 even nos in 1 to 96....
now the remaining part (n+1) is divisible by 8 for 12 odd numbers....such as 7, 15, 23, 31, 39.... to find this u
can divide 96 by 8....
so totally 48 + 12 = 60 numbers are there between 1 to 96 for which n(n+1)(n+2) is divisible 8....
now when calculate the % it would be ( 60 / 96 ) * 100 = 62.5 %....
Thursday, September 09, 2010
During a behavioral experiment in a psychology class, each student is asked to compute his or her lucky number by
raising 7 to the power of the student's favorite day of the week (numbered 1 through 7 for Monday through Sunday
respectively), multiplying the result by 3, and adding this to the doubled age of the student in years, rounded to
the nearest year. If a class consists of 28 students, what is the probability that the median lucky number in the
class will be a non-integer?
(A) 0%
(B) 10%
(C) 20%
(D) 30%
(E) 40%
Since any power of 7 is odd, the product of this power and 3 will always be odd. Adding this odd number to the
doubled age of the student (an even number, since it is the product of 2 and some integer) will always yield an odd
integer. Therefore, all lucky numbers in the class will be odd.
The results of the experiment will yield a set of 28 odd integers, whose median will be the average of the 14th and
15th greatest integers in the set. Since both of these integers will be odd, their sum will always be even and
their average will always be an integer. Therefore, the probability that the median lucky number will be a non-
integer is 0%.
raising 7 to the power of the student's favorite day of the week (numbered 1 through 7 for Monday through Sunday
respectively), multiplying the result by 3, and adding this to the doubled age of the student in years, rounded to
the nearest year. If a class consists of 28 students, what is the probability that the median lucky number in the
class will be a non-integer?
(A) 0%
(B) 10%
(C) 20%
(D) 30%
(E) 40%
Since any power of 7 is odd, the product of this power and 3 will always be odd. Adding this odd number to the
doubled age of the student (an even number, since it is the product of 2 and some integer) will always yield an odd
integer. Therefore, all lucky numbers in the class will be odd.
The results of the experiment will yield a set of 28 odd integers, whose median will be the average of the 14th and
15th greatest integers in the set. Since both of these integers will be odd, their sum will always be even and
their average will always be an integer. Therefore, the probability that the median lucky number will be a non-
integer is 0%.
According to the directions on a can of frozen orange juice concentrate, 1 can of concentrate is to be mixed with 3 cans of water to make orange juice. How many 12-ounce cans of the concentrate are required to prepare 200 6-ounce servings of orange juice?
A. 25
B. 34
C. 60
D. 67
E. 100
4 cans of orange juice need 1 can of concentrate.
200 cans 6-ounce of orange juice need 50 cans 6-ounce concentrate.
so 12-ounce concentrate = 50/2 = 25 cans
OR
Let number of 12 ounce cans required = x
number of 12 ounce cans of water required = 3x
Total required servings of Orange juice = 200*6 ounce
Hence
x(12) + 3x(12) = 200*6
48x= 200*6
x=25
Hence A
A. 25
B. 34
C. 60
D. 67
E. 100
4 cans of orange juice need 1 can of concentrate.
200 cans 6-ounce of orange juice need 50 cans 6-ounce concentrate.
so 12-ounce concentrate = 50/2 = 25 cans
OR
Let number of 12 ounce cans required = x
number of 12 ounce cans of water required = 3x
Total required servings of Orange juice = 200*6 ounce
Hence
x(12) + 3x(12) = 200*6
48x= 200*6
x=25
Hence A
E is a collection of four odd integers and the greatest difference between any two integers in E is 4. The standard
deviation of E must be one of how many numbers?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
Suppose the integers are 1, 3 and 5. Therefore the four integers can be:
1, 5, 5, 5
1, 3, 5, 5
1, 3, 3, 5
1, 1, 5, 5
1, 1, 1, 5
1, 1, 3, 5
Here two pairs have the same standard deviation. thus in all we have four different standard deviations. Hence B
deviation of E must be one of how many numbers?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
Suppose the integers are 1, 3 and 5. Therefore the four integers can be:
1, 5, 5, 5
1, 3, 5, 5
1, 3, 3, 5
1, 1, 5, 5
1, 1, 1, 5
1, 1, 3, 5
Here two pairs have the same standard deviation. thus in all we have four different standard deviations. Hence B
If eleven consecutive integers are listed from least to greatest, what is the average (arithmetic mean) of the
eleven integers?
(1) The average of the first nine integers is 7.
(2) The average of the last nine integers is 9.
Let the numbers be a, b, c, d, e, f, g, h , i, j, k
i) For odd number of consecutive integers median = mean
ii)We also know that the median is the "middle" number in a group (when arranged in ascending or descending order)
consisting of an odd number of numbers
We have to find f
From statement (1): It is given that average of first nine numbers = 7
Hence this implies e = 7 ..since it is given numbers are consecutive hence f = 8
Thus sufficient
From statement (2): It is given that average of last nine numbers = 9
Hence this implies g = 9..since it is given numbers are consecutive hence f = 8
Thus sufficient
eleven integers?
(1) The average of the first nine integers is 7.
(2) The average of the last nine integers is 9.
Let the numbers be a, b, c, d, e, f, g, h , i, j, k
i) For odd number of consecutive integers median = mean
ii)We also know that the median is the "middle" number in a group (when arranged in ascending or descending order)
consisting of an odd number of numbers
We have to find f
From statement (1): It is given that average of first nine numbers = 7
Hence this implies e = 7 ..since it is given numbers are consecutive hence f = 8
Thus sufficient
From statement (2): It is given that average of last nine numbers = 9
Hence this implies g = 9..since it is given numbers are consecutive hence f = 8
Thus sufficient
List K consists of 12 consecutive integers, if -4 is the least integer in list K, what is the range of the positive
integers in the list K?
A. 5
B. 6
C. 7
D. 11
E. 12
The least number in the list is -4, thus the list is: -4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6, 7
Positive integers in the above list: 1, 2, 3, 4, 5, 6, 7
Therefore the range of the positive integers is 7-1 = 6
integers in the list K?
A. 5
B. 6
C. 7
D. 11
E. 12
The least number in the list is -4, thus the list is: -4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6, 7
Positive integers in the above list: 1, 2, 3, 4, 5, 6, 7
Therefore the range of the positive integers is 7-1 = 6
A certain characteristic in a large population has a distribution that is symmetric about the mean m.If 68 percent
of the distribution lies within one Standard Deviation d of the mean, what percent of the distribution is less than
m+d?
A. 16%
B. 32%
C. 48%
D. 84%
E. 92%
In a normal bell curved distribution, 50% are below the mean and 50% are over it
If 68% are distributed within 1 S.D of the mean then this implies that 34% are 1 S.D above the mean and 34% are 1
S.D below the mean i.e 34% between m and m+d and 34% between m-d and m
The distribution is symmetric about m also => 32/2 = 16% between 0 and m-d and 16% m+d and above.
Hence total that is less than m+d = 100-16 = 84%
OR
Distribution is symmetric around mean => 68/2 = 34% =>(Mean-S.D, Mean) = (Mean, Mean+S.D] = 34% . Thus below Mean
+S.D = 50+34 = 84%
NOTE: For a normal bell-curve distribution, the percentage is approx 34% between the mean and 1 S.D. the percentage
is approximately 13.6% between 1 SD and 2 SD, the percentage is approximately 2% between 2 S.D and on...
of the distribution lies within one Standard Deviation d of the mean, what percent of the distribution is less than
m+d?
A. 16%
B. 32%
C. 48%
D. 84%
E. 92%
In a normal bell curved distribution, 50% are below the mean and 50% are over it
If 68% are distributed within 1 S.D of the mean then this implies that 34% are 1 S.D above the mean and 34% are 1
S.D below the mean i.e 34% between m and m+d and 34% between m-d and m
The distribution is symmetric about m also => 32/2 = 16% between 0 and m-d and 16% m+d and above.
Hence total that is less than m+d = 100-16 = 84%
OR
Distribution is symmetric around mean => 68/2 = 34% =>(Mean-S.D, Mean) = (Mean, Mean+S.D] = 34% . Thus below Mean
+S.D = 50+34 = 84%
NOTE: For a normal bell-curve distribution, the percentage is approx 34% between the mean and 1 S.D. the percentage
is approximately 13.6% between 1 SD and 2 SD, the percentage is approximately 2% between 2 S.D and on...
The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical A present and inversely proportional to the concentration of chemical B present. If the concentration of chemical B is increased by 100 percent, which of the following is closest to the percent change in the concentration of chemical A required to keep the reaction rate unchanged?
A. 100% decrease
B. 50% decrease
C. 40% decrease
D. 40% increase
E. 50% increase
rate = k*(A1^2)/B
If concentration of chemical B is increased by 100 percent then
rate = k*(A2^2)/2B
(A2/A1) ^ 2 = 2
A2/A1 = Square root (2)
(A2/A1) - 1 = Square root (2)-1 = 0.414
(A2-A1)/A1 = 0.414 = 40% approximately
A. 100% decrease
B. 50% decrease
C. 40% decrease
D. 40% increase
E. 50% increase
rate = k*(A1^2)/B
If concentration of chemical B is increased by 100 percent then
rate = k*(A2^2)/2B
(A2/A1) ^ 2 = 2
A2/A1 = Square root (2)
(A2/A1) - 1 = Square root (2)-1 = 0.414
(A2-A1)/A1 = 0.414 = 40% approximately
Of the 800 companies in Company X, 70% have been with the company for at least 10 years. If y of these "long-term"
members were to retire, and no other employee changes were to occur, what value of y would reduce the percent of
"long-term" employees in the company to 60%.
A) 200
B) 160
C) 112
D) 80
E) 56
Assume y=x
The number of people working more than 10 years = 70% of 800 = 560
Hence (560-x)/(800-x)=60%
Thus x=200
OR
The number of long term workers = 70% of 800 = 560
Now if y of the long term workers retired, then long term workers left are 560-y and the number of total employees
= 800-y
Thus (560-y)/(800-y) * 100 = 60
560-y = 480 - 0.6y
80 = 0.4y
y = 200 Ans
members were to retire, and no other employee changes were to occur, what value of y would reduce the percent of
"long-term" employees in the company to 60%.
A) 200
B) 160
C) 112
D) 80
E) 56
Assume y=x
The number of people working more than 10 years = 70% of 800 = 560
Hence (560-x)/(800-x)=60%
Thus x=200
OR
The number of long term workers = 70% of 800 = 560
Now if y of the long term workers retired, then long term workers left are 560-y and the number of total employees
= 800-y
Thus (560-y)/(800-y) * 100 = 60
560-y = 480 - 0.6y
80 = 0.4y
y = 200 Ans
5 pieces of wood have an average (arithmetic mean) length of 124 centimeters and a median length of 140
centimeters. What is the maximum possible length in centimeters of the shortest piece of wood?
A. 90
B. 100
C. 110
D. 130
E. 140
Shortest to Longest length ---- L1, L2, L3 = 140, L4, L5
L1 + L2 + 140 + L4 + L5 = 5 * 124
For L1 to be the maximum, L4 and L5 should be minimum
L1=L2
Thus, L1 + L2 = (5*124) - (140*3) = 620 - 420 = 200
Hence L1 = L2 = 100
centimeters. What is the maximum possible length in centimeters of the shortest piece of wood?
A. 90
B. 100
C. 110
D. 130
E. 140
Shortest to Longest length ---- L1, L2, L3 = 140, L4, L5
L1 + L2 + 140 + L4 + L5 = 5 * 124
For L1 to be the maximum, L4 and L5 should be minimum
L1=L2
Thus, L1 + L2 = (5*124) - (140*3) = 620 - 420 = 200
Hence L1 = L2 = 100
Is n less than 1 ?
(1) nx – n less than 0
(2) x–1 = –2
From stat(1): (n^x) – n less than 0 -- no information about x --- insufficient
From stat(2): x^–1 = –2 -- no information about n --- insufficient
Taking both the statements together we get: From stat 2 we get x = -1/ 2
Substituting value of x in stat 1 we get (n^-1/2) -n less than 0
(n^3/2)>1
=> n will be greater than 1 ---- sufficient
Hence C
(1) nx – n less than 0
(2) x–1 = –2
From stat(1): (n^x) – n less than 0 -- no information about x --- insufficient
From stat(2): x^–1 = –2 -- no information about n --- insufficient
Taking both the statements together we get: From stat 2 we get x = -1/ 2
Substituting value of x in stat 1 we get (n^-1/2) -n less than 0
(n^3/2)>1
=> n will be greater than 1 ---- sufficient
Hence C
How many odd integers are greater than integer X and less than the integer Y?
1). there are 12 even integers greater than X and less than Y
2). there are 24 integers greater than X and less than Y
From statement 1) -- We cannot determine about both X, Y being even or odd. ... hence insufficient
From statement 2) -- There are 24 integers between X and Y
Let it start with an even integer ...if so then it will end with an odd integer or if it starts with an odd integer
then it will end with an even integer ...hence there will always be 12 odd and 12 even integers in the total of 24
consecutive integers
e.g consider X=2, Y=27 thus the series will be is 2 ... 26, 27
X=3, Y=28, thus the series will be 3, ... 27, 28
In each case, the total number of odd integers is the same ... hence sufficient
1). there are 12 even integers greater than X and less than Y
2). there are 24 integers greater than X and less than Y
From statement 1) -- We cannot determine about both X, Y being even or odd. ... hence insufficient
From statement 2) -- There are 24 integers between X and Y
Let it start with an even integer ...if so then it will end with an odd integer or if it starts with an odd integer
then it will end with an even integer ...hence there will always be 12 odd and 12 even integers in the total of 24
consecutive integers
e.g consider X=2, Y=27 thus the series will be is 2 ... 26, 27
X=3, Y=28, thus the series will be 3, ... 27, 28
In each case, the total number of odd integers is the same ... hence sufficient
A certain clothing manufacturer makes only two types of men's blazer: cashmere and mohair. Each cashmere blazer requires 4 hours of cutting and 6 hours of sewing. Each mohair blazer requires 4 hours of cutting and 2 hours of sewing. The profit on each cashmere blazer is $40 and the profit on each mohair blazer is $35. How many of each type of blazer should the manufacturer produce each week in order to maximize its potential weekly profit on blazers?
1) The company can afford a maximum of 200 hours of cutting per week and 200 hours of sewing per week.
2) The wholesale price of cashmere cloth is twice that of mohair cloth.
First, let c be the number of cashmere blazers produced in any given week and let m be the no of mohair blazers
produced in any given week.Let p be the total profit on blazers for any given week.Since the profit on cashmere
blazers is $40 per blazer and the profit on mohair blazers is $35 per blazer, we can form the equation p = 40c +
35m. In order to know the maximum potential value of p, we need to know the maximum values of c and m.
Statement (1) tells us that the maximum number of cutting hours per week is 200 and that the maximum number of
sewing hours per week is 200.
Since it takes 4 hours of cutting to produce a cashmere blazer and 4 hours of cutting to produce a mohair blazer,
we can construct the following inequality: 4c + 4m < = 200. Since it takes 6 hours of sewing to produce a cashmere blazer and 2 hours of sewing to produce a mohair blazer, we can construct the following inequality: 6c + 2m < = 200 . In order to maximize the number of blazers produced, the company should use all available cutting and sewing time. So we can construct the following equations: 4c + 4m = 200 6c + 2m = 200 Since both equations equal 200, we can set them equal to each other and solve: 4c + 4m = 6c + 2m -->
2m = 2c -->
m = c -->
4m + 4(m) = 200 -->
8m = 200 -->
m = 25 -->
m = c -->
c = 25
So when m = 25 and c = 25, all available cutting and sewing time will be used. If p = 40c + 35m, the profit in this
scenario will be 40(25) + 35(25) or $1,875. Is this the maximum potential profit?
Since the profit margin on cashmere is higher, might it be possible that producing only cashmere blazers would be
more profitable than producing both types? If no mohair blazers are made, then the largest number of cashmere
blazers that could be made will be the value of c that satisfies 6c = 200 (remember, it takes 6 hours of sewing to
make a cashmere blazer). So c could have a maximum value of 33 (the company cannot sell 1/3 of a blazer). So
producing only cashmere blazers would net a potential profit of 40(33) or $1,320. This is less than $1,875, so it
would not maximize profit.
Since mohair blazers take less time to produce, perhaps producing only mohair blazers would yield a higher profit.
If no cashmere blazers are produced, then the largest number of mohair blazers that could be made will be the value
of m that satisfies 4m = 200 (remember, it takes 4 hours of cutting to produce a mohair blazer). So m would have a
maximum value of 50 in this scenario and the profit would be 35(50) or $1,750. This is less than $1,875, so it
would not maximize profit.
So producing only one type of blazer will not maximize potential profit, and producing both types of blazer
maximizes potential profit when m and c both equal 25.
Statement (1) is sufficient.
Statement (2) tells us that the wholesale cost of cashmere cloth is twice that of mohair cloth. This information is
irrelevant because the cost of the materials is already taken into account by the profit margins of $40 and $35
given in the question stem.
Statement (2) is insufficient.
1) The company can afford a maximum of 200 hours of cutting per week and 200 hours of sewing per week.
2) The wholesale price of cashmere cloth is twice that of mohair cloth.
First, let c be the number of cashmere blazers produced in any given week and let m be the no of mohair blazers
produced in any given week.Let p be the total profit on blazers for any given week.Since the profit on cashmere
blazers is $40 per blazer and the profit on mohair blazers is $35 per blazer, we can form the equation p = 40c +
35m. In order to know the maximum potential value of p, we need to know the maximum values of c and m.
Statement (1) tells us that the maximum number of cutting hours per week is 200 and that the maximum number of
sewing hours per week is 200.
Since it takes 4 hours of cutting to produce a cashmere blazer and 4 hours of cutting to produce a mohair blazer,
we can construct the following inequality: 4c + 4m < = 200. Since it takes 6 hours of sewing to produce a cashmere blazer and 2 hours of sewing to produce a mohair blazer, we can construct the following inequality: 6c + 2m < = 200 . In order to maximize the number of blazers produced, the company should use all available cutting and sewing time. So we can construct the following equations: 4c + 4m = 200 6c + 2m = 200 Since both equations equal 200, we can set them equal to each other and solve: 4c + 4m = 6c + 2m -->
2m = 2c -->
m = c -->
4m + 4(m) = 200 -->
8m = 200 -->
m = 25 -->
m = c -->
c = 25
So when m = 25 and c = 25, all available cutting and sewing time will be used. If p = 40c + 35m, the profit in this
scenario will be 40(25) + 35(25) or $1,875. Is this the maximum potential profit?
Since the profit margin on cashmere is higher, might it be possible that producing only cashmere blazers would be
more profitable than producing both types? If no mohair blazers are made, then the largest number of cashmere
blazers that could be made will be the value of c that satisfies 6c = 200 (remember, it takes 6 hours of sewing to
make a cashmere blazer). So c could have a maximum value of 33 (the company cannot sell 1/3 of a blazer). So
producing only cashmere blazers would net a potential profit of 40(33) or $1,320. This is less than $1,875, so it
would not maximize profit.
Since mohair blazers take less time to produce, perhaps producing only mohair blazers would yield a higher profit.
If no cashmere blazers are produced, then the largest number of mohair blazers that could be made will be the value
of m that satisfies 4m = 200 (remember, it takes 4 hours of cutting to produce a mohair blazer). So m would have a
maximum value of 50 in this scenario and the profit would be 35(50) or $1,750. This is less than $1,875, so it
would not maximize profit.
So producing only one type of blazer will not maximize potential profit, and producing both types of blazer
maximizes potential profit when m and c both equal 25.
Statement (1) is sufficient.
Statement (2) tells us that the wholesale cost of cashmere cloth is twice that of mohair cloth. This information is
irrelevant because the cost of the materials is already taken into account by the profit margins of $40 and $35
given in the question stem.
Statement (2) is insufficient.
IS x^4 + y^4 > z^4 ?
a) x^2 + y^2 > z^2
b) x + y > z
(1) x^2 + y^2 > z^2, when squared, gives the stated equation.
From this we cannot conclude definitively whether x^4 + y^4 > z^4 because the equation contains (2x^2 * y^2).
If this is removed, then x^4 + y^4 may or may not be > z^4.
Thus insufficient..
e.g - 2+3+4 > 5 --- if we remove 1 number from the left hand side of the inequality then the inequality may or may
not hold true..
OR
Statement (1) ---- x^2 + y^2 > z^2
Let x = {(2) ^ 1/2}, y = {(3) ^ 1/2}, z = {(4) ^1/2}
Hence 2+3>4 but at the same time 4+9 < 16
Let x = 2, y= 4, z=3
4+16>9
And 16 + 256 > 81
Thus (1) is insufficient.
Statement (2) ---- x+y> z
Let x=2, y=3, z=6
Hence 2+3<6
Now let x=2, y=6, z=3
Then 2+6>3
Both 1 and 2 together:insufficient
Hence answer E.
a) x^2 + y^2 > z^2
b) x + y > z
(1) x^2 + y^2 > z^2, when squared, gives the stated equation.
From this we cannot conclude definitively whether x^4 + y^4 > z^4 because the equation contains (2x^2 * y^2).
If this is removed, then x^4 + y^4 may or may not be > z^4.
Thus insufficient..
e.g - 2+3+4 > 5 --- if we remove 1 number from the left hand side of the inequality then the inequality may or may
not hold true..
OR
Statement (1) ---- x^2 + y^2 > z^2
Let x = {(2) ^ 1/2}, y = {(3) ^ 1/2}, z = {(4) ^1/2}
Hence 2+3>4 but at the same time 4+9 < 16
Let x = 2, y= 4, z=3
4+16>9
And 16 + 256 > 81
Thus (1) is insufficient.
Statement (2) ---- x+y> z
Let x=2, y=3, z=6
Hence 2+3<6
Now let x=2, y=6, z=3
Then 2+6>3
Both 1 and 2 together:insufficient
Hence answer E.
In the xy-plane, the line k passes through the origin and through the point (a, b), where ab is not equal to 0. Is
b positive ?
1). The slope of line k is negative.
2). a is less than b
It is given that the line passes through (0,0) and (a,b)
So it's slope = (b-0)/(a-0) = b/a
Statement (1) ---- Slope of line k is (-ve).
=> (b/a) less than 0 implies a and b are of opposite signs.
From the above we cannot conclude that b is less than 0.
Statement (2) ---- a is less than b ...again this alone cannot help us to infer that whether b is greater than 0 or
less than 0 as b can take the value either way round.
Combining Statement (1) and (2) ---- we know that a and b are of opposite signs and that a is less than b.
Therefore, clearly a is negative and hence b is positive.
Hence the answer is C.
b positive ?
1). The slope of line k is negative.
2). a is less than b
It is given that the line passes through (0,0) and (a,b)
So it's slope = (b-0)/(a-0) = b/a
Statement (1) ---- Slope of line k is (-ve).
=> (b/a) less than 0 implies a and b are of opposite signs.
From the above we cannot conclude that b is less than 0.
Statement (2) ---- a is less than b ...again this alone cannot help us to infer that whether b is greater than 0 or
less than 0 as b can take the value either way round.
Combining Statement (1) and (2) ---- we know that a and b are of opposite signs and that a is less than b.
Therefore, clearly a is negative and hence b is positive.
Hence the answer is C.
Each of the 25 balls in a certain box is either red, blue or white and has a number from 1 to 10 painted on it. If one ball is to be selected at random from the box, what is the probability that the ball selected will either be
white or have an even number painted on it?
1) The probability that the ball will both be white and have an even number painted on it is 0.
2) The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2.
The question is asking for P(W) or P(E)
=>P(W) or P(E) = P(W) + P(E) - P(W and E)
From statement (1) --- it is given that P(W and E) = 0 --- insufficient as we do not know individual probabilities.
From statement (2) --- it is given that P(W) - P(E) = 0.2 --- insufficient
Combining both statement (1) and (2) it is still insufficient as P(W) + P(E) cannot be calculated....P(W U E) = P
(W) + P(E) - P( W and E)
NOTE : P(A and B) = P(A) * P(B) --- This is true only iff both A & B are Independent. Otherwise, it would be -- P(A
and B) = P (A) * P(B/A) = P(B) * P(A/B).
From the question we do not know that if both A and B are independent.
white or have an even number painted on it?
1) The probability that the ball will both be white and have an even number painted on it is 0.
2) The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 0.2.
The question is asking for P(W) or P(E)
=>P(W) or P(E) = P(W) + P(E) - P(W and E)
From statement (1) --- it is given that P(W and E) = 0 --- insufficient as we do not know individual probabilities.
From statement (2) --- it is given that P(W) - P(E) = 0.2 --- insufficient
Combining both statement (1) and (2) it is still insufficient as P(W) + P(E) cannot be calculated....P(W U E) = P
(W) + P(E) - P( W and E)
NOTE : P(A and B) = P(A) * P(B) --- This is true only iff both A & B are Independent. Otherwise, it would be -- P(A
and B) = P (A) * P(B/A) = P(B) * P(A/B).
From the question we do not know that if both A and B are independent.
Is x = square root (x^2) if
(1) x = even
(2) 13 is less than x is less than 17
From Statement (1) -- x = even . Always remember square root (x^2) = mod x.
E.g - squareroot (-4 ^ 2) = 4 ; suareroot (4^ 2) = 4. Hence 1 is insufficient.
From Statement (2) --13 is less than x is less than 17 implies that x can be 14, 15 or 16 ..hence X IS
POSITIVE...Hence sufficient.
Note : Is x = square root (x^2) is equivalent to stating Is x = mod x
(1) x = even
(2) 13 is less than x is less than 17
From Statement (1) -- x = even . Always remember square root (x^2) = mod x.
E.g - squareroot (-4 ^ 2) = 4 ; suareroot (4^ 2) = 4. Hence 1 is insufficient.
From Statement (2) --13 is less than x is less than 17 implies that x can be 14, 15 or 16 ..hence X IS
POSITIVE...Hence sufficient.
Note : Is x = square root (x^2) is equivalent to stating Is x = mod x
Series of A(n) is such that A(n) = A(n-1) / n. How many elements of the series are bigger than 1/2 ?
(1) A(2) = 5
(2) A(1) - A(2) = 5
From statement (1) --- A(2) = 5 . It is given that A(n) = A(n-1) / n.
Hence A(2) = A(1) / 2
=> A(1) = 10
A(2) = 5
=A(3) = A(2) / 3 = 5 /3
A(4) = A(3) / 4 = 5 /( 3 * 4) = 5 /12
Hence statement (1) alone is sufficient to aswer the question.
From statement (2) --- A(1) - A(2) = 5
=> A(1) - A(1) /2 = 5
=> A(1) = 10
Hence A(2) = 5
A(3) = A(2) / 3 = 5 /3 ...
Hence statement (2) alone is sufficient to answer the question.
(1) A(2) = 5
(2) A(1) - A(2) = 5
From statement (1) --- A(2) = 5 . It is given that A(n) = A(n-1) / n.
Hence A(2) = A(1) / 2
=> A(1) = 10
A(2) = 5
=A(3) = A(2) / 3 = 5 /3
A(4) = A(3) / 4 = 5 /( 3 * 4) = 5 /12
Hence statement (1) alone is sufficient to aswer the question.
From statement (2) --- A(1) - A(2) = 5
=> A(1) - A(1) /2 = 5
=> A(1) = 10
Hence A(2) = 5
A(3) = A(2) / 3 = 5 /3 ...
Hence statement (2) alone is sufficient to answer the question.
n is a positive integer. What is the remainder when n is divided by 6?
(1) n is a multiple of 3.
(2) When n is divided by 2, the remainder is 1.
From statement (1) - n can be 3, 6, 9, 12...
Therefore:
(i) If n is odd, the reminder is 3 on dividing by 6.
(ii) If n is even, the reminder is 0 on dividing by 6.
Hence Insufficient.
From statement (2) - When n is divided by 2, the remainer is 1. Hence this is an odd integer as when an odd number
is divided by 2, the remainder is 1.
n = 2*k + 1
(i) If n = 7 then n/6 = 6*1 + 1
(ii) If n = 9 then n/6 = 6*1 + 3
Hence insufficient.
taking statement (1) and (2) together - From Statement (1) we know that we need to know that whether n is odd or
even in order to be able to conclude.Statement (2) gives us this information.
Hence Sufficient
Hence, C is the answer.
(1) n is a multiple of 3.
(2) When n is divided by 2, the remainder is 1.
From statement (1) - n can be 3, 6, 9, 12...
Therefore:
(i) If n is odd, the reminder is 3 on dividing by 6.
(ii) If n is even, the reminder is 0 on dividing by 6.
Hence Insufficient.
From statement (2) - When n is divided by 2, the remainer is 1. Hence this is an odd integer as when an odd number
is divided by 2, the remainder is 1.
n = 2*k + 1
(i) If n = 7 then n/6 = 6*1 + 1
(ii) If n = 9 then n/6 = 6*1 + 3
Hence insufficient.
taking statement (1) and (2) together - From Statement (1) we know that we need to know that whether n is odd or
even in order to be able to conclude.Statement (2) gives us this information.
Hence Sufficient
Hence, C is the answer.
Triangle A has one side of length x. If (x^8) ^ 1/2 = 81 , what is the perimeter of Triangle A?
1) Triangle A has sides whose lengths are consecutive integers
2) Triangle A is NOT a right triangle
OE - By simplifying the equation given in the question stem, we can solve for x as follows:
(x^8) ^ 1/2 = 81
x^4 = 81
x = 3
Thus, we know that one side of Triangle A has a length of 3.
Statement (1) tells us that Triangle A has sides whose lengths are consecutive integers. Given that one of the
sides of Triangle A has a length of 3, this gives us the following possibilities: (1, 2, 3) OR (2, 3, 4) OR (3, 4,
5).
However, the first possibility is NOT a real triangle, since it does not meet the following condition, which is
true for all triangles: The sum of the lengths of any two sides of a triangle must always be greater than the
length of the third side. Since 1 + 2 is not greater than 3, it is impossible for a triangle to have side lengths
of 1, 2 and 3.
Thus, Statement (1) leaves us with two possibilities. Either Triangle A has side lengths 2, 3, 4 and a perimeter of
9 OR Triangle A has side lengths 3, 4, 5 and a perimeter of 12. Since there are two possible answers, Statement (1)
is not sufficient to answer the question.
Statement (2) tells us that Triangle A is NOT a right triangle. On its own, this is clearly not sufficient to
answer the question, since there are many non-right triangles that can be constructed with a side of length 3.
Taking both statements together, we can determine the perimeter of Triangle A.
From Statement (1) we know that Triangle A must have side lengths of 2, 3, and 4 OR side lengths of 3, 4, and 5.
Statement (2) tells us that Triangle A is not a right triangle; this eliminates the possibility that Triangle A has
side lengths of 3, 4, and 5 since any triangle with these side lengths is a right triangle (this is one of the
common Pythagorean triples). Thus, the only remaining possibility is that Triangle A has side lengths of 2, 3, and
4, which yields a perimeter of 9.
The correct answer is C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
1) Triangle A has sides whose lengths are consecutive integers
2) Triangle A is NOT a right triangle
OE - By simplifying the equation given in the question stem, we can solve for x as follows:
(x^8) ^ 1/2 = 81
x^4 = 81
x = 3
Thus, we know that one side of Triangle A has a length of 3.
Statement (1) tells us that Triangle A has sides whose lengths are consecutive integers. Given that one of the
sides of Triangle A has a length of 3, this gives us the following possibilities: (1, 2, 3) OR (2, 3, 4) OR (3, 4,
5).
However, the first possibility is NOT a real triangle, since it does not meet the following condition, which is
true for all triangles: The sum of the lengths of any two sides of a triangle must always be greater than the
length of the third side. Since 1 + 2 is not greater than 3, it is impossible for a triangle to have side lengths
of 1, 2 and 3.
Thus, Statement (1) leaves us with two possibilities. Either Triangle A has side lengths 2, 3, 4 and a perimeter of
9 OR Triangle A has side lengths 3, 4, 5 and a perimeter of 12. Since there are two possible answers, Statement (1)
is not sufficient to answer the question.
Statement (2) tells us that Triangle A is NOT a right triangle. On its own, this is clearly not sufficient to
answer the question, since there are many non-right triangles that can be constructed with a side of length 3.
Taking both statements together, we can determine the perimeter of Triangle A.
From Statement (1) we know that Triangle A must have side lengths of 2, 3, and 4 OR side lengths of 3, 4, and 5.
Statement (2) tells us that Triangle A is not a right triangle; this eliminates the possibility that Triangle A has
side lengths of 3, 4, and 5 since any triangle with these side lengths is a right triangle (this is one of the
common Pythagorean triples). Thus, the only remaining possibility is that Triangle A has side lengths of 2, 3, and
4, which yields a perimeter of 9.
The correct answer is C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
If x is a positive integer, then how many factors does x have ?
(1) x is divisible by one more positive integer than 3^4 is.
(2) x is the product of three different prime numbers
From statement (1) - x has 5 factors 1, 3, 9, 27, 81. Adding one more factor from statement (1) makes total number
of factors to be six.
Thus x has 6 factors.
Hence sufficient
From statement (2) - x = abc where a, b, c are prime numbers. Thus in all x has 8 factors - 1, a, b, c, ab, bc, ac,
abc.
Thus x has 8 factors.
Hence sufficient
Note: Here each condition gives a different answer but still satifies the sufficient conditions.
(1) x is divisible by one more positive integer than 3^4 is.
(2) x is the product of three different prime numbers
From statement (1) - x has 5 factors 1, 3, 9, 27, 81. Adding one more factor from statement (1) makes total number
of factors to be six.
Thus x has 6 factors.
Hence sufficient
From statement (2) - x = abc where a, b, c are prime numbers. Thus in all x has 8 factors - 1, a, b, c, ab, bc, ac,
abc.
Thus x has 8 factors.
Hence sufficient
Note: Here each condition gives a different answer but still satifies the sufficient conditions.
What is the 999th term of the series S ?
(1) The first 4 four terms of S are (1 + 1)² , (2 + 1)² , (3 + 1)² , and (4 + 1)².
(2) For every x, the xth term of S is (x + 1)².
From statement (1) - It is given that the 1st 4 four terms of S are (1 + 1)² , (2 + 1)² , (3 + 1)² , and (4 + 1)².
We only know the first four terms. We cannot assume that terms following the first 4 terms will be like -- nth term
= (n+1)^2.
There can be a sequence that follows a different rule for any term in it
From statement (2) - It is given that for every x, the xth term of S is (x + 1)². Hence sufficient as it implies
"for any number x that refers to the term in S."
(1) The first 4 four terms of S are (1 + 1)² , (2 + 1)² , (3 + 1)² , and (4 + 1)².
(2) For every x, the xth term of S is (x + 1)².
From statement (1) - It is given that the 1st 4 four terms of S are (1 + 1)² , (2 + 1)² , (3 + 1)² , and (4 + 1)².
We only know the first four terms. We cannot assume that terms following the first 4 terms will be like -- nth term
= (n+1)^2.
There can be a sequence that follows a different rule for any term in it
From statement (2) - It is given that for every x, the xth term of S is (x + 1)². Hence sufficient as it implies
"for any number x that refers to the term in S."
Is 5^k less than 1,000?
(1) 5^(k-1) greater than 3,000
(2) 5^(k-1) = 5^k - 500
From statement (1) - 5^(k-1) is greater than 3000
=> 5^k/5 is greater than 3000
=> 5^k is greater than 15000
Hence sufficient
From statement (2) - 5^(k-1) = 5^k - 500
=> 5^(k-1) = 5^k - 500
=> 5^k - 5^(k-1) = 500
=> 5^k(1- 5^-1) = 500
=> 5^k(4/5) = 500
=> 5^k = 2500/4 = 625
Therefore 5^k is less than 1000
Hence sufficient
(1) 5^(k-1) greater than 3,000
(2) 5^(k-1) = 5^k - 500
From statement (1) - 5^(k-1) is greater than 3000
=> 5^k/5 is greater than 3000
=> 5^k is greater than 15000
Hence sufficient
From statement (2) - 5^(k-1) = 5^k - 500
=> 5^(k-1) = 5^k - 500
=> 5^k - 5^(k-1) = 500
=> 5^k(1- 5^-1) = 500
=> 5^k(4/5) = 500
=> 5^k = 2500/4 = 625
Therefore 5^k is less than 1000
Hence sufficient
Edwin is planning to drive from Boston to New Orleans. By what percent would his travel time be reduced if Edwin
decides to split the driving time equally with his friend George, instead of making the trip alone?
(1) The driving distance from Boston to New Orleans is 1500 miles.
(2) George’s driving speed is 1.5 times Edwin’s driving speed.
OE - The question asks for the percent decrease in Edwin’s travel time. To determine this, we need to be able to
find the ratio between, T1 (the travel time if Edwin drives alone) and T2 (the travel time if Edwin and George
drive together). Note that we do NOT need to determine specific values for T1 and T2; we only need to find the
ratio between them.
Percentage change is defined as follows: Difference/Original = (T1 - T2)/ T1 = 1 - (T2/ T1)
Ultimately, we can solve the percentage change equation above by simply determining the value of T2 /T1
Using the formula Rate × Time = Distance, we can write equations for each of the 2 possible trips
T1 = Travel time if Edwin drives alone
T2 = Travel time if Edwin and George drive together
E = Edwin’s Rate
G = George’s Rate
D = Distance of the trip
If Edwin travels alone: ET1 = D
If Edwin and George travel together: .5(E + G)T2 = D
(Since Edwin and George split the driving equally, the rate for the trip is equal to the average of Edwin and
George’s individual rates).
Since both trips cover the same distance (D), we can combine the 2 equations as follows:
ET1 = .5(E + G)T2
Then, we can isolate the ratio of the times (T2/T1) as follows:
E/ .5(E + G) = T2/ T1
Now we look at the statements to see if they can help us to solve for the ratio of the times.
Statement (1) gives us a value for D, the distance, which does not help us since D is not a variable in the ratio
equation above.
Statement (2) tells us that George’s rate is 1.5 times Edwin’s rate. Thus, G = 1.5E. We can substitute this
information into the ratio equation above:
E/ .5(E + G) = T2/ T1 ---> E/ .5(E + 1.5E) = T2/ T1 ---> E/ .5E + .75E = T2/T1
---> E/ 1.25E = T2/ T1 ---> 1/ 1.25 = T2/ T1 ---> .8 = T2/ T1
Thus, using this ratio we can see that Edwin’s travel time for the trip will be reduced as follows:
1 - (T2/T1) = 1 - .8 = .2 ---> 20%
Statement (2) alone is sufficient to answer the question.
The correct answer is B.
decides to split the driving time equally with his friend George, instead of making the trip alone?
(1) The driving distance from Boston to New Orleans is 1500 miles.
(2) George’s driving speed is 1.5 times Edwin’s driving speed.
OE - The question asks for the percent decrease in Edwin’s travel time. To determine this, we need to be able to
find the ratio between, T1 (the travel time if Edwin drives alone) and T2 (the travel time if Edwin and George
drive together). Note that we do NOT need to determine specific values for T1 and T2; we only need to find the
ratio between them.
Percentage change is defined as follows: Difference/Original = (T1 - T2)/ T1 = 1 - (T2/ T1)
Ultimately, we can solve the percentage change equation above by simply determining the value of T2 /T1
Using the formula Rate × Time = Distance, we can write equations for each of the 2 possible trips
T1 = Travel time if Edwin drives alone
T2 = Travel time if Edwin and George drive together
E = Edwin’s Rate
G = George’s Rate
D = Distance of the trip
If Edwin travels alone: ET1 = D
If Edwin and George travel together: .5(E + G)T2 = D
(Since Edwin and George split the driving equally, the rate for the trip is equal to the average of Edwin and
George’s individual rates).
Since both trips cover the same distance (D), we can combine the 2 equations as follows:
ET1 = .5(E + G)T2
Then, we can isolate the ratio of the times (T2/T1) as follows:
E/ .5(E + G) = T2/ T1
Now we look at the statements to see if they can help us to solve for the ratio of the times.
Statement (1) gives us a value for D, the distance, which does not help us since D is not a variable in the ratio
equation above.
Statement (2) tells us that George’s rate is 1.5 times Edwin’s rate. Thus, G = 1.5E. We can substitute this
information into the ratio equation above:
E/ .5(E + G) = T2/ T1 ---> E/ .5(E + 1.5E) = T2/ T1 ---> E/ .5E + .75E = T2/T1
---> E/ 1.25E = T2/ T1 ---> 1/ 1.25 = T2/ T1 ---> .8 = T2/ T1
Thus, using this ratio we can see that Edwin’s travel time for the trip will be reduced as follows:
1 - (T2/T1) = 1 - .8 = .2 ---> 20%
Statement (2) alone is sufficient to answer the question.
The correct answer is B.
The area of a parallelogram is 100. What is the perimeter of the parallelogram ?
1) the base of the parallelogram is 10
2) one of the angles of the parallelogram is 45 degree
From (1) - For a parallelogram, Area = Base*Height => Height = 10 - There are infinite ways to draw a parallelogram
with 100 as area, as long as the height is 10 units.( parallelograms with varying slants from 1 to 179 degrees) --
insufficent
From (2) - one of the angles is 45 degrees - hence the opposite angle is 45 degrees too, and the two remaining
angles wil be 135 degrees each. But this doesn't tell us the measure of the sides to determine the perimeter -
insufficient
From(1) and (2) together
From (2) we know that there is only one of such parallelograms that has an angle 45 to the base.
Base = 10. One angle = 45 degrees
In Parallelogram opposite angles are equal ..
Hence there are 2*45 degrees
Sum of all angles = 360
Thus 2*45 + 2 x = 360 which gives each other angle as 135 degrees. Hence C
1) the base of the parallelogram is 10
2) one of the angles of the parallelogram is 45 degree
From (1) - For a parallelogram, Area = Base*Height => Height = 10 - There are infinite ways to draw a parallelogram
with 100 as area, as long as the height is 10 units.( parallelograms with varying slants from 1 to 179 degrees) --
insufficent
From (2) - one of the angles is 45 degrees - hence the opposite angle is 45 degrees too, and the two remaining
angles wil be 135 degrees each. But this doesn't tell us the measure of the sides to determine the perimeter -
insufficient
From(1) and (2) together
From (2) we know that there is only one of such parallelograms that has an angle 45 to the base.
Base = 10. One angle = 45 degrees
In Parallelogram opposite angles are equal ..
Hence there are 2*45 degrees
Sum of all angles = 360
Thus 2*45 + 2 x = 360 which gives each other angle as 135 degrees. Hence C
Is the three-digit number n less than 550?
1) The product of the digits in n is 30
2) The sum of the digits in n is 10
From statement (1) : the factors of 30 are 1,2,3,5,6,10,15,30
Now because the number must be digits (single number) we do not need to consider 10,15 and 50
Now if the hundreds digit of the 3-digit number = any digit among the 1,2,3 digits ---> answer to the question is
clearly Yes.
But if the hundreds digit = 5 or 6 ---> answer to the question is No.
Hence (1) alone is insufficient.
From statement (2) : There are different combinations where the sum of digits can be equal to 10. e.g 541 and 145.
Hence (2) alone is insufficient.
Combining statements (1) and (2) we get :
If the hundreds digit= 5 , the second digit can only be 1,2,3 because if the digit is equal to 6 it violates
statement (2) Hence in this case answer to the question is Yes.
If hundreds digit = 6 then the number n must contain 5 so as to satisfy the first statement but then it will
violate statement (2) as the total of digits of n will exceed 10
Thus no 3-digit number exists with the hundreds digit equal to 6 satisfying both the statements together.
Hence the number n will always be less than 550 as it will be the combination of 1, 2, 3 and 5
1) The product of the digits in n is 30
2) The sum of the digits in n is 10
From statement (1) : the factors of 30 are 1,2,3,5,6,10,15,30
Now because the number must be digits (single number) we do not need to consider 10,15 and 50
Now if the hundreds digit of the 3-digit number = any digit among the 1,2,3 digits ---> answer to the question is
clearly Yes.
But if the hundreds digit = 5 or 6 ---> answer to the question is No.
Hence (1) alone is insufficient.
From statement (2) : There are different combinations where the sum of digits can be equal to 10. e.g 541 and 145.
Hence (2) alone is insufficient.
Combining statements (1) and (2) we get :
If the hundreds digit= 5 , the second digit can only be 1,2,3 because if the digit is equal to 6 it violates
statement (2) Hence in this case answer to the question is Yes.
If hundreds digit = 6 then the number n must contain 5 so as to satisfy the first statement but then it will
violate statement (2) as the total of digits of n will exceed 10
Thus no 3-digit number exists with the hundreds digit equal to 6 satisfying both the statements together.
Hence the number n will always be less than 550 as it will be the combination of 1, 2, 3 and 5
If x and y are integers, does IxI = y ?
(1) (y^2 - x^2) = 0
(2) xy/(x+y) = 0
Statement (1) insufficient -- y can take any value i.e can be +ve or -ve. We cannot assume y to be +ve. Therefore y
may or may not be equal to x
Statement (2) sufficient -- xy/(x+y) = 0 => xy = 0 => x = 0 or y = 0
In case x = 0, then y cannot be equal to 0. Hence y cannot be equal to x.
In case y = 0, then x cannot be equal to 0. Hence y cannot be equal to x.
Therefore sufficient.
Hence B
(1) (y^2 - x^2) = 0
(2) xy/(x+y) = 0
Statement (1) insufficient -- y can take any value i.e can be +ve or -ve. We cannot assume y to be +ve. Therefore y
may or may not be equal to x
Statement (2) sufficient -- xy/(x+y) = 0 => xy = 0 => x = 0 or y = 0
In case x = 0, then y cannot be equal to 0. Hence y cannot be equal to x.
In case y = 0, then x cannot be equal to 0. Hence y cannot be equal to x.
Therefore sufficient.
Hence B
If q is a multiple of prime numbers, is q a multiple of r?
1) r is less than 4
2) q = 18
From Statement (1) -- q can be positive and r can be negative.
r can also be a real number but not necessarily an integer.
Hence insufficient.
From Statement (2) -- q = 18.
But r can be negative or can be positive.
r can also be a real number not an integer
Hence insufficient
Taking both statements (1) and (2) together -- Again r can be positive or negative. And again r can also be a real
number.
Hence insufficient
1) r is less than 4
2) q = 18
From Statement (1) -- q can be positive and r can be negative.
r can also be a real number but not necessarily an integer.
Hence insufficient.
From Statement (2) -- q = 18.
But r can be negative or can be positive.
r can also be a real number not an integer
Hence insufficient
Taking both statements (1) and (2) together -- Again r can be positive or negative. And again r can also be a real
number.
Hence insufficient
What is the value of x?
1) (-x)^3 = -x^3
2) (-x)^2 = -x^2
From statement (1):
if x = 0 both sides are equal
if x = 1 both sides are again equal {(-1)^3 = -1 & -1^3 = 1}
=> x = 0 or x = 1
Hence insufficient
From statement (2):
x can only be zero because the square of a number other than zero cannot be negative
{(-1)^2 = 1 which is not equal to -(1)^2)}
=> from above it is sufficient to say that x = 0
Hence sufficient
1) (-x)^3 = -x^3
2) (-x)^2 = -x^2
From statement (1):
if x = 0 both sides are equal
if x = 1 both sides are again equal {(-1)^3 = -1 & -1^3 = 1}
=> x = 0 or x = 1
Hence insufficient
From statement (2):
x can only be zero because the square of a number other than zero cannot be negative
{(-1)^2 = 1 which is not equal to -(1)^2)}
=> from above it is sufficient to say that x = 0
Hence sufficient
Is |x - 1| less than 1 ?
1). (x - 1)^2 less than and equal to 1
2). x^2 - 1 greater than 0
|x-1| less than 1 is only true when 0 less than x less than 1
From statement (1): (x-1)^2<=1
True when 0<=x<=2
If x=0.5, then |x-1| less than 1 is true
If x=2, then |x-1| less than 1 is not true
Hence insufficient
From statement (2): x^2>1 means x>1 and x<-1
True when x=1.5, but not when x=3
Hence insufficient
Statement (1) and (2) together: 1 is less than x is less than and equal to 2
Taking x=1.5 and x=2
Hence insufficient
1). (x - 1)^2 less than and equal to 1
2). x^2 - 1 greater than 0
|x-1| less than 1 is only true when 0 less than x less than 1
From statement (1): (x-1)^2<=1
True when 0<=x<=2
If x=0.5, then |x-1| less than 1 is true
If x=2, then |x-1| less than 1 is not true
Hence insufficient
From statement (2): x^2>1 means x>1 and x<-1
True when x=1.5, but not when x=3
Hence insufficient
Statement (1) and (2) together: 1 is less than x is less than and equal to 2
Taking x=1.5 and x=2
Hence insufficient
In the xy-plane, at what two points does the graph of y=(x+a)(x+b) intersect the x-axis?
1). a+b= -1
2). The graph intersects the y-axis at (0,-6)
From Statement (1) -- a+b = -1...no information about a and b ...hence insufficient
From Statement (2) -- If x = 0 the y = -6 thus ab = 6....insufficient
Taking statements (1) and (2) together: (x+a)*(x+b)=0
x^2+(a+b)x+ab=0
Hence x=-3, x=2
Thus the answer C
1). a+b= -1
2). The graph intersects the y-axis at (0,-6)
From Statement (1) -- a+b = -1...no information about a and b ...hence insufficient
From Statement (2) -- If x = 0 the y = -6 thus ab = 6....insufficient
Taking statements (1) and (2) together: (x+a)*(x+b)=0
x^2+(a+b)x+ab=0
Hence x=-3, x=2
Thus the answer C
A student worked 20 days. For each of the amount shown (see attached table) in the first row of the table, second row gives the number of days the student earned that amount. Median amount of money earned per day for 20 days is?
A) 96
B) 84
C) 80
D) 70
E) 48
Median day = 20+1)/2 = 10.5 th -- money earned was 84 = Average value of 10th and 11th day in the sequence = Median amount of money Average value of 10th day = 84 Average value of 11th day = 84 Average value of 10th and 11th day = 84 ans
Strategies for Averages and Statistics
1. Mean Average = total of quantities / number of quantities
2. The median is the "middle" number in a group (when arranged in ascending or descending order) consisting of an odd number of numbers, and the average of the two middle numbers if there are an even number of numbers
3. For a set of consecutive integers, the median is the the average of the first and the last integer
4. Mode is the most frequently recurring number/numbers among the given set of numbers. It can be more than one
5. Range is the difference between the largest number and smallest number is a set
6. Calculation of Standard Deviation (SD):
1. Find the mean, \scriptstyle\overline{x}, of the values.
2. For each value xi calculate its deviation (\scriptstyle x_i - \overline{x}) from the mean.
3. Calculate the squares of these deviations.
4. Find the mean of the squared deviations. This quantity is the variance σ2.
5. Take the square root of the variance.
7. Variance is the square of the standard deviation
8. SD does not change when the same constant is added or subtracted to all the members of the set
9. If mean = maximum value it means that all values are equal and SD is 0
10. A set of numbers with range of zero means that all of the numbers are the same, hence the dispersion of the numbers from its mean is zero
11. For data with approximately the same mean, the greater the spread, the greater the SD.
12. SD is the square root of the average of the sum of square of the variation from the mean
13. The more uneven members are dispersed around their arithmetic average, the more their SD
14. You only need to know the difference between values and total number of values to compute SD
15. If we know all the numbers of the list, there is a definite SD, regardless of what it is, we can compute it and get an answer – this is helpful for DS questions
16. If the range is 0, then the SD must also be 0, because there is no variance
17. The SD of any list is not dependent on the average, but on the deviation of the numbers from the average. So just by knowing that two lists having different averages doesn't say anything about their standard deviation - different averages can have the same SD
18. The sum of the deviations of the elements from the mean must be 0
19.Closer the more values to the MEAN, lower the SD
20. If Range or SD of a list is 0, then the list will contain all identical elements
21. Standard Deviation is also useful when comparing the spread of two separate data sets that have approximately the same mean. The data set with the smaller Standard Deviation has a narrower spread of measurements around the mean and therefore usually has comparatively fewer high or low values.
In general, the more widely spread the values are, the larger the Standard Deviation is.
22. If you multiply all terms by x then SD =x times old SD and mean = x times old mean
23. For comparing the SD for two sets any information about mean ,median,mode and range are insufficient unless you can determine the individual terms from the given data
24. Symmetric about the mean means that the shape of the distribution on the right and left side of the curve are mirror-images of each other
25. For a given set of consecutive even numbers.. mean = median
26. When you have a set of consecutive numbers (integers, evens, odds, multiples), the mean is equal to the median
if you have the averages of all the FRACTIONS or PERCENTAGES of a group, then you'll be able to calculate the overall average of the group. This is a worthwhile fact to memorize for the data sufficiency problems.
- Note ----- Make sure you know that, when ALL numbers in a set are multiplied or divided by some number,** the mean and standard deviation are multiplied/divided by the same number
**This includes increasing or decreasing all the numbers in the set by some percentage (which can be accomplished by multiplication: e.g., 30% increase = multiplication by 1.3).
2. The median is the "middle" number in a group (when arranged in ascending or descending order) consisting of an odd number of numbers, and the average of the two middle numbers if there are an even number of numbers
3. For a set of consecutive integers, the median is the the average of the first and the last integer
4. Mode is the most frequently recurring number/numbers among the given set of numbers. It can be more than one
5. Range is the difference between the largest number and smallest number is a set
6. Calculation of Standard Deviation (SD):
1. Find the mean, \scriptstyle\overline{x}, of the values.
2. For each value xi calculate its deviation (\scriptstyle x_i - \overline{x}) from the mean.
3. Calculate the squares of these deviations.
4. Find the mean of the squared deviations. This quantity is the variance σ2.
5. Take the square root of the variance.
7. Variance is the square of the standard deviation
8. SD does not change when the same constant is added or subtracted to all the members of the set
9. If mean = maximum value it means that all values are equal and SD is 0
10. A set of numbers with range of zero means that all of the numbers are the same, hence the dispersion of the numbers from its mean is zero
11. For data with approximately the same mean, the greater the spread, the greater the SD.
12. SD is the square root of the average of the sum of square of the variation from the mean
13. The more uneven members are dispersed around their arithmetic average, the more their SD
14. You only need to know the difference between values and total number of values to compute SD
15. If we know all the numbers of the list, there is a definite SD, regardless of what it is, we can compute it and get an answer – this is helpful for DS questions
16. If the range is 0, then the SD must also be 0, because there is no variance
17. The SD of any list is not dependent on the average, but on the deviation of the numbers from the average. So just by knowing that two lists having different averages doesn't say anything about their standard deviation - different averages can have the same SD
18. The sum of the deviations of the elements from the mean must be 0
19.Closer the more values to the MEAN, lower the SD
20. If Range or SD of a list is 0, then the list will contain all identical elements
21. Standard Deviation is also useful when comparing the spread of two separate data sets that have approximately the same mean. The data set with the smaller Standard Deviation has a narrower spread of measurements around the mean and therefore usually has comparatively fewer high or low values.
In general, the more widely spread the values are, the larger the Standard Deviation is.
22. If you multiply all terms by x then SD =x times old SD and mean = x times old mean
23. For comparing the SD for two sets any information about mean ,median,mode and range are insufficient unless you can determine the individual terms from the given data
24. Symmetric about the mean means that the shape of the distribution on the right and left side of the curve are mirror-images of each other
25. For a given set of consecutive even numbers.. mean = median
26. When you have a set of consecutive numbers (integers, evens, odds, multiples), the mean is equal to the median
if you have the averages of all the FRACTIONS or PERCENTAGES of a group, then you'll be able to calculate the overall average of the group. This is a worthwhile fact to memorize for the data sufficiency problems.
- Note ----- Make sure you know that, when ALL numbers in a set are multiplied or divided by some number,** the mean and standard deviation are multiplied/divided by the same number
**This includes increasing or decreasing all the numbers in the set by some percentage (which can be accomplished by multiplication: e.g., 30% increase = multiplication by 1.3).
- list of consecutive integers: you can just take the average of the first and last numbers, and that's the same as the average of the entire list.
Strategies for Set Theory
Formulas for three-component set problems:u = union
n = intersection
1. For 3 sets A, B, and C: P(AuBuC) = P(A) + P(B) + P(C) – P(AnB) – P(AnC) – P(BnC) + P(AnBnC)
2. No of persons in exactly one set:
P(A) + P(B) + P(C) – 2P(AnB) – 2P(AnC) – 2P(BnC) + 3P(AnBnC)
3. No of persons in exactly two of the sets: P(AnB) + P(AnC) + P(BnC) – 3P(AnBnC)
4. No of persons in exactly three of the sets: P(AnBnC)
5. No of persons in two or more sets: P(AnB) + P(AnC) + P(BnC) – 2P(AnBnC)
6. No of persons in atleast one set:
P(A) + P(B) + P(C) - P(AnB) - P(AnC) - P(BnC) + 2 P(AnBnC)
1. For three sets A, B, and C, P(AuBuC): (A+B+C+X+Y+Z+O)
2. Number of people in exactly one set: ( A+B+C)
3. Number of people in exactly two of the sets: (X+Y+Z)
4. Number of people in exactly three of the sets: O
5. Number of people in two or more sets: ( X+Y+Z+O)
6. Number of people only in set A: A
7. P(A): A+X+Y+O
8. P( AnB): X+O
Notes :
* In Datasufficiency problems, do not assume that overlap between sets (using double matrix problems) does not exists!!
Larry, Michael, and Doug have five donuts to share. If any one of the men can be given any whole number of donuts from 0 to 5, in how many different ways can the donuts be distributed?
(A) 21
(B) 42
(C) 120
(D) 504
(E) 5040
For n = 6 it will be 28. For n = 7 it will be 36.....
However here is the explanation based on the counting method...
1) 1 person gets all 5 donuts -
Possibility: 3.
2). 2 persons get all 5 donuts -
The one without donuts - possibility - 3.
The other 2 persons have 4 ways.
Hence in total -- 3*4 = 12.
3). All have donuts -
The way to divide can only be - 1, 2, 2; 1, 3, 1
But this arrangement can get interchanged midst 3 persons, hence it becomes --
(3!)/ (2) + (3!)/ (2) = 6.
(3!)/ (2) is needed because donuts are all same without difference.
3! is counting the same arrangement twice.
Hence the answer - 3 + 12 + 6 = 21.
(A) 21
(B) 42
(C) 120
(D) 504
(E) 5040
For n = 6 it will be 28. For n = 7 it will be 36.....
However here is the explanation based on the counting method...
1) 1 person gets all 5 donuts -
Possibility: 3.
2). 2 persons get all 5 donuts -
The one without donuts - possibility - 3.
The other 2 persons have 4 ways.
Hence in total -- 3*4 = 12.
3). All have donuts -
The way to divide can only be - 1, 2, 2; 1, 3, 1
But this arrangement can get interchanged midst 3 persons, hence it becomes --
(3!)/ (2) + (3!)/ (2) = 6.
(3!)/ (2) is needed because donuts are all same without difference.
3! is counting the same arrangement twice.
Hence the answer - 3 + 12 + 6 = 21.
Six mobsters have arrived at the theater for the premiere of the film “Goodbuddies.” One of the mobsters, Frankie, is an informer, and he's afraid that another member of his crew, Joey, is on to him. Frankie, wanting to keep Joey in his sights, insists upon standing behind Joey in line at the concession stand. How many ways can the six arrange themselves in line such that Frankie’s requirement is satisfied?
(A) 6
(B) 24
(C) 120
(D) 360
(e) 720
Frankie wants to keep Joe in his sights, therefore Joe will always be ahead of F in the queue. (take care this doesnot implies that Joe and Frankie will be together)
1. Frankie is in last position in the queue, then Joe can be in any position from 1 to 5 = 5!
2. Frankie is in the 5th position in the queue then Joe can be in any of the positions 1 to 4 (positions ahead of Frankie) i.e 4 ways and rest of mobsters will be positioned in rest 4 places (4!) ways. = 4*4!
3. Frankie is in the 4th position then Joe can be take any place between 1 to 3 (positions ahead of Frankie) 3 ways and rest of the mobsters will be positioned in rest of 4 places i.e 4!ways. = 3*4!
4. Frankie is in the 3rd position then Joe can be in 1 or 2 places (i.e positions ahead of Frankie) 2 ways and rest of mobsters will be positioned in rest of the 4 places (4!) ways. = 2*4!
5. Frankie is in placed in the 2nd position then Joe can only be in 1st position i.e only position ahead him i.e 1 way and rest of mobsters will be placed in the rest of the 4 places (4!) ways. = 1*4!
Thus total no of ways = 5! + ( 4 * 4! ) + ( 3 * 4! ) + ( 2 * 4! ) + ( 1 * 4! ) = 360
(A) 6
(B) 24
(C) 120
(D) 360
(e) 720
Frankie wants to keep Joe in his sights, therefore Joe will always be ahead of F in the queue. (take care this doesnot implies that Joe and Frankie will be together)
1. Frankie is in last position in the queue, then Joe can be in any position from 1 to 5 = 5!
2. Frankie is in the 5th position in the queue then Joe can be in any of the positions 1 to 4 (positions ahead of Frankie) i.e 4 ways and rest of mobsters will be positioned in rest 4 places (4!) ways. = 4*4!
3. Frankie is in the 4th position then Joe can be take any place between 1 to 3 (positions ahead of Frankie) 3 ways and rest of the mobsters will be positioned in rest of 4 places i.e 4!ways. = 3*4!
4. Frankie is in the 3rd position then Joe can be in 1 or 2 places (i.e positions ahead of Frankie) 2 ways and rest of mobsters will be positioned in rest of the 4 places (4!) ways. = 2*4!
5. Frankie is in placed in the 2nd position then Joe can only be in 1st position i.e only position ahead him i.e 1 way and rest of mobsters will be placed in the rest of the 4 places (4!) ways. = 1*4!
Thus total no of ways = 5! + ( 4 * 4! ) + ( 3 * 4! ) + ( 2 * 4! ) + ( 1 * 4! ) = 360
How many 5 digit numbers can be created if the following terms apply: the leftmost digit is even, the second is odd, the third is a non even prime and the fourth and fifth are two random digits not used before in the number?
a) 2520
b) 3150
c) 3360
d) 6000
e) 7500
Answer: Given OA - A is incorrect ... ans is 2688 which is not an option in the choices...
When 2nd and 3rd digit gets repeated:
The first digit will be a non zero even (2, 4, 6, 8) = 4 ways
3rd digit is a non even prime = (3, 5, 7) = 3 ways
2nd digit is a REPEAT of that prime: 1 way
the fourth digit has not been used: 8 ways
the fifth digit has not been used: 7 ways
Hence 4*3*8*7 = 672 ways
Now, the non repeating case:
1st digit will be a non zero even (2, 4, 6, 8) = 4 ways
3rd digit (3, 5, 7) = 3 ways
2nd (no repeat and odd) = 4 ways
4th digit = 7
5th digit = 6
Hence 4*3*4*7*6 = 2016
Total number of ways = 2016 + 672 = 2688 ways
Note:
1). 1 is neither a prime nor a composite number.
2). 2 is the only even prime number.
a) 2520
b) 3150
c) 3360
d) 6000
e) 7500
Answer: Given OA - A is incorrect ... ans is 2688 which is not an option in the choices...
When 2nd and 3rd digit gets repeated:
The first digit will be a non zero even (2, 4, 6, 8) = 4 ways
3rd digit is a non even prime = (3, 5, 7) = 3 ways
2nd digit is a REPEAT of that prime: 1 way
the fourth digit has not been used: 8 ways
the fifth digit has not been used: 7 ways
Hence 4*3*8*7 = 672 ways
Now, the non repeating case:
1st digit will be a non zero even (2, 4, 6, 8) = 4 ways
3rd digit (3, 5, 7) = 3 ways
2nd (no repeat and odd) = 4 ways
4th digit = 7
5th digit = 6
Hence 4*3*4*7*6 = 2016
Total number of ways = 2016 + 672 = 2688 ways
Note:
1). 1 is neither a prime nor a composite number.
2). 2 is the only even prime number.
A certain stock exchange designates each stock with a one- , two-,or three-letter code ,where each letter is selected from the 26 letters of the alphabet. If the letter may be repeated and if the same letters used in a different order constitute a different code, how many different stocks is it possible to uniquely designate with these codes?
A) 2951
B) 8125
C) 15600
D) 15302
E) 18278
Total ways to pick a 3 letter code = 26*26*26 = 26^3
Total ways to pick a 2 letter code = 26*26 = 26^2
Total ways to pick a 1 letter code = 26
Thus total stocks: 26^3 + 26^2 + 26 = 26(26^2 + 26 + 1) = 18278
A) 2951
B) 8125
C) 15600
D) 15302
E) 18278
Total ways to pick a 3 letter code = 26*26*26 = 26^3
Total ways to pick a 2 letter code = 26*26 = 26^2
Total ways to pick a 1 letter code = 26
Thus total stocks: 26^3 + 26^2 + 26 = 26(26^2 + 26 + 1) = 18278
Which of the following fractions has a decimal equivalent that is a terminating decimal?
A. 10/189
B. 15/196
C. 16/225
D. 25/144
E. 39/ 128
RULE: If denominator of a fraction has just the prime factors of 2 or 5 or both it is terminating otherwise not
128 = 2*2*2*2*2*2*2 => has only factors of 2..hence the ans E.
A. 10/189
B. 15/196
C. 16/225
D. 25/144
E. 39/ 128
RULE: If denominator of a fraction has just the prime factors of 2 or 5 or both it is terminating otherwise not
128 = 2*2*2*2*2*2*2 => has only factors of 2..hence the ans E.
If set S consist of the numbers 1, 5, -2, 8, and n, is 0 less than n less than 7 ?
1). the median of the numbers in S is less than 5.
2). the median of the numbers in S is greater than 1
From statement (1): Median will be less than 5 only if n is located below 5
Thus the median will either be 1 if n less than 1 or n if 1 less than n less than 5
Hence in both cases n is less than 5 but it can also be n less than 0 ....insufficient
From statement (2): Median will be greater than 1 only if n is located above 1
Thus median will either be 5 if n greater than 5 or n if 1 less than n less than 5
Hence in both cases n greater than 1 but it can also be n greater than 7 ....insufficient
Taking statements (1) and (2) together: 1 less than n less than 5 which lies within the given interval 0 less than n less than 7 ...thus possible values for n are be 2, 3, or 4
Hence the answer C
1). the median of the numbers in S is less than 5.
2). the median of the numbers in S is greater than 1
From statement (1): Median will be less than 5 only if n is located below 5
Thus the median will either be 1 if n less than 1 or n if 1 less than n less than 5
Hence in both cases n is less than 5 but it can also be n less than 0 ....insufficient
From statement (2): Median will be greater than 1 only if n is located above 1
Thus median will either be 5 if n greater than 5 or n if 1 less than n less than 5
Hence in both cases n greater than 1 but it can also be n greater than 7 ....insufficient
Taking statements (1) and (2) together: 1 less than n less than 5 which lies within the given interval 0 less than n less than 7 ...thus possible values for n are be 2, 3, or 4
Hence the answer C
At least 100 students at a certain high school study Japanese. If 4 percent of the students who study French also study Japanese, do more students at the school study French than Japanese?
1). 16 students at the school study both French and Japanese.
2). 10 percent of the students at the school who study Japanese also study French.
From statement (1):16 students study both French and Japanese, so 16/0.04=400 students study French. But we don't know how many the total number of students are so the number Japanese student can be at least 100 or more than 400....insufficient
From statement (2): 10% Japanese studying students = 4% French studying students
10/100 study Japanese = 4/100 study French
25 study Japanese = 10 study French
Hence study French > study Japanese ... thus more students at the school study French than study Japanese....sufficient
1). 16 students at the school study both French and Japanese.
2). 10 percent of the students at the school who study Japanese also study French.
From statement (1):16 students study both French and Japanese, so 16/0.04=400 students study French. But we don't know how many the total number of students are so the number Japanese student can be at least 100 or more than 400....insufficient
From statement (2): 10% Japanese studying students = 4% French studying students
10/100 study Japanese = 4/100 study French
25 study Japanese = 10 study French
Hence study French > study Japanese ... thus more students at the school study French than study Japanese....sufficient
If q is a integer, is q^4 a multiple of 64?
(1) q^4 is not a multiple of 128.
(2) q^2 has 27 factors, 7 of which are less than or equal to 10
OE:
From statement (1): Given q is an integer, thus q can be written as product of distinct prime factors, where the power of 2 must be a whole number(a non-negative integer). This implies that the power of 2 in the prime factorization of q^4 must be a multiple of 4. As 2^7 is not a factor of q^4, the highest power of 2 that could be a factor of q^4 is 2^4.
Hence q^4 is not a multiple of 64=2^6......sufficient
From statement (2): Given that q^2 has 27 factors. Thus q^2 could be of the form
(i) a^26, or
(ii)b^2*c^8 or
(iii) a^2*b^2*c^2 where a b and c are distinct prime numbers.
Because q^2 has 7 factors less than 11, (i) is impossible.
As for (ii) 2^8*3^2 has exactly 7 factors under 11 (1,2,3,4,6,8,9), in which case q^4 would be a multiple of 64. 3^8*2*2 is not a possibility for q^2 (it only has 6 factors under 1: 1,2,3,4,6,9)
Regarding (iii) if q^2=2^2*3^2*5^2, q^2 would have 8 factors under 11- 1,2,3,4,5,6,9,10 and q^2=2^2*3^2*7^2 would have 7 factors under 11: 1,2,3,4,6,7,9) In this case, q^4 would not be a multiple of 64. Thus (2) is insufficient
hence the answer A
(1) q^4 is not a multiple of 128.
(2) q^2 has 27 factors, 7 of which are less than or equal to 10
OE:
From statement (1): Given q is an integer, thus q can be written as product of distinct prime factors, where the power of 2 must be a whole number(a non-negative integer). This implies that the power of 2 in the prime factorization of q^4 must be a multiple of 4. As 2^7 is not a factor of q^4, the highest power of 2 that could be a factor of q^4 is 2^4.
Hence q^4 is not a multiple of 64=2^6......sufficient
From statement (2): Given that q^2 has 27 factors. Thus q^2 could be of the form
(i) a^26, or
(ii)b^2*c^8 or
(iii) a^2*b^2*c^2 where a b and c are distinct prime numbers.
Because q^2 has 7 factors less than 11, (i) is impossible.
As for (ii) 2^8*3^2 has exactly 7 factors under 11 (1,2,3,4,6,8,9), in which case q^4 would be a multiple of 64. 3^8*2*2 is not a possibility for q^2 (it only has 6 factors under 1: 1,2,3,4,6,9)
Regarding (iii) if q^2=2^2*3^2*5^2, q^2 would have 8 factors under 11- 1,2,3,4,5,6,9,10 and q^2=2^2*3^2*7^2 would have 7 factors under 11: 1,2,3,4,6,7,9) In this case, q^4 would not be a multiple of 64. Thus (2) is insufficient
hence the answer A
A certain library assesses fines for overdue books as follows. On the first day that a book is overdue, the total fine is $0.10. For each additional day that the book is overdue the total fine is either increased by $0.30 or double, whichever results in the lesser amount. What is the total fine for a book on the fourth day it is overdue?
A. $0.60
B. $0.70
C. $0.80
D. $0.90
E. $1.00
Day 1 = 0.10
Day 2 = 0.20
Day 3 = 0.20*2 = 0.40
Thus on 4th day the total fine is: 0.10 + 0.20 + 0.40 = 0.70
A. $0.60
B. $0.70
C. $0.80
D. $0.90
E. $1.00
Day 1 = 0.10
Day 2 = 0.20
Day 3 = 0.20*2 = 0.40
Thus on 4th day the total fine is: 0.10 + 0.20 + 0.40 = 0.70
If *triangle* denotes one of the four arithmetic operations addition, subtraction, multiplication and division, what is the value of 1 *triangle* 2 ?
(1) n *triangle* 0 = n for all integers n.
(2) n *triangle* n = 0 for all integers n.
From statement (1): *triangle* can be both positive or negative as
n-0 = n
n+0 = n
Hence insufficient
From statement (2): *triangle* can only be negative in this case as
n-n = 0
Hence sufficient
(1) n *triangle* 0 = n for all integers n.
(2) n *triangle* n = 0 for all integers n.
From statement (1): *triangle* can be both positive or negative as
n-0 = n
n+0 = n
Hence insufficient
From statement (2): *triangle* can only be negative in this case as
n-n = 0
Hence sufficient
In XY plane, does the line with equation y=3x+2 contain point (r,s)?
1) (3r + 2 - s)(4r + 9 - s) = 0
2) (4r - 6 - s)(3r + 2 - s) = 0
Given that y = 3x+2 implies that does 3x+2-y = 0 contains the point (r,s) implies is (3r+2-s) = 0 ?
From statement (1): (3r+2-s)(4r+9-s) = 0 implies either (3r+2-s) = 0 or (4r+9-s) = 0.
Now when (3r+2-s)...the line passes through (r,s)
When (4r+9-s) = 0 ...we cannot determine that whether the line passes through (r,s) or not.
Hence insufficient
From statement (2): (4r-6-s)(3r+2-s) = 0 implies either (4r-6-s) = 0 or (3r+2-s) = 0
Now when (4r-6-s) = 0 ... we cannot determine that whether the line passes through (r,s) or not
When (3r+2-s) = 0..the line passes through (r,s)
Hence insufficient
Taking statement (1) and (2) together: (3r+2-s)(4r+9-s) = 0 and (4r-6-s)(3r+2-s) =0... We cannot have both 4r+9-s=0 and 4r-6-s=0 so it is (3r+2-s) = 0 ... only this equation makes both the equations to be 0
Hence sufficient
1) (3r + 2 - s)(4r + 9 - s) = 0
2) (4r - 6 - s)(3r + 2 - s) = 0
Given that y = 3x+2 implies that does 3x+2-y = 0 contains the point (r,s) implies is (3r+2-s) = 0 ?
From statement (1): (3r+2-s)(4r+9-s) = 0 implies either (3r+2-s) = 0 or (4r+9-s) = 0.
Now when (3r+2-s)...the line passes through (r,s)
When (4r+9-s) = 0 ...we cannot determine that whether the line passes through (r,s) or not.
Hence insufficient
From statement (2): (4r-6-s)(3r+2-s) = 0 implies either (4r-6-s) = 0 or (3r+2-s) = 0
Now when (4r-6-s) = 0 ... we cannot determine that whether the line passes through (r,s) or not
When (3r+2-s) = 0..the line passes through (r,s)
Hence insufficient
Taking statement (1) and (2) together: (3r+2-s)(4r+9-s) = 0 and (4r-6-s)(3r+2-s) =0... We cannot have both 4r+9-s=0 and 4r-6-s=0 so it is (3r+2-s) = 0 ... only this equation makes both the equations to be 0
Hence sufficient
Sania has a circular garden in her backyard. She puts poles A,B and C on the circumference of her garden. Then she ties ropes between these poles. Is length of one of the ropes is equal to the diameter of her garden?
1. Slope of line joining pole A and B is 3/4 and slope of line joining poles B and C is -4/3
2. Length of line joining pole A and B is 12 and length of line joining B and C is 5
From statement (1): Given that the slope of line AB is 3/4 and slope of line BC is -4/3. This implies that the product of slopes = -1. Hence AB perpendicular BC and B is a right angle. Thus AC is a diameter which implies ABC form a semi-circle.
Hence sufficient
From statement (2): Given that length of line AB is 12 and length of BC is 5. However this does not imply that ABC is a right angled triangle. We can draw number of different triangles with the same given two sides but with different third side.
Hence insufficient
1. Slope of line joining pole A and B is 3/4 and slope of line joining poles B and C is -4/3
2. Length of line joining pole A and B is 12 and length of line joining B and C is 5
From statement (1): Given that the slope of line AB is 3/4 and slope of line BC is -4/3. This implies that the product of slopes = -1. Hence AB perpendicular BC and B is a right angle. Thus AC is a diameter which implies ABC form a semi-circle.
Hence sufficient
From statement (2): Given that length of line AB is 12 and length of BC is 5. However this does not imply that ABC is a right angled triangle. We can draw number of different triangles with the same given two sides but with different third side.
Hence insufficient
Is a-3b an even number?
1). b=3a+3
2). b-a is an odd number
From statement (1): Given that b=3a+3
Thus a-3b=a-3(3a+3) = -8a-9 which may be even, odd, integer, non-integer, rational etc ... Hence insufficient
From statement (2): Given that b-a is an odd number implies b is of the form b=(2k+1)+a where k is an integer
Thus a-3b= a-3[(2k+1)+a] = -2a -6k-3 which may be even, odd, integer, non-integer, rational etc ..Hence insufficient
Taking statement (1) and (2) together: -8a-9=-2a-6k-3 for some integer k
or -6a=-6k+6=-6(k+1) implies a=k+1
Thus a is an integer, either odd or even
Now statement (2) tells us that b is also an integer and that exactly one of {a,b} is even
If a is even and b is odd, a-3b is odd
If b is even and a is odd a-3b is odd
Thus (1) and (2) combined tell us that a-3b is an odd number...hence sufficient
1). b=3a+3
2). b-a is an odd number
From statement (1): Given that b=3a+3
Thus a-3b=a-3(3a+3) = -8a-9 which may be even, odd, integer, non-integer, rational etc ... Hence insufficient
From statement (2): Given that b-a is an odd number implies b is of the form b=(2k+1)+a where k is an integer
Thus a-3b= a-3[(2k+1)+a] = -2a -6k-3 which may be even, odd, integer, non-integer, rational etc ..Hence insufficient
Taking statement (1) and (2) together: -8a-9=-2a-6k-3 for some integer k
or -6a=-6k+6=-6(k+1) implies a=k+1
Thus a is an integer, either odd or even
Now statement (2) tells us that b is also an integer and that exactly one of {a,b} is even
If a is even and b is odd, a-3b is odd
If b is even and a is odd a-3b is odd
Thus (1) and (2) combined tell us that a-3b is an odd number...hence sufficient
In the figure shown, what is the value of x?(1) The length of line segment of QR is equal to the length of line segment RS
(2) The length of line segment of ST is equal to the length of line segment TU
From statement (1): Length of line segment of QR is equal to the length of line segment RS ..this implies angle RQS = angle RSQ = p(say)
From statement (2): Length of line segment of ST is equal to the length of line segment TU .. this implies angle TUS = angle TSU = q(say)
Hence p+p+angle QRS = 180 --- eq(1) and q+q+angle UTS = 180 --- eq(2)
Thus, p+q+x = 180
Now because angle RPT = 90, QRS+UTS= 90
adding eq(1) and eq(2) we get:
2p+2q+QRS+UTS = 360
2p+2q+90=360
p+q = 270/2 = 135
Now x = 180-p-q..hence the answer C
x = 180 - (p+q) = 180 - 135 = 45
In triangle ABC, AB has a length of 10 and D is the midpoint of AB. What is the length of line segment DC?
(1) Angle C= 90
(2) Angle B= 45
From statement (1): it is given that angle C = 90 degrees ...this implies that ABC is a right angle triangle with AB as the hypotenuse and DC as the median. We know that --- In all right triangles, the median on the hypotenuse is the half of the hypotenuse. Hence DC=5. The ans is A.
(1) Angle C= 90
(2) Angle B= 45
From statement (1): it is given that angle C = 90 degrees ...this implies that ABC is a right angle triangle with AB as the hypotenuse and DC as the median. We know that --- In all right triangles, the median on the hypotenuse is the half of the hypotenuse. Hence DC=5. The ans is A.
p^a * q^b * r^c * s^d = x, where x is a perfect square.
p^a * q^b * r^c * s^d = x, where x is a perfect square. If p, q, r, and s are prime integers, are they distinct?
(1) 18 is a factor of ab and cd
(2) 4 is not a factor of ab and cd
OE: When a perfect square is broken down into its prime factors, those prime factors always come in "pairs." For example, the perfect square 225 (which is 15 squared) can be broken down into the prime factors 5 * 5 * 3 * 3. Notice that 225 is composed of a pair of 5's and a pair of 3's.
The problem states that x is a perfect square. The prime factors that build x are p, q, r, and s. In order for x to be a perfect square, these prime factors must come in pairs. This is possible if either of the following two cases hold:
Case One: The exponents a, b, c, and d are even. In the example 3^2 5^4 7^2 11^6, all the exponents are even so all the prime factors come in pairs.
Case Two: Any odd exponents are complemented by other odd exponents of the same prime. In the example 3^1 5^4 3^3 11^6, notice that 3^1 and 3^3 have odd exponents but they complement each other to create an even exponent (3^4), or "pairs" of 3's. Notice that this second case can only occur when p, q, r, and s are NOT distinct. (In this example, both p and r equal 3.)
Statement (1) tells us that 18 is a factor of both ab and cd. This does not give us any information about whether the exponents a, b, c, and d are even or not.
Statement (2) tells us that 4 is not a factor of ab and cd. This means that neither ab nor cd has two 2's as prime factors. From this, we can conclude that at least two of the exponents (a, b, c, and d) must be odd. As we know from Case 2 above, if paqbrcsd is a perfect square but the exponents are not all even, then the primes p, q, r and s must NOT be distinct.
The correct answer is B: Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(1) 18 is a factor of ab and cd
(2) 4 is not a factor of ab and cd
OE: When a perfect square is broken down into its prime factors, those prime factors always come in "pairs." For example, the perfect square 225 (which is 15 squared) can be broken down into the prime factors 5 * 5 * 3 * 3. Notice that 225 is composed of a pair of 5's and a pair of 3's.
The problem states that x is a perfect square. The prime factors that build x are p, q, r, and s. In order for x to be a perfect square, these prime factors must come in pairs. This is possible if either of the following two cases hold:
Case One: The exponents a, b, c, and d are even. In the example 3^2 5^4 7^2 11^6, all the exponents are even so all the prime factors come in pairs.
Case Two: Any odd exponents are complemented by other odd exponents of the same prime. In the example 3^1 5^4 3^3 11^6, notice that 3^1 and 3^3 have odd exponents but they complement each other to create an even exponent (3^4), or "pairs" of 3's. Notice that this second case can only occur when p, q, r, and s are NOT distinct. (In this example, both p and r equal 3.)
Statement (1) tells us that 18 is a factor of both ab and cd. This does not give us any information about whether the exponents a, b, c, and d are even or not.
Statement (2) tells us that 4 is not a factor of ab and cd. This means that neither ab nor cd has two 2's as prime factors. From this, we can conclude that at least two of the exponents (a, b, c, and d) must be odd. As we know from Case 2 above, if paqbrcsd is a perfect square but the exponents are not all even, then the primes p, q, r and s must NOT be distinct.
The correct answer is B: Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
What is the greatest common divisor of positive integers a and b?
(1) a and b share exactly one common factor
(2) a and b are both prime numbers
From statement (1): we know that a and b have only one common factor, and we also know that all positive integers share the common factor 1 only, so we know it must be 1...hence sufficient
From Statement (2): we know that a and b are both prime, this implies the greatest common factor will have to be 1 or if a = b could be the same prime number then the GCF would be a (=b). ...hence insufficient
NOTE: You cannot assume that a and b are different integers if the question stem does not states the same
(1) a and b share exactly one common factor
(2) a and b are both prime numbers
From statement (1): we know that a and b have only one common factor, and we also know that all positive integers share the common factor 1 only, so we know it must be 1...hence sufficient
From Statement (2): we know that a and b are both prime, this implies the greatest common factor will have to be 1 or if a = b could be the same prime number then the GCF would be a (=b). ...hence insufficient
NOTE: You cannot assume that a and b are different integers if the question stem does not states the same
If Line K in the XY-Plane has equation y=mx+b, where m and b are constants, what is the slope of K?
1. K is parallel to the line with equation y=(1-m)x+(b+1)
2. K intersects the line with equation y=2x+3 at the point (2,7)
From statement (1): y=(1-m)x+(b+1) has the same slope as y=mx+b. (Parallel lines have same slope)
Thus 1-m = m
implies Slope of K=m=1/2 ---- Hence sufficient
From statement (2): just says line y=2x+3 is not parallel to K, these two lines can have any angle between them other than 0, 180, 360 degrees ---- hence insufficient
1. K is parallel to the line with equation y=(1-m)x+(b+1)
2. K intersects the line with equation y=2x+3 at the point (2,7)
From statement (1): y=(1-m)x+(b+1) has the same slope as y=mx+b. (Parallel lines have same slope)
Thus 1-m = m
implies Slope of K=m=1/2 ---- Hence sufficient
From statement (2): just says line y=2x+3 is not parallel to K, these two lines can have any angle between them other than 0, 180, 360 degrees ---- hence insufficient
In the number line, are x and y on different sides of zero point?
1). The distance from x to zero is equal to the distance from y to 1
2). The sum of the distance from x to zero and the distance from y to 1 is less than 1
From statement (1): |x-0|=|y-1|
x = y-1
x = 1-y ....hence insufficient
From statement (2): |x-0|+|y-1| less than 1
= x+y-1 less than 1
= x+1-y less than 1
= -x+y-1 less than 1
= -x+1-y less than 1 ....hence insufficient
Taking statements (1) and (2) together: still insufficient
1). The distance from x to zero is equal to the distance from y to 1
2). The sum of the distance from x to zero and the distance from y to 1 is less than 1
From statement (1): |x-0|=|y-1|
x = y-1
x = 1-y ....hence insufficient
From statement (2): |x-0|+|y-1| less than 1
= x+y-1 less than 1
= x+1-y less than 1
= -x+y-1 less than 1
= -x+1-y less than 1 ....hence insufficient
Taking statements (1) and (2) together: still insufficient
Does the curve (x-a)^2 + (y-b)^2=16 intersect the y-axis ?
(1) a^2+b^2>16
(2) a=|b|+5
The given curve will intersect the y-axis when x=0
Thus we get a^2 + (y-b)^2 = 16
<=> a^2 + y^2 + b^2 - 2yb = 16
<=> y^2 - 2yb + a^2 + b^2 -16 = 0
In order to have real roots b^2 - 4ac >= 0
=> 4b^2 - 4(1)(a^2 + b^2 -16) >=0
=> a^2 <=16
From statement (1): Given that a^2 + b^2 > 16
No information about a^2 <=16 --- hence insufficient
From statement (2): Given a = |b|+5
=> |b| is positive or zero.
=> a is atleast equal to 5 and value of a^2 is atleast 25.
But we know that a^2 <=16 ====> does not intersect the y axis ---- hence sufficient
Hence the answer B
(1) a^2+b^2>16
(2) a=|b|+5
The given curve will intersect the y-axis when x=0
Thus we get a^2 + (y-b)^2 = 16
<=> a^2 + y^2 + b^2 - 2yb = 16
<=> y^2 - 2yb + a^2 + b^2 -16 = 0
In order to have real roots b^2 - 4ac >= 0
=> 4b^2 - 4(1)(a^2 + b^2 -16) >=0
=> a^2 <=16
From statement (1): Given that a^2 + b^2 > 16
No information about a^2 <=16 --- hence insufficient
From statement (2): Given a = |b|+5
=> |b| is positive or zero.
=> a is atleast equal to 5 and value of a^2 is atleast 25.
But we know that a^2 <=16 ====> does not intersect the y axis ---- hence sufficient
Hence the answer B
What is the median number of employees assigned per project for the projects at Company Z?
(1) 25 percent of the projects at Company Z have 4 or more employees assigned to each project.
(2) 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project.
From Statement 1): It is given that 25 percent of the projects at Company Z have 4 or more employees assigned to each project - but we donot know the percentage of projects who have employees less than 4 or in other words we do not have any information about the rest 75% projects ---- hence insufficient
From Statement 2): It is given that 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project - but we donot know the percentage of projects who have employees more than 2 or in other words we do not have any information about the rest 65% projects---- hence insufficient
Taking both the statements together:
25 percent of the projects at Company Z have employees 4 , 5, 6..
35 percent of the projects at Company Z have employees 2, 1, 0
=> 40% of projects have 3 employees = > median value is 3
(1-35)employees -- (36-75)employees -- (76-100)employees
2 or less than 2 ---------3, 3, 3, ------------ 4 or more than 4
(1) 25 percent of the projects at Company Z have 4 or more employees assigned to each project.
(2) 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project.
From Statement 1): It is given that 25 percent of the projects at Company Z have 4 or more employees assigned to each project - but we donot know the percentage of projects who have employees less than 4 or in other words we do not have any information about the rest 75% projects ---- hence insufficient
From Statement 2): It is given that 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project - but we donot know the percentage of projects who have employees more than 2 or in other words we do not have any information about the rest 65% projects---- hence insufficient
Taking both the statements together:
25 percent of the projects at Company Z have employees 4 , 5, 6..
35 percent of the projects at Company Z have employees 2, 1, 0
=> 40% of projects have 3 employees = > median value is 3
(1-35)employees -- (36-75)employees -- (76-100)employees
2 or less than 2 ---------3, 3, 3, ------------ 4 or more than 4
A store purchases 20 coats that each cost an equal amount and then sold each of the 20 coats at an equal price, what was the store's gross profit on the 20 coats?
1). If the selling price per coat had been twice as much, the store's gross profit on the 20 coats would have been 2400
2). If the store selling price per coat had been $2 more, the store's gross profit on the 20 coats would have been 440
Suppose the cost of each coat = x
Suppose sales price = y
=> Total cost = 20x and sales price = 20y
We are required to find the profit: 20(y-x).
From statement 1): It is given that 20(2y-x)=2400
Doesn't helps to find out 20(y-x) --- insufficient
From statement 2): It is given that 20(y+2-x) = 440,
=> 20(y-x) = 400 --- sufficient
Hence B
1). If the selling price per coat had been twice as much, the store's gross profit on the 20 coats would have been 2400
2). If the store selling price per coat had been $2 more, the store's gross profit on the 20 coats would have been 440
Suppose the cost of each coat = x
Suppose sales price = y
=> Total cost = 20x and sales price = 20y
We are required to find the profit: 20(y-x).
From statement 1): It is given that 20(2y-x)=2400
Doesn't helps to find out 20(y-x) --- insufficient
From statement 2): It is given that 20(y+2-x) = 440,
=> 20(y-x) = 400 --- sufficient
Hence B
If x is not equal to zero, is 1/x > 1 ?
1) y/ x > y
2) x^3 > x^2
From statement (1): y/x > y
=> y > xy
=> y(1-x) > 0
y>0 when x is less than 1 or y less than 0> when x is greater than 1 ---- hence insufficient
From statement (2): x^3 > x^2
=> x^2(x-1) > 0
=> x^2 >0 and x>1
Since x>1, 1/x can not be greater than 1 ---- hence sufficient
Hence B
1) y/ x > y
2) x^3 > x^2
From statement (1): y/x > y
=> y > xy
=> y(1-x) > 0
y>0 when x is less than 1 or y less than 0> when x is greater than 1 ---- hence insufficient
From statement (2): x^3 > x^2
=> x^2(x-1) > 0
=> x^2 >0 and x>1
Since x>1, 1/x can not be greater than 1 ---- hence sufficient
Hence B
DS-2
What is the average (aritmetic mean) hight of the n people in a certain group?
1 the average hight of the n/3 talles people in the group is 6 feet and 2 1/2 inches and the average height of rest of the people in the group is 5 feet and 10 inches.
2 The sum of the lenghts of the people is 178 feet and 9 inches.
Looking at statement (2) first, we see that it is not sufficient, because the average (arithmetic mean) of a group of numbers is defined as (sum of data) / (# of data points). With statement (2), we only have the numerator of this expression (the # of people in the group is unknown), so we can't figure out the average.
Looking at statement (1) alone, we can set up the average as follows:
Average = (sum of data points) / (# of data points)
= [(n/3)(74.5) + (2n/3)(70)] / (n) <-- note that I used inches here, so I won't have to write in more fractions than necessary (trying to write fractions on this forum is not fun) = [(1/3)(74.5) + (2/3)(70)] / (n) There's no need to simplify further, because the 'n' is gone: you get one number. Therefore, this statement is sufficient. Answer = A Note that, if you have the averages of all the FRACTIONS or PERCENTAGES of a group, then you'll be able to calculate the overall average of the group. This is a worthwhile fact to memorize for the data sufficiency problems
1 the average hight of the n/3 talles people in the group is 6 feet and 2 1/2 inches and the average height of rest of the people in the group is 5 feet and 10 inches.
2 The sum of the lenghts of the people is 178 feet and 9 inches.
Looking at statement (2) first, we see that it is not sufficient, because the average (arithmetic mean) of a group of numbers is defined as (sum of data) / (# of data points). With statement (2), we only have the numerator of this expression (the # of people in the group is unknown), so we can't figure out the average.
Looking at statement (1) alone, we can set up the average as follows:
Average = (sum of data points) / (# of data points)
= [(n/3)(74.5) + (2n/3)(70)] / (n) <-- note that I used inches here, so I won't have to write in more fractions than necessary (trying to write fractions on this forum is not fun) = [(1/3)(74.5) + (2/3)(70)] / (n) There's no need to simplify further, because the 'n' is gone: you get one number. Therefore, this statement is sufficient. Answer = A Note that, if you have the averages of all the FRACTIONS or PERCENTAGES of a group, then you'll be able to calculate the overall average of the group. This is a worthwhile fact to memorize for the data sufficiency problems
Speed Time and Distance -1
A boat traveled upstream a distance of 90 miles at an average speed of (v-3) miles per hour and then traveled the same distance downstream at an average speed of (v+3) miles per hour. If the trip upstream took half an hour longer than the trip downstream, how many hours did it take the boat to travel downstream?
A: 2.5
B: 2.4
C: 2.3
D: 2.2
E: 2.1
90/(V-3) = 1/2 + 90/(V+3)
90/(V-3) - 90/(V+3) = 1/2
[90(V+3) - 90(V-3)] / (V+3) (V-3) = 1/2
[90V + 270 - 90V +270] = [V^2 - 9]/2
540 *2 = V^2 - 9
1080 + 9 = V^2
1089 = V^2
V = 3*11 = 33
Downstream = 90/(v+3) = 90/36 = 10/4 = 2.5
A: 2.5
B: 2.4
C: 2.3
D: 2.2
E: 2.1
90/(V-3) = 1/2 + 90/(V+3)
90/(V-3) - 90/(V+3) = 1/2
[90(V+3) - 90(V-3)] / (V+3) (V-3) = 1/2
[90V + 270 - 90V +270] = [V^2 - 9]/2
540 *2 = V^2 - 9
1080 + 9 = V^2
1089 = V^2
V = 3*11 = 33
Downstream = 90/(v+3) = 90/36 = 10/4 = 2.5
Data Sufficiency - 1
From statement (1): Square root of a number less than 1 is also less than 1 - sufficient
From statement (2): Reciprocal of any number less than 1 is greater than the number - sufficient
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