Test: Arithmetic - GMAT MCQ

Statement (1): X > Y > Z
This statement provides information about the relative values of X, Y, and Z, but it doesn't give us any specific values or equations. We can't determine the exact values of X, Y, or Z from this statement alone. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): None of them is equal to 4
This statement tells us that X, Y, and Z are not equal to 4. It doesn't provide any other information about their values or relationships. Again, we can't determine the exact values of X, Y, or Z from this statement alone. Thus, statement (2) alone is not sufficient to answer the question.

Combining the statements:
By combining the statements, we still don't have any specific values or equations. The relative ordering provided in statement (1) doesn't help us determine the value of Z, and statement (2) only rules out the value 4 for X, Y, and Z. Therefore, even when considered together, the statements do not provide enough information to determine the value of Z.

Hence, the correct solution is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): There are a total of five people at the table, including Byrne.

This statement tells us that the total number of people in the group is five, including Byrne. However, it does not provide any information about the gender distribution or how much each person spent. Therefore, we cannot determine the number of men in the group based on this statement alone.

Statement (2): The women order meals that cost an average of $19, and the men order meals that cost an average of $27.

This statement gives us information about the average cost of meals for women and men separately. However, it does not provide the total number of people in the group or the total amount spent. Without knowing the total amount spent or the number of people, we cannot determine the number of men in the group based on this statement alone.

Since neither statement alone is sufficient to answer the question, we need to consider both statements together:

By combining both statements, we know that there are five people in total (including Byrne) and that the average cost of meals for women is $19, while the average cost for men is $27. However, we still don't have enough information to determine the exact number of men in the group.

For example, it's possible that there are three women and two men in the group, or two women and three men. In both cases, the total number of people and the average costs of meals for women and men would be consistent with the given information. Therefore, even when considering both statements together, we cannot determine the number of men in the group.

Since both statements together are not sufficient to answer the question, the correct answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Let's analyze each statement separately:

Statement (1): x = (3y)/4
This statement provides a relationship between x and y, stating that x is equal to three-fourths of y. However, it does not provide any information about the specific values of x or y. Without knowing the value of y, we cannot determine the value of x. Thus, statement (1) alone is not sufficient.

Statement (2): y = 2
This statement gives us a specific value for y, which is 2. However, it does not provide any information about the relationship between x and y or the value of x. Without knowing the value of x or the relationship between x and y, we cannot determine the value of x. Thus, statement (2) alone is not sufficient.

Now let's consider both statements together:

Combining both statements, we have:
x = (3y)/4 (from statement 1)
y = 2 (from statement 2)

We can substitute the value of y from statement 2 into the equation from statement 1:
x = (3(2))/4
x = 6/4
x = 3/2

By combining both statements, we were able to determine the value of x as 3/2. Therefore, both statements together are sufficient to answer the question.

Hence, the correct answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

To determine if the sum of 7x and 7y is divisible by 14, we can simplify the expression. Since both terms have a common factor of 7, we can factor it out:

7x + 7y = 7(x + y)

So, the question becomes whether 7(x + y) is divisible by 14.

Now let's analyze the given statements:

Statement (1): x = y
If x = y, then we can substitute y for x in the expression 7(x + y) and get 7(2y) = 14y. Since 14y is divisible by 14, we can conclude that the sum is divisible by 14. Statement (1) alone is sufficient to answer the question.

Statement (2): x and y are both even integers
If both x and y are even integers, then we can write x = 2a and y = 2b, where a and b are integers. Substituting these values into the expression 7(x + y), we get 7(2a + 2b) = 14(a + b). Since 14(a + b) is divisible by 14, we can conclude that the sum is divisible by 14. Statement (2) alone is also sufficient to answer the question.

Therefore, each statement alone is sufficient to determine that the sum of 7x and 7y is divisible by 14. The answer is (D).

We are given that at least 10 cars have tinting windows and fog lights. Let's denote the number of cars with tinting windows as "T" and the number of cars with fog lights as "F."

From the information given, we can deduce the following:

• 40% of cars with tinting windows also have fog lights: 0.4T.
• 80% of cars with fog lights also have tinting windows: 0.8F.
• The total number of cars with tinting windows or fog lights or both is 52.

Now, let's analyze the statements:

Statement (1): 80% cars which have fog light also have tinting windows.

This statement alone does not provide us with any specific information about the number of cars with tinting windows or fog lights. We cannot determine if the number of cars with tinting windows is larger than the number of cars with fog lights based on this statement alone. Therefore, statement (1) is not sufficient to answer the question.

Statement (2): 52 cars have tinting windows or fog light or both.

This statement tells us the total number of cars with tinting windows or fog lights or both, which is 52. However, it doesn't provide any information about the specific number of cars with tinting windows or fog lights individually. Therefore, statement (2) alone is not sufficient to answer the question.

By examining both statements together, we can combine the information from both statements and determine the answer:

Combining the information from both statements, we know that there are 52 cars with tinting windows or fog lights or both. However, we still don't have enough information to determine the specific number of cars with tinting windows or fog lights individually.

Therefore, the correct answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): The tens digit of x is one less than the tens digit of z.

This statement provides information about the relationship between the tens digits of x and z. Let's assume the tens digit of x is represented by 'a' and the tens digit of z is represented by 'b'. According to statement (1), we have the equation:

a = b - 1

However, this equation alone does not provide enough information to determine the value of y.
For example, if x = 205 and z = 216, then a = 2 and b = 3.
In this case, there are multiple possible values for y, such as 13 or 22.
Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): The sum of digits of y is equal to the tens digit of y.

This statement provides information about the relationship between the sum of digits and the tens digit of y.
Let's assume the tens digit of y is represented by 'c'.
According to statement (2), we have the equation:

c = sum of digits of y

However, this equation alone does not provide enough information to determine the value of y.
For example, if c = 3, there are multiple possible values for y that satisfy this condition, such as 12, 21, or 30.
Therefore, statement (2) alone is not sufficient to answer the question.

By analyzing the statements together, we can combine the information from both statements.
However, even when considering both statements, we still cannot determine the exact value of y.
Both statements provide some constraints on the possible values of y, but they do not uniquely determine its value.

Therefore, the correct answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

To determine whether the hundreds' digit of number "a" is equal to the sum of the hundreds' digits of numbers "b" and "c," let's evaluate the two given statements:

Statement (1): Tens' digit of "a" = tens' digit of "b" + tens' digit of "c."

This statement alone does not provide any information about the hundreds' digit. The tens' digit and hundreds' digit are independent of each other, so we cannot conclude whether the hundreds' digit of "a" is equal to the sum of the hundreds' digits of "b" and "c" based on this statement alone.

Statement (2): Units' digit of "a" = units' digit of "b" + units' digit of "c."

Similar to statement (1), this statement also does not provide any information about the hundreds' digit of the numbers. The units' digit and hundreds' digit are unrelated, so we cannot determine whether the hundreds' digit of "a" is equal to the sum of the hundreds' digits of "b" and "c" based on this statement alone.

Since neither statement alone provides the required information, we need to evaluate both statements together.

By considering both statements, we can determine the relationship between the hundreds' digit of "a" and the hundreds' digits of "b" and "c." However, we do not have any information about the tens' and units' digits, which are necessary to solve the problem.

Therefore, the correct solution is (A): Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.

To determine the number of elements in set A, we need to evaluate whether the sum of elements in set A is less than 6. Let's analyze each statement separately:

Statement (1): The average of the greatest 3 elements is 150.
This statement provides information about the three largest elements in set A, but it doesn't provide any details about the remaining elements. We cannot determine the number of elements or their values based on this statement alone.

Statement (2): No elements of set A are less than 125.
This statement gives a lower bound for the elements in set A. It guarantees that all elements in set A are greater than or equal to 125. However, it doesn't provide any upper bound or information about the sum of the elements.

Now let's consider both statements together:
From statement (2), we know that all elements in set A are at least 125. If the number of elements in set A is less than 6, the minimum sum of the elements would be 125 * 6 = 750. However, statement (1) states that the sum of the elements in set A is 700, which is less than the minimum sum. Therefore, it is impossible for the number of elements in set A to be less than 6.

Since statement (2) alone is sufficient to answer the question, but statement (1) alone is not, the correct solution is (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

To determine the number of positive values among x, y, and z, we need to analyze both statements:

Statement (1): x + y + z = 20

This statement provides the sum of the three numbers.
However, it does not give us any specific information about the individual values of x, y, and z.
For example, the numbers could be (-5, 10, 15), which would result in only one positive number.
Alternatively, they could be (7, 7, 6), resulting in three positive numbers.
Therefore, statement (1) alone is not sufficient to determine the number of positive values.

Statement (2): x + y = 14

This statement gives the sum of two numbers, but it doesn't provide any information about the value of z.
Again, we can have different scenarios.
For instance, if x = 10 and y = 4, both numbers are positive.
However, if x = 20 and y = -6, only one number would be positive.
Thus, statement (2) alone is also not sufficient to determine the number of positive values.

Combining both statements, we can try to find a unique solution.
From statement (2), we have x + y = 14.
If we subtract this equation from statement (1) (x + y + z = 20), we get z = 20 - 14 = 6.
However, this still does not give us enough information to determine the number of positive values.
For example, x = 3, y = 11, and z = 6 would result in two positive numbers, while x = 15, y = -1, and z = 6 would only give us one positive number.

Since neither statement alone nor both statements together provide enough information to determine the number of positive values among x, y, and z, the correct answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

To determine whether Mrs. K's total compensation for the three late contracts was reduced by more than 11%, let's analyze each statement separately:

Statement (1): Without any reduction, she would have received $550 for the last of the three late contracts, and at least $1200 for each of the others.

This statement provides information about the contract prices but does not provide any details about the number of contracts completed in a month or the compensation reduction rates. It does not give enough information to determine whether the total compensation was reduced by more than 11%.

Statement (2): Without any reduction, she would have received $1500 for the first of the three late contracts.

Similar to statement (1), this statement only provides information about the contract price for the first late contract. It does not provide information about the number of contracts completed in a month or the compensation reduction rates. Thus, it is also insufficient to determine whether the total compensation was reduced by more than 11%.

Since neither statement alone provides enough information to answer the question, the correct answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

To fully answer the question, we would need additional information about the number of contracts completed in a month, the reduction rates for subsequent contracts, and the prices for all three contracts.