SAT Test Prep

CHAPTER 9
SPECIAL MATH PROBLEMS

Lesson 2: Mean/Median/Mode Problems

Average (Arithmetic Mean) Problems


Just about every SAT will include at least one question about averages, otherwise known as arithmetic means. These won’t be simplistic questions like “What is the average of this set of numbers?” You will have to really understand the concept of averages beyond the basic formula.


You probably know the procedure for finding an average of a set of numbers: add them up and divide by how many numbers you have. For instance, the average of 3, 7, and 8 is . You can describe this procedure with the “average formula”:

Since this is an algebraic equation, you can manipulate it just like any other equation, and get two more formulas:


All three of these formulas can be summarized in one handy little “pyramid”:


This is a great tool for setting up tough problems. To find any one of the three quantities, you simply need to find the other two, and then perform the operation between them. For instance, if the problem says, “The average (arithmetic mean) of five numbers is 30,” just write 30 in the “average” place and 5 in the “how many” place. Notice that there is a multiplication sign between them, so multiply  to find the third quantity: their sum.

Medians


median is something that splits a set into two equal parts. Just think of the median of a highway:
it splits the highway exactly in half. The median of a set of numbers, then, is the middle number when they are listed in increasing order. For instance, the median of {–3, 7, 65} is 7, because the set has just as many numbers bigger than 7 as less than 7. If you have an even number of numbers, like {2, 4, 7, 9}, then the set doesn’t have one “middle” number, so the median is the average of the two middle numbers. (So the median of {2, 4, 7, 9} is 


When you take standardized tests like the SAT, your score report often gives your score as a percentile, which shows the percentage of students whose scores were lower than yours. If your percentile score is 50%, this means that you scored at the median of all the scores: just as many (50%) of the students scored below your score as above your score.


The average (arithmetic mean) and the median are not always equal, but they are equal whenever the numbers are spaced symmetrically around a single number.


Example:

Consider any set of numbers that is evenly spaced, like 4, 9, 14, 19, and 24:

Notice that these numbers are spaced symmetrically about the number 14. This implies that the mean and the median both equal 14. This can be helpful to know, because finding the median of a set is often much easier than calculating the mean.

Modes


Occasionally the SAT may ask you about the mode of a set of numbers. A mode is the number that appears the most frequently in a set. (Just remember: MOde = MOst.) It’s easy to see that not every set of numbers has a mode. For instance, the mode of [-3, 4, 4, 1, 12] is 4, but [4, 9, 14, 19, 24] doesn’t have a mode.


Concept Review 2: Mean/Median/Mode Problems

1. Draw the “average pyramid.”

2. Explain how to use the average pyramid to solve a problem involving averages.

3. Define a median.

4. Define a mode.

5. In what situations is the mean of a set of numbers the same as its median?

6. The average (arithmetic mean) of four numbers is 15. If one of the numbers is 18, what is the average of the remaining three numbers?

7. The average (arithmetic mean) of five different positive integers is 25. If none of the numbers is less than 10, then what is the greatest possible value of one of these numbers?

8. Ms. Appel’s class, which has twenty students, scored an average of 90% on a test. Mr. Bandera’s class, which has 30 students, scored an average of 80% on the same test. What was the combined average score for the two classes?

SAT Practice 2: Mean/Median/Mode Problems

1. If , what is the average (arithmetic mean) of 2x, 2x, y, and 3y, in terms of x?

(A) 2x

(C) 3 x

2. The average (arithmetic mean) of seven integers is 11. If each of these integers is less than 20, then what is the least possible value of any one of these integers?

(A) –113

(B) –77

(C) –37

(D) –22

(E) 0

3. The median of 8, 6, 1, and k is 5. What is k?

4. The average (arithmetic mean) of two numbers is z. If one of the two numbers is x, what is the value of the other number in terms of x and z?

5. A set of n numbers has an average (arithmetic mean) of 3k and a sum of 12m, where k and m are positive. What is the value of n in terms of k and m?

(E) 36 km

6. The average (arithmetic mean) of 5, 8, 2, and k is 0. What is the median of this set?

(A) 0

(B) 3.5

(C) 3.75

(D) 5

(E) 5.5

7. A die is rolled 20 times, and the outcomes are as tabulated above. If the average (arithmetic mean) of all the rolls is a, the median of all the rolls is b, and the mode of all the rolls is c, then which of the following must be true?

(A) I only

(B) II only

(C) I and II only

(C) I and II only

(D)II and III only

(E) I, II, and III only

8. If a 30% salt solution is added to a 50% salt solution, which of the following could be the concentration of the resulting mixture?

I. 40%

II. 45%

III. 50%

(A) I only

(B) I and II only

(C) I and III only

(D) I and III only

(E) I, II, and III

9. Set A consists of five numbers with a median of m. If Set B consists of the five numbers that are two greater than each of the numbers in Set A, which of the following must be true?

I. The median of Set B is greater than m..

II. The average (arithmetic mean) of Set B is greater than m..

III. The greatest possible difference between two numbers in Set B is greater than the greatest possible difference between two numbers in Set A.

(A) I only

(B) I and II only

(C) I and III only

(D) I and III only

(E) I, II, and III

Answer Key 2: Mean/Median/Mode Problems

Concept Review 2

1. It should look like this:

2. When two of the three values are given in a problem, write them in the pyramid and perform the operation between them. The result is the other value in the pyramid.

3. A median is the “middle” number when the numbers are listed in order. If there are an even number of numbers in the set, it is the average of the two middle numbers.

4. The number that appears the most frequently in a set.

5. When the numbers are evenly spaced, the mean is always equal to the median. This is true more generally if the numbers are distributed “symmetrically” about the mean, as in [–10, –7, 0, 7, 10].

6. If the average of four numbers is 15, then their sum must be . If one of the numbers is 18, then the sum of the other three is . So the average of the other three is .

7. You need to read this problem super-carefully. If the average of the five numbers is 25, then their sum is . If none of the numbers is less than 10, and since they are all different integers, the least that four of them can be is 10, 11, 12, and 13. Therefore, if x is the largest possible number in the set,

8. If the 20 students in Ms. Appel’s class averaged 90%, then they must have scored a total of  points. Similarly, Mr. Bandera’s class scored a total of . The combined average is just the sum of all the scores divided by the number of scores: .

Notice, too, that you can get a good estimate by just noticing that if there were an equal number of students in each class, the overall average would simply be the average of 80 and 90, which is 85. But since there are more students in Mr. Bandera’s class, the average must be weighted more heavily toward 80.

SAT Practice 2

1. D The average of 2x, 2x, y, and 3y is . Substituting  for y gives .

2. C If the average of seven integers is 11, their sum is . If each of these integers is less than 20, then the greatest any can be is 19. The question doesn’t say that the integers must be different, so if x is the least possible of these integers, .

3. 4 The median is the average of the two middle numbers. A little trial and error shows that 1, 4, 6, and 8 have a median of 5, so k must be 4.

4. C Call the number you are looking for y. The average of x and y is z, so set up the equation and solve:

5. A Just fill in the pyramid: .

6. B The average is 0, so . Solving for k gives us . So we put the numbers in order: –15, 2, 5, 8. Since there are an even number of numbers, the median is the average of the two middle numbers: .

7. B The most frequent number is 1, so . This means that statement III is untrue, and you can eliminate choices (D) and (E). To find the median, you need to find the average of the 10th and 11th numbers, when you arrange them in order. Since both of these are 3, . Therefore, statement II is true, and you can eliminate choice (A). To find the average, just divide the sum by , so . Therefore, statement I is not true, so the answer is (B).

8. B When a 30% solution and a 50% solution are combined, the concentration must be anywhere between 30% and 50%, depending on how much of each you added. It can’t be 50%, though, because the 30% solution dilutes it.

9. A If Set A were {0, 0, 10, 10, 10}, then its median, m, would be 10. Set B would be {2, 2, 12, 12, 12}. Inspection of Set B shows that it is a counterexample to statements II and III, leaving (A) as a possible answer.