GMAT Classic Mock Test - 7 - GMAT MCQ

Assign variables to pieces of the problem:
Let n = number of computer game units sold with the old strategy
Let p = price of computer game per unit with the old strategy
We know that before the new strategy, the total revenue was $100. Based on the number of copies sold at the price of the old strategy:
np = $100.
We know that with the new strategy, the total revenue was $100. Based on 10 more copies sold and a $5 reduction in the price per unit:
(n+10)(p-$5) = $100.
Expand this second equation:
np - 5n + 10p - 50 = 100.
From the first equation, we know that np = $100, so plug that in for np:
100 - 5n + 10p - 50 = 100.
From the first equation, if we divide by n, we end up with p = $100/n. Plug this value in for p:
100 - 5n + 10(100/n) - 50 = 100.
Combine like terms:
50 - 5n + 1000/n = 100.
Subtract 50 from each side:
1000/n - 5n = 50.
Divide by 5:
200/n - n = 10.
Multiply through by n:
200 - n² = 10n
Subtract 10n from both sides:
200 - n² - 10n = 0
Rearrange:
-n² - 10n + 200 = 0
Multiply by -1:
n² + 10n - 200 = 0
Factor the quadratic:
(n+20)(n-10) = 0
Thus n= -20, or n=10. Since n cannot be negative, n in this case must be 10. However, do not be tricked into choosing A at this point. We want the number sold with the new strategy, not the old.
n=10 means 10 copies were sold with the old strategy. The new strategy sold 10 more, resulting in a total of 20 copies sold.
n_new = n_old + 10 = 10 + 10 = 20
The answer is C.

Evaluate each statement one by one, starting with the first.
Evaluate Statement I.
In order to be divisible by 6, a number must have the prime factors of 6, which are 2 and 3.
In a set of 3 consecutive integers, at least one of them will be even. Any even number has a 2 as one of its prime factors. Thus, the product of the 3 consecutive positive integers will have a factor of 2.
Any consecutive series of 3 integers has a multiple of 3 in it, since every third integer is a multiple of 3. Thus, either x, y, or z is a multiple of 3 and therefore has 3 as one of its prime factors. Thus, the product xyz will have a factor of 3.
Since xyz will have a 3 and a 2 in its prime factorization tree, xyz must be divisible by 6. Therefore, I is always true.
Evaluate Statement II.
Since x, y, and z are consecutive integers such that x < y < z, we can rewrite y and z in terms of x: y=x+1, and z=x+2.
Substitute these values in the equation:
(z-x)(y-x+1)=4
([x+2]-x)([x+1]-x+1)=4
Simplify the equation:
(x-x+2)([x-x+1+1)=4
(2)(1+1) = 2(2) = 4
It is clear that II will always be true.
Evaluate Statement III.
Since x, y, and z are consecutive numbers such that x < y < z, if y is even, then x and z will both be odd. The product of two odd numbers is odd so xz would be odd in this case. But, xy = (odd)(even) = even.
But, if y is odd, then x and z will be even. The product of two even numbers is even, so xz is even in this case and xy is also even.
Since there is no way to guarantee that both x and y are odd, we cannot conclude that statement III is always true.
Note: Since x and y are consecutive integers, either x or y will always be even. Consequently, xy will always be even: either (even)(odd) = even OR (odd)(even) = even.
Since I and II must be true, but III is not always true, the correct answer is D.

This question can be solved using either real numbers or algebra.
Method 1: Use Real Numbers.
Assume that the worker in question, call him John, earned $50 by working 10 hours for $5 per hour.
After the pay-raise, the total amount earned will be the same at $50, but the hourly rate will increase by 20% to $5(1.2) = $6 per hour.
Using the intuition that salary=(hours)(wage), you can find that $50=(hours)($6) and therefore the number of hours worked after the change in pay will equal (50/6) = 8.33 hours
Since John worked 10 hours before the change in wage and he now works 8.33 hours, John reduced the number of hours he worked by (10-8.33)/10 = .1666666% = 17%
Note: It does not matter what numbers you use, as long as the original pay equals the post wage-change pay and the wage is increased by 20%.
Method 2: Use Algebra.
Assign variables to pieces of the problem:
Let p = the percentage of old hours worked after the raise
if p = 95%, the worker labors 95% of the old hours after the raise (i.e., 5% fewer hours after the raise)
Let r = hourly rate that employees were paid before the raise
Let R = hourly rate that employees were paid after the raise
Let n = number of hours worked before pay-raise
Let N = number of hours worked after pay-raise
We know that the total pay before and after the pay-raise equal each other. The pay before was based on the rate r with hours n while the pay afterwards was based on rate R and hours N:
rn = RN
The rate after the raise was 20% more than the rate before the raise:
R = 1.20r
Plug this value into the equation for R:
rn = RN
rn = 1.20rN
We are looking for the percentage of hours worked after the raise, so substitute N in terms of n and percent p.
N = pn; the new number of hours equals the old number of hours multiplied by the percent of old hours worked after the raise
rn = (1.2r)(pn).
Divide both sides by r and n:
1 = 1.2p
Divide by 1.2 to solve for p:
p = 1/1.2 = 10/12 = 5/6 = .8333 = 83.33%.
Be careful: the question asks what percent the worker would reduce his hours, so subtract from 100% to yield 16.66%, or 17%. The correct answer is D.

It is essential to recognize that the remainder when an integer is divided by 10 is simply the units digit of that integer. To help see this, consider the following examples:
4/10 is 0 with a remainder of 4
14/10 is 1 with a remainder of 4
5/10 is 0 with a remainder of 5
105/10 is 10 with a remainder of 5
It is also essential to remember that the z is a positive integer and multiple of 2. Any integer that is a multiple of 2 is an even number. So, z must be a positive even integer.
With these two observations, the question can be simplified to: "what is the units digit of 4 raised to an even positive integer?"
The units digit of 4 raised to an integer follows a specific repeating pattern:
4¹ = 4
4² = 16
4³ = 64
4⁴ = 256
4^{(odd number)} → units digit of 4
4^{(even number)} → units digit of 6
There is a clear pattern regarding the units digit. 4 raised to any odd integer has a units digit of 4 while 4 raised to any even integer has a units digit of 6.
Since z must be an even integer, the units digit of p=4z will always be 6. Consequently, the remainder when p=4z is divided by 10 will always be 6.
In case this is too theoretical, consider the following examples:
z=2 → p=4^z=16 → p/10 = 1 with a remainder of 6
z=4 → p=4^z=256 → p/10 = 25 with a remainder of 6
z=6 → p=4^z=4096 → p/10 = 409 with a remainder of 6
z=8 → p=4^z=65536 → p/10 = 6553 with a remainder of 6

1. The shortest distance of line xy will occur when x and y are at the same point (i.e., the point where the two circles come together). In this instance, the line xy will be 0 units long.
2. In order to determine the longest possible distance for xy, we must first recall that the longest line across a circle is the circle's diameter. In other words, it is impossible to construct a line from one point on a circle to another point on the same circle that is longer than the circle's diameter.
3. The longest distance of line xy will occur when x and y are at exact opposite sides of the two circles. In other words, when x is at the far left of A and y is at the far right of B. More technically, line xy will be the combined diameter of circles A and B. This makes sense given that the diameter of a circle is the longest possible line from one point on the circle to another point on the same circle.
   
4. Since the length of xy is the length of the diameter of A plus the diameter of B, we need to find the diameter of each circle.
   Area of A = 144π = πr²
   rA = 12 = radius of circle A
   dA = 2(12) = 24 = diameter of circle A
   Area of B = 169π = πr²
   rB = 13 = radius of circle B
   dB = 2(13) = 26 = diameter of circle B
5. Maximum distance of xy = 24 + 26 = 50

Order the members of set S in ascending order. Leave out n since you do not yet know where it will fall.
-27, -3, -2, 2, 3, 15, 17
At this point, you do not know where n will be. However, it is not essential to know the exact placement of n in order to find which answer choice could be the median of set S.
When n is included in set S, set S has an even number of terms. Consequently, the median will be the average of the middle two terms of set S.
It is important to reiterate that set S consists of "unique integers." Consequently, n cannot equal any of the existing values of S.
By logically examining set S, there are three options for the median:
Option 1: n > 3 and Median = 2.5
If n > 3, the median will be 2.5
-27, -3, -2, 2, 3, n, 15, 17
-27, -3, -2, 2, 3, 15, n, 17
-27, -3, -2, 2, 3, 15, 17, n
In each case, the median will be between 2 and 3 → Median will be 2.5
Option 2: n < -3 and Median = 0
If n < -3, the median will be 0
-27, n, -3, -2, 2, 3,15, 17
n, -27, -3, -2, 2, 3,15, 17
In each case, the median will be between -2 and 2 → Median will be 0
Option 3: Median Is (n+2)/2
If -2 < n < 2, n could be -1, 0, 1 (remember n must be a unique integer, so it cannot be 2 or -2 since these numbers are already used).
The series is now ordered: -27, -3, -2, n, 2, 3, 15, 17
Since the median will be the average of the middle two terms, we can simplify the series to:
-2, n, 2, 3, which will simplify to:
n, 2
The median of the series will be the average of n and 2.
Remember that n can equal -1, 0, or 1.
If n = -1 → Median will be 0.5
If n = 0 → Median will be 1
If n = 1 → Median will be 1.5
Since 1 is the only possible median that is included as an answer choice, it is the correct answer.

In this problem, order is important. Jane standing to the left of Mary is a different arrangement than Mary standing to the left of Jane. Because order is important, the permutations equation is used.
The formula for a permutation is:
nPr= n!/(n-r)!
where n is the total number of selections available and r is the number of items to be selected.
For each set of two models from each continent, there are 2P2ways to arrange them. From the permutation formula with n equal to 2 and r equal to 2 this results in:
₂P₂= 2! = 2
Since there three groups of two there are:
₂P₂*₂P₂*₂P₂ or 2*2*2 or 8 ways to arrange each group within each group.
Since there are three groups of models that can be placed in three different positions there are:
₃P₃ways to arrange the three groups.
The value of₃P₃is (3*2*1)/(3-3)! = 6/0! = 6, or 6 ways to arrange the different groups. Note that 0! is defined to be equal to one.
Since there are 6 ways to arrange the groups and 8 ways to arrange the models within their own groups there are:
8X6 or 48 different ways to arrange the models. So the correct answer is A.

1. Seating students in the order A, B, C, D, E, F, G is not the same as seating them B, C, D, E, F, G, A. Since the order in which the students are seated does matter, this is a permutations problem. To solve such a problem, determine the number of possibilities for each slot and then multiply these numbers together to get the total number of permutations.
2. There are seven empty seats and seven students to fill them. However, you are told that two of the students must sit in an end seat, either the first seat or the seventh seat. That means that there are 2 possibilities for the first seat – either of these two students. Therefore there is only 2 – 1 = 1 possibility for the seventh seat (i.e., the student who must sit in an end seat who is not sitting in the first seat).
3. Since two of the students must sit in end seats, there are 7 – 2 = 5 possibilities for the second seat, 7 – 3 = 4 possibilities for the third seat, 7 – 4 = 3 possibilities for the fourth seat, 7 – 5 = 2 possibilities for the fifth seat, and 7 – 6 = 1 possibility for the sixth seat.
4. Now that you have determined the number of possibilities for each slot, multiply them together to determine the total number of permutations:
5. (2)(5)(4)(3)(2)(1)(1) = 240
6. So the correct answer is choice (D).

If x takes any value greater than 93, the expression will definitely be positive.
Therefore, the set of values that x takes should be from the set of positive integers upto 93
The expression(x - 1)(x - 3)(x - 5)....(x - 93) has a total of 47 terms.

When x = 1, 3, 5, 7, .... 93 the value of the expression will be zero. i.e., for odd values of x, the expression will be zero.
We need to evaluate whether the expression will be negative for all even numbers upto 93.

Let x = 2: First term (x - 1) > 0; remaining 46 terms < 0.
The product of one positive number and 46 negative numbers will be positive (product of even number of negative terms will be positive).
So, x = 2 does not satisfy the condition.

Let x = 4: (x - 1) and (x - 3) are positive; the remaining 45 terms < 0.
The product of two positive numbers and 45 negative numbers will be negative. So, x = 4 satisfies the condition.

Let x = 6: First 3 terms (x - 1), (x - 3), and (x - 5) are positive and the remaining 44 terms are negative.
Their product > 0. So, x = 6 does not satisfy the condition.

Let x = 8: First 4 terms positive; remaining 43 terms < 0. Their product < 0. So, x = 8 satisfies the condition.
Extrapolating what we have oberved with these 4 terms, we could see a pattern.
The expression takes negative values when x = 4, 8, 12, 16, ..... 92 (multiples of 4)

92, the last value that x can take is the 23rd multiple of 4.
Hence, number of such values of x = 23

Choice B is the correct answer.

What kind of numbers have exactly 3 factors?
Any positive integer will have '1' and the number itself as factors. That makes it a minimum of 2 factors (except '1' which has only one factor). If the positive integer has only one more factor, then in addition to 1 and the number, the square root of the number should be the only other factor.

There are two key points in the above finding. The number has to be a perfect square. And the only factor other than 1 and the number itself should be its square root.

Therefore, if a positive integer has only 3 factors, then it should be a perfect square and it should be the square of a prime number.

How many numbers from {1, 2, 3, 4, .... 100} have exactly 3 factors?
Let us look at an example. 4 has the following factors: 1, 2, and 4 (exactly 3 factors). It is the square of '2' which a prime number.
Squares of numbers that are not prime numbers will have more than 3 factors. For instance, 36 is a perfect square. But it has 9 factors.

Number of squares of prime numbers from 1 to 100 that have exactly 3 factors are 4, 9, 25, and 49. i.e., 4 numbers

Step 1: Compute the total number of possibilities
Number of ways of selecting two distinct integers from the set of first 100 positive integers = 100C2 ways.
i.e., 100C2 = 100 × 992
Step 2: Compute the number of favourable outcomes
The product of two distinct numbers 'a' and 'b' will be 4 when one of the numbers is 1 and the other is 4. There is only one set that will result in this product.
The same holds good for the other 3 numbers as well. Product of two distinct numbers 'a' and 'b' will be 9 when one of the numbers is 1 and the other is 9 and so on.
Therefore, there are 4 outcomes in which the product of the two numbers will result in a number that has exactly 3 factors.

Step 3: Compute the required Probability

Choice B is the correct answer.
 

Working alone at a constant rate if A takes 'a' days to complete a task, A will complete 1/a of the task in a day.
Step 1: Translate words into mathematical expressions
A will complete 1/a of the task in a day.
Therefore, in 2 days A will complete 2/a of the task in a day.
Similarly, A will complete 5/a of the task in 5 days.

If A starts, they finish the task in exactly 10 days.
A starts and works for 2 days. So, A will work on day 1 and day 2.
Then B will work for the next 2 days. B will work on day 3 and day 4.
A will continue for the next 2 days. i.e., on day 5 and day 6.
B will work on day 7 and day 8.
A will for the last 2 days i.e., day 9 and day 10.
Therefore, A will work on day 1, day 2, day 5, day 6, day 9, and day 10. i.e., for 6 days.
And B will work on day 3, day 4, day 7, and day 8. i.e., for 4 days.
In 6 days A will complete 6/a of the task.
In 4 days B will complete 4/a of the task.
With A working 6 days and B working 4 days, the task is completed.

If B starts, they finish the task they take half a day more. i.e., 10.5 days.
Therefore, B will work on day 1, day 2, day 5, day 6, day 9, and day 10. i.e., for 6 days.
And A will work on day 3, day 4, day 7, day 8 and half a day on day 11. i.e., for 4.5 days.
In 6 days B will complete 6/b of the task.
In 4 days A will complete 4.5/a of the task.
With B working 6 days and A working 4.5 days, the task is completed.

Step 2: Solve the two equations

Solving the two equations we get a = 9 days and b = 12 days.
The question: How long does it take to complete the task if they both work together?
Working together A and B will complete  th of the task in a day.
Hence, they will complete the task in 36/7 days.
Choice D is the correct answer.
 

Step 1: Divide each semicircle into a triangle and the shaded region
P and Q are the centers of the two semicircles.
Draw BP perpendicular to AC.
BP is radius to the semi-circle. So are AP and PC.
Therefore, BP = AP = PC = 2 units.
In semicircle ABC, area of the shaded portion is the difference between the area of half the semicircle PBC and the area of the triangle PBC.
Triangle PBC is a right triangle because PB is perpendicular to PC. PB and PC are radii to the circle and are equal. So, triangle PBC is an isosceles triangle.
Therefore, triangle PBC is a right isosceles triangle.

Step 2: Compute areas of half the semicircle and the triangle

Area of half the semicircle − Area of region PBC
Area of the semicircle ABC = 1/2 area of the circle of radius 2.
So, area of half the semicircle, PBC = 1/4 area of the circle of radius 2.
Area of half the semicircle, PBC = 1/4 × π × 2²
Area of half the semicircle, PBC = π sq units
Area of right isosceles triangle PBC
Area of right triangle PBC = 1/2 × PC × PB
Area of triangle PBC = 1/2 × 2 × 2 = 2 sq units

Area of shaded region
Area of shaded region in one of the semi circles ABC = (π - 2) sq units
Therefore, area of the overall shaded region = 2(π - 2) sq units = 2π - 4 sq units

Choice C is the correct answer.

Understanding Absolute Values
= 1 When a is positive.
= 1 When a is negative.
Step 1: Compute the Maximum Value of S
When will the value of the expression be maximum?

The value of the expression will be maximum when all of the terms become positive.
i.e., a, b, and ab should be positive and c should be negative.
When a is positive, 
When b is positive, 
When a and b are positive, as required in the previous two steps, ab will be positive and the expression, 
When c is negative,
Therefore, the maximum value = 1 + 1 + 2 + 3 -(-4) = 11
Step 2: Compute the Minimum Value of S
When will the value of the expression be minimum?

 

The value of the expression will be minimum if we make as many terms negative as possible.
Higher the magnitude of the terms made negative, lower the value of the expression.
c has to be positive for S to be minimum. The last term will then be -4.
If ab is negative, then 
If ab has to be negative, one of a or b has to be positive and the other has to be negative.
Possibility 1: If a > 0 and b < 0,
The value of the expression is 1 + 1 - 2 - 3 - 4 = -7.
Possibility 2: If a < 0 and b > 0, 
The value of the expression is 1 - 1 + 2 - 3 - 4 = -5.
Therefore, the minimum value is -7
Step 3: Compute the difference
Maximum value of S = 11.
Minimum value of S = -7.
The difference is 18.
Choice E is the correct answer.

Step 1: Key Data from the Question Stem
Set S has 6 elements.
The elements of set S are distinct.
x and y are prime numbers. Because 2 is already an element in S, both x and y have to be odd.
0 < x < 40 and 0 < y < 40
Step 2: Check Statement I
I. The maximum possible range of the set is greater than 33.
The key word in this entire statement is maximum. We have to determine whether the maximum value possible for the range exceeds 33.
We know x and y are prime numbers. The largest prime number less than 40 is 37.
If either x or y is 37, the largest number in the set will be 37 and the smallest number is 2.
Therefore, the maximum range of the set will be 37 - 2 = 35. It is greater than 33.

Statement I is true. So, eliminate choices that do not contain I.
Eliminate choice D

Step 3: Check Statement II
II. The median can never be an even number.
There are 6 numbers in the set. Therefore, the median is the arithmetic mean of the 3rd and the 4th term when the numbers are written in ascending or descending order.
The elements are {2, 4, 6, 8, x, y}, where x and y are prime numbers.
If x and y take 3 and 5 as values, the median is 4.5
If x = 3, y = 7 or greater, the median is 5.
If x = 5, y = 7 or greater, the median is 5.5
If x = 7, y = 11 or greater, the median is 6.5
If x = 11 or greater and y = 13 or greater, the median is 7.
It is quite clear that the median is either an odd number or is not an interger. So, the median can never be an even integer.

Statement II is true. Eliminate choices that do not contain II.
Eliminate choices A and C as well.

Step 4: Check Statement III
III. If y = 37, the average of the set will be greater than the median.
If y = 37, the set will be {2, 4, 6, 8, x, 37}, where x is a prime number greater than 2 and less than 37.
The average will be
If x = 3, median = 5 and average = 10. Average > median.
If x = 5, median = 5.5 and average = 10.33. Average > median
If x = 7, median = 6.5 and average = 10.66. Average > medain
If x = 11 or greater, the median = 7. Average will be definitely greater than 10. So, Average > Median.
It is true that the average is greater than the median if y = 37.
Statement III is also true.
Statements I, II, and III are true.
Choice E is the correct answer.

x and y are integers and |x - y| = 12

Approach: Square both sides and solve.
Squaring both sides, we get (x - y)² = 144
x² + y² - 2xy = 144
Add, 4xy to both sides of the equation.
x² + y² - 2xy + 4xy = 144 + 4xy
x² + y² + 2xy = 144 + 4xy
Or (x + y)² = 144 + 4xy
(x + y)2 will NOT be negative for real values of x and y.
i.e., (x + y)² ≥ 0
∴ 144 + 4xy ≥ 0
Or 4xy ≥ -144
So, xy ≥ -36

The least value that xy can take is -36.

Choice D is the correct answer.

Note that this question asks for a specific number, not a ratio. Consequently, keep in mind that knowing y percent of the total staff is composed of women from outside the United States is not sufficient.
Evaluate Statement (1) alone.
If 25% of the staff are men, 75% must be women.
Men    Women    Total
From U.S.            
From Outside U.S.            
.25(x)    .75(x)    x
There is not enough information to determine the number of women from outside the United States. Statement (1) alone is NOT SUFFICIENT.
Evaluate Statement (2) alone.
Since 20 men from the U.S. represent 20% of the staff, the total staff is 100. We also know that there are 20 men from the U.S. and 2(20)=40 women from the U.S. for a total of 20+40=60 employees from the U.S. Consequently, 100-60=40 employees must be from outside the U.S.
Men    Women    Total
From U.S.    20    40    60
From Outside U.S.            40
x=100
Since we cannot determine the breakdown of the 40 employees from outside the U.S., it is impossible to determine the number of women from outside the U.S.; Statement (2) alone is NOT SUFFICIENT.
Evaluate Statements (1) and (2) together.
Fill in as much information as possible from Statements (1) and (2). We now know that there are a total .25(x)=.25(100)=25 men and .75(x)=.75(100)=75 women.
Men    Women    Total
From U.S.    20    40    60
From Outside U.S.    5    35    40
25    75    x=100
35 members of the staff of Advanced Computer Technology Consulting are women from outside the United States.
Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, but Statements (1) and (2), when taken together, are SUFFICIENT, answer C is correct.

Evaluate Statement (1) alone.
Take the cube root of both sides:
8x³ = 27y³
2x = 3y
Rearrange in order to find a ratio of 2x to y.
2x/y = 3
Consequently, 2x is 3 times y.
Statement (1) alone is SUFFICIENT.
Evaluate Statement (2) alone.
Take the square root of both sides:
4x² = 9y²
2x = 3y
2x/y = 3
However, we must also consider that in taking the square root, a negative root is possible. To illustrate this, consider the following example:
Let x = 3 and y = 2 → 4x² = 9y²
Let x = -3 and y = 2 → 4x² = 9y²
Let x = -3 and y = -2 → 4x² = 9y²
Let x = 3 and y = -2 → 4x² = 9y²
In the four examples above, although 4x² = 9y², there is no consistent ratio of 2x to y since the negative numbers cause ratios to be negative. Consequently, Statement (2) is NOT SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer A is correct.

Evaluate Statement (1) alone.
1. It is possible that XZ is a straight line with Y as the midpoint, making ZY=YX.
2. However, just because ZY = YX does not mean Y must always be the midpoint; XYZ could be an equilateral triangle.

Statement (1) alone is NOT SUFFICIENT.
Evaluate Statement (2) alone.
By definition, the center of a circle is the midpoint of a diameter. Consequently, XZ runs through point Y and XY = YZ since both are radii and all radii must be the same length.
Statement (2) alone is SUFFICIENT.
Since Statement (1) alone is NOT SUFFICIENT but Statement (2) alone is SUFFICIENT, answer B is correct.

Be aware that simply because you have two equations with two unknowns does not mean that a solution exists. You must have two unique equations with two unknowns in order for a solution to exist.
Evaluate Statement (1) alone.
There are two possible ways to solve this problem:
Method (1): Substitute b from Statement (1) into the original equation.
15a + 6(5 – 2.5a) = 30
15a + 30 - 15a = 30
30 = 30
0 = 0
Based upon this answer, the equation in Statement (1) is the equation in the original question solved for b. Consequently, we only have one equation and two unknowns. There is not enough information to determine a-b.

Method (2): Rearrange the equation in Statement (1) and subtract this equation from the original equation.
b = 5 – 2.5a
b + 2.5a = 5
2.5a + b = 5
Multiply by 6 so b's cancel:15a + 6b = 30
This method also shows that the equation in Statement (1) is nothing more than the original equation rearranged. Consequently, we only have one equation and two unknowns. There is not enough information to determine a-b.
Statement (1) is NOT SUFFICIENT.
Evaluate Statement (2) alone.
Try to line up the two equations so that you can subtract them:
9b = 9a – 81
81 + 9b = 9a
81 = 9a - 9b
Statement (2) Equation: 9a - 9b = 81
Original Question Equation: 15a + 6b = 30
At this point, you can stop since you know that you have two unique equations and two unknowns. Consequently, there will be a solution for a and for b, which means there will be one unique value for a-b. Statement (2) is SUFFICIENT.
If you want to solve to see this (Note: Do not solve this in a test as it takes too much time and is not necessary):
Multiply (2) by 4: 36a - 36b = 324
Multiply Original by 6: 90a + 36b = 180

6*Original + 2*Statement(2): (90a + 36a) + (36b + -36b) = 180 + 324
126a = 204
a = 4

Solve for b:
9b = 9(4) - 81 = -45
b = -5

a - b = 4 - (-5) = 4 + 5 = 9
Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.

Evaluate Statement (1) alone.
Try to solve for n:
n² - 6n = -9
n² - 6n + 9 = 0
(n - 3)² = 0
n - 3 = 0
n = 3
With one value for n, we can find a single value for (n + 1)2
Statement (1) alone is SUFFICIENT.
Evaluate Statement (2) alone.
Expand the terms and simplify them:
n² - 2n + 1 = n² – 5
-2n + 1 = -5
-2n + 6 = 0
6 = 2n
n = 3
With one value for n, we can find a single value for (n + 1)2
Statement (2) alone is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

Revenue = Cost_store-front*Quantity_store-front + Cost_bulk*Quantity_bulk
Evaluate Statement (1) alone.
Store-Front: 75% of 100 packages is 75 packages at the store-front rate.
Bulk: The remainder (or 25%) of the 100 packages (i.e., 100-75 = 25) is 25 packages at the bulk-rate.
Without any dollar amounts (such as the cost of the bulk-rate and the cost of the store-front rate), it is impossible to calculate John’s total revenue.
Statement (1) alone is NOT SUFFICIENT.
Evaluate Statement (2) alone.
Translate Statement (2) into algebra:
Store-Front > 2(Bulk)
$4 > 2(Bulk)
Bulk < $2
Although Statement (2) tells us the dollar amount of each shipping rate, without information about the number of packages shipped at each rate, it is impossible to calculate John’s revenue.
Statement (2) alone is NOT SUFFICIENT.
Evaluate Statements (1) and (2) together.
Store-Front: 75 packages {from Statement (1)} shipped at $4 each {from Statement (2)} -> $300 in revenue from the store-front rate.
Bulk: 25 packages shipped at less than $2 each; no more than $50 in revenue from the bulk-rate.
However, you still cannot calculate the total revenue definitively.
Revenue = Cost_store-front*Quantity_store-front + Cost_bulk*Quantity_bulk
Filling in what we found thus far:
Revenue = $4*75 + Cost_bulk*25
Statements (1) and (2), even when taken together, are NOT SUFFICIENT.
Since Statement (1) alone is NOT SUFFICIENT, Statement (2) alone is NOT SUFFICIENT, and Statements (1) and (2), even when taken together are NOT SUFFICIENT, answer E is correct.

Any positive integer that is divided by 2 will have a remainder of 1 if it is odd. However, it will not have a remainder if it is even.
N/2 → Remainder = 0 if N is even
N/2 → Remainder = 1 if N is odd
Evaluate Statement (1) alone.
If a number contains only odd factors, it will be odd (and will have a remainder of 1 when divided by 2). If a number contains at least one even factor, it will be even (and divisible by 2).
15 = 3*5 {only odd factors; not divisible by 2; remainder of 1}
21 = 3*7 {only odd factors; not divisible by 2; remainder of 1}
63 = 3*3*7 {only odd factors; not divisible by 2; remainder of 1}

30 = 3*5*2 {contains an even factor; divisible by 2}
42 = 3*7*2 {contains an even factor; divisible by 2}
50 = 5*5*2 {contains an even factor; divisible by 2}
Simply because "N contains odd numbers as factors" does not mean that all of N's factors are odd. Consequently, it is entirely possible that N contains an even factor, in which case N is even and N is divisible by 2. Possible values for N:
18 = 2*3*3 {contains odd factors, but is divisible by 2; remainder = 0}
30 = 2*5*3 {contains odd factors, but is divisible by 2; remainder = 0}
But:
27 = 3*3*3 {contains odd factors, but is not divisible by 2; remainder = 1}
15 = 3*5 {contains odd factors, but is not divisible by 2; remainder = 1}
Since some values of N that meet the conditions of Statement (1) are divisible by 2 while other values that also meet the conditions of Statement (1) are not divisible by 2, Statement (1) does not provide sufficient information to definitively determine whether N is divisible by 2.
Statement (1) alone is NOT SUFFICIENT.
Evaluate Statement (2) alone.
Since "N is a multiple of 15", possible values for N include:
15, 30, 45, 60, 75, 90
Possible values for N give different remainders when divided by 2:
15/2 → Remainder = 1
30/2 → Remainder = 0
45/2 → Remainder = 1
60/2 → Remainder = 0
75/2 → Remainder = 1
90/2 → Remainder = 0
Since different legitimate values of N give different remainders when divided by 2, Statement (2) is not sufficient for determining the remainder when N is divided by 2.
Statement (2) alone is NOT SUFFICIENT.
Evaluate Statements (1) and (2).
Since "N is a multiple of 15" and "N contains odd numbers as factors", possible values for N include:
15, 30, 45, 60, 75, 90
Adding Statement (1) to Statement (2) does not provide any additional information since any number that is a multiple of 15 must also have odd numbers as factors.
Possible values for N give different remainders when divided by 2:
15/2 → Remainder = 1
30/2 → Remainder = 0
45/2 → Remainder = 1
60/2 → Remainder = 0
75/2 → Remainder = 1
90/2 → Remainder = 0
Since different legitimate values of N give different remainders when divided by 2, Statements (1) and (2) are not sufficient for determining the remainder when N is divided by 2.
Statements (1) and (2), even when taken together, are NOT SUFFICIENT.
Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer E is correct.

Step 1: Decode the Question Stem and Get Clarity
Q1. What kind of an answer will the question fetch?
An "Is" question will fetch an "Yes" or a "No" as an answer.
The data provided in the statements will be considered sufficient if the question is answered with a conclusive Yes or a conclusive No.

Q2. When is the answer an "Yes"?
If mn > nn, the answer to the question is a conclusive Yes.

Q3. When is the answer a "No"?
If mn ≤ nn, the answer to the question is a conclusive No.
Note: When mn = nn, the answer is No.

Q4. What values can m and n take?
From the information available from the question stem, both m and n can take only integer values.
So, we need not worry about values such as 0.5 or 1.2.
However, both m and n can be either positive or negative. Neither can be 0.

Step 2: Evaluate Statement 1 ALONE
Statement 1: |m| = n
We can infer the following information about m and n from the statement.

The modulus of a number is always positive. n = |m|. Hence, n is positive
m can take either positive or negative values.
Example: Let m = -3 and n = 3. |m| = n holds good.
(-3)3 < 33. So, the answer to the question is NO.

Counter Example: Let m = -2 and n = 2. |m| = n still holds good.
(-2)2 = 22. So, the answer to the question is NO.

Notice that we are not able to come up with a counter example. Both examples returned NO as answer.
Not finding a counter example might be our limitation. Let us reason why we seem to be getting NO as answer and will it hold good for all values satisfying statement 1.

When a negative number is raised to an odd power, the result is negative. So, LHS < RHS. Answer is NO.
When a negative number is raised to an even power, the result is positive. So, LHS = RHS. It is still not greater. So, the answer will still be NO.

The values that m and n takes based on statement 1 gives a conclusive answer to the question.
Hence, statement 1 is sufficient.
Eliminate answer options B, C, and E.

Step 3: Evaluate Statement 2 ALONE
Statement 2: m < n
We need to check whether we get a conclusive Yes or No using this statement to determine whether statement 2 alone is sufficient.
Let us look for counter examples

Example: Let m = 2 and n = 3
23 < 33. So, the answer to the question is NO.

Counter Example: Let m = -3 and n = 2
(-3)2 > 22. So, the answer to the question is YES.

The values that m and n takes based on statement 2 do not give a conclusive answer to the question.
Hence, statement 2 is not sufficient.
Eliminate answer option D.

Choice A is the correct answer.

Step 1: Decode the Question Stem and Get Clarity
Q1. What kind of an answer will the question fetch?
The question asks us to find the remainder when x is divided by 6.
The data provided in the statements will be considered sufficient if we get a unique value for the remainder.

Q2. When is the data not sufficient?
If after using the information given in the statements, we are not able to determine a unique remainder when x is divided by 6, the data given in the statements is not sufficient to answer the question.

Q3. What information do we have about x from the question stem?
x is a positive integer.

Step 2: Evaluate Statement 1 ALONE
Statement 1: When x is divided by 7, the remainder is 5.

Approach 1: x can therefore, be expressed as 7k + 5
If k = 0, x = 5. The remainder when x is divided by 6 will be 5.
If k = 1, x = 12. The remainder when x is divided by 6 will be 0.
The remainder varies as the value of k varies.

Approach 2: List numbers (about 10 to 12) that satisfy the condition given in statement 1 and check whether you get a unique remainder.
x could be 5, 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89 .....
The remainders that we get correspondingly are 5, 0, 1 and so on
As seen with Approach 1, we are not getting a unique remainder.

We are not able to get a unique remainder using statement 1.
Hence, statement 1 is not sufficient.
Eliminate answer options A and D.

Step 3: Evaluate Statement 2 ALONE
Statement 2: When x is divided by 9, the remainder is 3.

Approach 1:: x can be expressed as 9p + 3
If p = 0, x = 3. Hence, the remainder when x is divided by 6 will be 3.
If p = 1, x = 12. The remainder when x is divided by 6 will be 0.
The remainder varies as the value of k varies.

Approach 2:: List numbers (about 10 to 12) that satisfy the condition given in statement 2 and check whether you get a unique remainder.
x could be 3, 12, 21, 30, 39, 48, 57, 66, 75, 84, 93 .....
The remainders that we get correspondingly are 3, 0, 3 and so on.
As seen with Approach 1, we are not getting a unique remainder.

We are not able to get a unique remainder using statement 2.
Hence, statement 2 is not sufficient.
Eliminate answer option B.

Step 4: Evaluate Statements TOGETHER
Statements: When x is divided by 7, the remainder is 5 & When x is divided by 9, the remainder is 3.

Approach 1:: Equating information from both the statements, we can conclude that 7k + 5 = 9p + 3.
Or 7k = 9p - 2.
i.e., 9p - 2 is a multiple of 7. When p = 1, 9p - 2 = 7. So, one instance where the conditions are satisfied is when k = 1 and p = 1.
x will be 12 and the remainder when x is divided by 6 is 0.

When p = 2, 9p - 2 is not divisible by 7. Proceeding by incrementing values for p, when p = 8, 9p - 2 = 70, which is divisible by 7.
When p = 8, x = 75.
The remainder when 75 is divided by 6 is 3.
The remainder when x was 12 was 0. The remainder when x is 75 is 3.

Approach 2:: The values of x that satisfy statement 1 are 5, 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89 ...
The values of x that satisfy statement 2 are 3, 12, 21, 30, 39, 48, 57, 66, 75, 84, 93 .....
Values of x that are present in both sets are 12, and 75 from the set listed above.
The remainders when 12 and 75 are divided by 6 are 0 and 3 respectively.

We are not able to get a unique remainder despite combining the two statements, the data provided is NOT sufficient.
Hence, statements together are not sufficient.
Eliminate answer option C.

Choice E is the correct answer.

Step 1: Decode the Question Stem and get clarity
Q1. What kind of an answer will the question fetch?
The question asks us to find the change in per kilogram cost of the candy in cents after the weight of each candy was reduced.
The data provided in the statements will be sufficient if we get a unique value in cents.

Q2. When is the data not sufficient?
If after using the information given in the statements, we are not able to determine a unique value in cents for the change in cost per kilogram of the candy, the data is NOT sufficient.

Let us assign the following variables.
Let the initial cost per kilogram be x cents; let the cost per kilogram after reducing the weight be y cents, where y > x.
We need to find (y − x).

Step 2: Evaluate Statement 1 ALONE
Statement 1: The weight of each piece of candy bar reduced by 9 grams.

We do not know either x or y from this statement.
We cannot find a unique value for (y - x).

Hence, statement 1 is not sufficient.
Eliminate answer options A and D.

Step 3: Evaluate Statement 2 ALONE
Statement 2: The weight of each piece of candy bar reduced by 9%.

All that we can deduce is that the new weight of the candies is 9% lesser than its original weight or that the new weight is 91% of the original weight.
Statement 2 also does not provide us with either x or y.

We are not able to get a unique value for (y - x) using statement 2, statement 2 is also NOT Sufficient.
Hence, statement 2 is not sufficient.
Eliminate answer options B.

Step 4: Evaluate Statements TOGETHER
Statements: "The weight of each piece of candy bar reduced by 9 grams" and "The weight of each piece of candy bar reduced by 9%"

9% reduction is 9 grams
We can deduce that the weight of each candy bar was 100 grams before the reduction.
We still do not have any information on x and y.

We are not able to get a unique value for (y - x) despite combining the two statements, the data provided is NOT sufficient..
Hence, statements together are not sufficient.
Eliminate answer option C.

Choice E is the correct answer.

Step 1: Decode Question Stem and Get Clarity
Q1. When is the data sufficient?
If we are able to get a UNIQUE value for the area of the triangle from the information given in the statements, the data is sufficient.

Q2. What should you watch out for?
1. Check to see whether more than one set of values are possible with the given data. For e.g., more than one right triangle can be described for a given hypotenuse of 10 units.
2. Remember that sides of a triangle need not necessarily be integers. Particularly, one might be tempted to think that only triangles whose sides are Pythagorean triplets are right triangles. All Pythagorean triplets are right triangles. But all right triangles need not be Pythagorean triplets. For instance 1, 1, √2 will form sides of a right triangle, where one of the sides is not an integer.

Step 2: Evaluate Statement 1 ALONE
Statement 1: a = 4

It is not possible to find the area of a right triangle with information about just one of its sides and no other information about the triangle.

We are not able to find the area using statement 1.
Hence, statement 1 is not sufficient.
Eliminate answer option A and D.

Step 3: Evaluate Statement 2 ALONE
Statement 2: a + b + c = 12

The temptation is to conclude that the sides are 3, 4, and 5. These values will form sides of a right triangle and the sum of a, b, and c is 12.

But before concluding that we have a definite answer, let us check whether any right triangle other than 3, 4, and 5 is possible.

For instance, it is possible to have a right isosceles triangle whose perimeter is 12.
The area of that triangle will be different from that of the triangle with sides 3, 4, and 5.

We are not able to find a UNIQUE area using statement 2.
Hence, statement 2 is not sufficient.
Eliminate answer option B.

Step 4: Evaluate Statements TOGETHER
Statements: a = 4 & a + b + c = 12

In a right triangle, the longest side is the hypotenuse.
Side that measures 4 is therefore, not the hypotenuse as 4 is the average of the 3 sides.
The hypotenuse has to necessarily be greater than the average of the 3 sides.
So, either b or c is the hypotenuse.

Let us assume c to be the hypotenuse. So, 4 and b are the two perpendicular sides of the right triangle.
4 + b + c = 12. So, b + c = 8. Therefore, b = 8 - c

Applying Pythagoras Theorem, c2 = 42 + (8 - c)2
Solving for c, c2 = 16 + 64 - 16c + c2
16c = 80. So, c = 5.
If c = 5, b = 8 - c = 3
Using the two statements, we could get a unique set of values for a, b, and c.
Hence, we will be able to find the area of the triangle.

Statements together are sufficient to find a UNIQUE value as the area of the right triangle.
Eliminate answer option E.

Choice C is the correct answer.

Step 1: Decode the Question Stem and Simplify it
What does "Is |x| > x?| mean?
The modulus of a number gives the magnitude of that number.
Substitute a positive value for x: |x| = x
Substitute zero for x: |x| = x
Substitute a negative value for x: |x| > x
So, the question ultimately boils down to Is x < 0?

Step 2: Evaluate Statement 1 ALONE
Statement 1: x² + y² = 4
Squares of real numbers are non-negative.
So, both x² and y² are non negative.
So, x² could be 0 or positive.

If x² = 0, x is 0.
Answer to the question "Is x < 0?" is NO.

If x² is positive, x may be positive or negative.
Answer to the question "Is x < 0?" is NO or YES

We are not able to find a conclusive answer to the question using statement 1 ALONE.
Statement 1 alone is NOT Sufficient.
Eliminate answer options A and D.

Step 3: Evaluate Statement 2 ALONE
Statement 2: x³ + y² = 0
Squares of real numbers are non-negative. So, y² is definitely not a negative number.

Two possibilities exist for x³ and y²
Possibility 1: Both x³ and y² are 0.
If x³ = 0, the value of x = 0.
The answer to the question "Is x < 0?" is NO.

Possibility 2: x³ is negative and y² is positive.
If x³ < 0, x < 0
The answer to the question "Is x < 0?" is YES.

We are not able to find a conclusive answer to the question using statement 2 ALONE.
Statement 2 alone is NOT Sufficient.
Eliminate answer option B.

Step 4: Evaluate Statements TOGETHER
Statement 1: x2 + y² = 4
Statement 2: x³ + y² = 0

From Statement 1, if x = 0, y² = 4
And from statement 2, if x = 0, y² = 0.
So, if x = 0, the statements contradict each other.
So, x cannot be 0.

Therefore, y² has to be positive and x³ has to be negative to satisfy both statements.
If x³ < 0, we can deduce that x < 0.
Answer to the question "Is x < 0?" is a conclusive YES.

We are able to answer the question by combining the two statements.

Choice C is the correct answer.

Step 1: Evaluate Statement 1 ALONE
Statement 1: All 3 sides are integers.
The ratio of the sides of an isosceles right triangle is 1 : 1 : √2
Either the sides with ratio 1 or the side with ratio √2 is not an integer.
From Statement 1 we know all sides are integers. If all three sides are integers, such a right triangle cannot be isosceles.

We are able to answer the question a conclusive NO using statement 1 ALONE.
Statement 1 ALONE is SUFFICIENT.
Eliminate answer options B, C, and D.

Step 2: Evaluate Statement 2 ALONE
Statement 2: The square of the hypotenuse is twice the product of the other two sides
Let the sides be x, y, and z with z as the hypotenuse.
By Pythagoras theorem: x² + y² = z²
From statement 2: z² = 2xy
⇒ x² + y² = 2xy
⇒ x² + y² - 2xy = 0
⇒ (x - y)² = 0
⇒ x – y = 0 ⇒ x = y

If x and y are same, the triangle is an isosceles right triangle.

We are able to answer the question a conclusive YES using statement 2.

Each statement is INDEPENDENTLY sufficient to answer the question.

Choice D is the correct answer.

Step 1: Decode the Question Stem
Is y > x/6?
We can rewrite the question as "Is 6y > 6?"

Step 2: Evaluate Statement 1 ALONE
Statement 1: 20 < x < 30
x takes values from 21 to 29.
For all these values of x, y = 2 × 2 = 4. So, 6y = 24

Approach: Counter Example
Example: x = 21, 6y is greater than x. Answer: Yes
Counter example: x = 25, 6y is not greater than x. Answer: No
Answer to the question "Is 6y > x?" is sometimes YES and sometimes NO.

We are not able to find a conclusive answer to the question using statement 1 ALONE.
Statement 1 alone is NOT Sufficient.
Eliminate answer options A and D.

Step 3: Evaluate Statement 2 ALONE
Statement 2: y = 10
So, the tens place of x is 5.
Values that x can take are 50 ≤ x ≤ 59
Because y = 10, 6y = 60
For all values of x in the range 50 ≤ x ≤ 59, 6y > x
The answer to the question "Is 6y > x?" is a conclusive YES.

We are able to answer the question a definite YES using statement 2 ALONE.
Statement 2 AlONE is Sufficient.
Eliminate answer options C and E.

Choice B is the correct answer.

Statement 1: The intercepts of Line K, perpendicular to L, are of the opposite signs.
Inference 1: If the intercepts of line K are of opposite signs, line K is a positive sloping line.
Inference 2: So, we can infer that line L is a negative sloping line.

Possibility 1: Negative sloping Line L with a positive x-intercept

 

In this possibility, answer to the question is YES

Possibility 2: Negative sloping Line L with a negative x-intercept

In this possibility, answer to the question is NO
Because both possibilities exist, we are not able to answer the question using statement 1.
Statement 1 alone is NOT sufficient.
Eliminate answer options A and D.
Step 2: Evaluate Statement 2 ALONE
Statement 2: Line L passes through the 4th quadrant.
Possibility 1: Line L passes through IV quadrant. Has a positive x-intercept.

 

In this possibility, answer to the question is YES

Possibility 2: Line L passes through IV quadrant. Has a negative x-intercept

 

In this possibility, answer to the question is NO

Because both possibilities exist, we are not able to answer the question using statement 2.

Statement 2 alone is NOT sufficient.
Eliminate answer option B.

Step 3: Evaluate Statements TOGETHER
Statement 1: The intercepts of Line K, perpendicular to L, are of the opposite signs.
Statement 2: Line L passes through the 4th quadrant.

Key Inference from Statement 1: Line L is a negative sloping line.

Possibility 1: Negative Sloping Line L passes through IV quadrant. Has a positive x-intercept.

 

In this possibility, answer to the question is YES

Possibility 2: Negative Sloping Line L passes through IV quadrant. Has a negative x-intercept

In this possibility, answer to the question is NO
Because both possibilities exist, we are not able to answer the question despite combining the statements.
Statement TOGETHER are NOT sufficient.
Eliminate answer option C.
Choice E is the correct answer.