GMAT Math : Arithmetic
Study concepts, example questions & explanations for GMAT Math
All GMAT Math Resources
Example Questions
Example Question #1173 : Data Sufficiency Questions
When assigning a score for the term, a professor takes the mean of all of a student's test scores.
Joe is trying for a score of 90 for the term. He has one test left to take. What is the minimum that Joe can score and achieve his goal?
Statement 1: He has a median score of 85 so far.
Statement 2: He has a mean score of 87 so far.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Knowing the median score is neither necessary nor helpful. What will be needed is the sum of the scores so far and the number of tests Joe has taken. But the number of tests taken is not given, and without this, there is no way to find the sum either.
Example Question #1174 : Data Sufficiency Questions
What is the mean of 
Statement 1:
Statement 2:
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
The mean of a data set requires you to know the sum of the elements and the number of elements; you know the latter, but neither statement alone provides any clues to the former.
However, if you know both, you can add both sides of the equations as follows:
Rewrite as:
and divide by 9:
Now you know the sum, so divide it by 6 to get the mean:
Example Question #1175 : Data Sufficiency Questions
A meteorologist is attempting to calculate the average (arithmetic mean) temperature highs for the past week. What is the arithmetic mean of the high temperatures for the last week?
1. The mean high temperature for the past 3 days is 75 degrees. The mean high temperature for the 4 days before that was 80 degrees.
2. The high temperatures for the last 7 days are as follows (in degrees): 80, 81, 79, 80, 77, 75, 73
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
EACH statement ALONE is sufficient.
Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.
EACH statement ALONE is sufficient.
Statement 1 can be used to find the arithmetic mean by combining the information. For statement 1, the average for the last seven days is found by reversing the arithmetic mean equation. Let 
Also, we know 
So
Statement 2 can be used to find the arithmetic mean using the arithmetic mean formula. This is the total sum, divided by the number of days. Thus,
Example Question #1176 : Data Sufficiency Questions
If 
It is impossible to tell from the information given.
Multiply the second equation by 2 on each side, and add it to the first equation.
Divide this sum by 5:
Example Question #1177 : Data Sufficiency Questions
Choose the answer that best describes sufficient data to solve the problem.
3 numbers are given in increasing order. The arithemetic mean of the first two is 5 less than the arithmetic mean of all three. The sum of the first two numbers is equal to the arithmetic mean of the last two. What is the first number?
I. The second number is given.
II. The arithmetic mean of the first and third numbers is given.
Statement I is sufficient to solve the problem, but Statement II alone is not.
Statement II is sufficient to solve the problem, but Statement I alone is not.
Both statements together are sufficient to solve the problem.
Neither statement is sufficient to solve the problem (additional information is needed).
Either statement alone is sufficient to solve the problem.
Statement I is sufficient to solve the problem, but Statement II alone is not.
3 numbers are given in increasing order.
I. The second number is given.
II. The arithmetic mean of the first and third numbers is given.
We are given a system of equations by the prompt:
The arithemetic mean of the first two is 5 less than the arithmetic mean of all three.
The sum of the first two numbers is equal to the arithmetic mean of the last two.
Combining these two, we get:
Thus, the second statement doesn't actually provide us with any information (we are still left with 2 equations and 3 variables, which cannot be solved for any particular number).
On the other hand, if y is determined, then 
So the first gives sufficient data while the second does not.
Example Question #1171 : Data Sufficiency Questions
1. If the arithmetic mean of five different numbers is 50, how many of the numbers are greater than 50?
(1) None of the five numbers is greater than 100.
(2) Three of the five numbers are 24, 25 and 26, respectively.
Statements (1) and (2) TOGETHER are NOT sufficient.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
EACH statement ALONE is sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
For statement (1), there are different combinations that satisfy the condition. For example, the five numbers can be 
For statement (2), even though we know three of them, the two unknown numbers can both be greater than 
Putting the two statements together, we know that the sum of the two unknown numbers is
Since none of them is greater than 100, both of them have to be greater than 50. Therefore when we combine the two statements, we know that there are two numbers that are greater than 50.
Example Question #1172 : Data Sufficiency Questions
Data sufficiency question
Determine the mean of a number set.
1. The mode is 7.
2. The median is 7.
Each statement alone is sufficient
Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient
Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question
Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question
Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question
The median gives information about the center of a set of numbers, but is insufficient for calculating the mean. Additionally, the mode merely indicates which number is the most repeated value. Therefore, more information is required to calculate the mean.
Example Question #12 : Dsq: Calculating Arithmetic Mean
What is the value of 
(1)
(2) The arithmetic mean of the numbers in the list is 
Both statements TOGETHER are not sufficient.
Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient.
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Each statement ALONE is sufficient.
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) provides the value of y in terms of x, which is not enough to determine the value of x in the list we are given.
Statement (2) gives the arithmetic mean of the list. We can the write the following equation:
However, we cannot find the value of x using the information in Statement (2) only.
Using the information in Statement (1), we can replace y by x-4 in the previous equation:
We need both statements to calculate the value of x.
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Example Question #13 : Dsq: Calculating Arithmetic Mean
A professor records the average class grade for each exam. The average class grades for the semester are respectively:
What is the average class grade for the semester?
(1)
(2)
Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
Each statement ALONE is SUFFICIENT
Both statements TOGETHER are not sufficient.
Each statement ALONE is SUFFICIENT
The mean class grade is the sum of all average class grades divided by the number of grades:
Using Statement (1):
Therefore Statement (1) is sufficient to calculate the arithmetic mean for these grades.
Using statement (2):
Therefore
Therefore Statement (2) is sufficient to calculate the arithmetic mean for these grades.
Each Statement ALONE IS SUFFICIENT to answer the question
Example Question #34 : Descriptive Statistics
During a particularly hectic week at Ballard High, Robert drank 5, 8, 3, 6, 2, 9, and 14 cans of Slurp Soda, respectively, on each of the 7 days. What is the product of Robert's mean and median soda consumption for that week?
To find the mean, we find the sum of all of the values, and divide by how many there are:
To find the mean we rearrange the values in ascending numerical order and select the middle value:
The product, then, is
All GMAT Math Resources
