### All PSAT Math Resources

## Example Questions

### Example Question #11 : Arithmetic Mean

The average for 24 students on a test is 81%. Two more students take the test, averaging 74% between the two of them. What is the total class average (to the closest hundreth) if these two students are added to the 24?

**Possible Answers:**

81.22%

79.12%

77.44%

80.46%

77.5%

**Correct answer:**

80.46%

The easiest way to solve this is to consider the total scores as follows:

Group 1: 81 * 24 = 1944

Group 2: 74 * 2 = 148

Therefore, the total percentage points earned for the class is 148 + 1944 = 2092. The new class average will be 2092/26 or 80.46. (For our purposes, this is 80.46%.)

### Example Question #11 : Statistics

Twenty students scored an average of 83% on a test. After three more took the exam, the class average was exactly 84%. Which of the following is a possible set of scores for these three students?

**Possible Answers:**

88%, 92%, 100%

92%, 96%, and 98%

72%, 90%, 98%

None of the other answers

84%, 92%, and 96%

**Correct answer:**

84%, 92%, and 96%

Let's think of the exams in terms of 100 points for a 100%. This means that the first 20 students received 20 * 83 or 1660 points.

Now, we must figure out how many points would be necessary for 23 students to have an average of exactly 84%. That would be found using the equation for a mean:

x / 23 = 84 → x = 1932

That means that our 3 students had to get a total of 1932 – 1660, or 272 points. The only answer that matches that among our answers is 84%, 92%, and 96%.

### Example Question #21 : Statistics

If a = 2b = 4c = 8d, what is the average of a, b, c, and d, in terms of a?

**Possible Answers:**

19a/4

4a

15a/16

a/8

15a/32

**Correct answer:**

15a/32

Convert each term, so a = a, b = a/2, c = a/4, and d = a/8

so then the average would be (a + a/2 + a/4 + a/8)/4

= (8a/8 + 4a/8 + 2a/8 + a/8)/4

= (15a/8)/4 = 15a/32

### Example Question #22 : Statistics

The median of a set of ten consecutive integers is –2.5. What is the mean of this set?

**Possible Answers:**

2.5

2

0

–1

–2.5

**Correct answer:**

–2.5

Let's let a, b, c, d, e, f, g, h, i, and j represent the integers in the set from least to greatest.

The set has an even number of integers. This means that the median will have to be the average of the two numbers in the middle. The middle numbers will have to be the fifth and the sixth numbers, which are represented by e and f. This means that the average of e and f must be –2.5. Because –2.5 is halfway between –3 and –2, this means e is –3 and f is –2. Our set looks like this so far:

a, b, c, d, -3, -2, g, h, i, j

Because the set consists of consecutive integers, each term must equal one more than the one before it. So, g, h, i, and j are the four numbers are –2, and a, b, c, and d are the four numbers before –3. Thus, our set is as follows:

–7, –6, –5, –4, –3, –2, –1, 0, 1, 2

The question asks us to find the mean, which is the sum of the elements in the set divided by the number of elements in the set.

The sum of the numbers in the set is –7 + –6 + –5 + . . . 1 + 2 = –25.

Because there are ten numbers in the set, the mean of the set is –25/10 = –2.5.

The answer is –2.5.

### Example Question #21 : Statistics

If the mean of z and s is 22, and the mean of b and f is 22, what is the mean of z, s, b, and f?

**Possible Answers:**

27

17

20

22

**Correct answer:**

22

If the average of two numbers is 22, and the average of two more numbers is 22, the average of all of them will be 22.

### Example Question #23 : Statistics

In a set of four numbers, the second is twice the size of the first, the third is two more than three times the size of the first, and the fourth is five times the third. In terms of x, what is the mean of these four numbers?

**Possible Answers:**

21*x*

(11*x*/4) + 0.5

None of the other answers

*x* + 1.5

(21*x*/4) + 3

**Correct answer:**

(21*x*/4) + 3

Begin by writing out the values of our numbers.

Number 1: *x* (since it is our reference point for all the other numbers)

Number 2: 2*x* (simple from the problem)

Number 3: 3*x* + 2 (two more [i.e. + 2] than three times the first [i.e. 3*x*])

Number 4: 5 * (Number 3) = 5 (3*x* + 2) = 15*x* + 10

Now, to find the mean, we need to add together all of these three values and divide them by four. Their sum is:

*x* + 2*x* + 3*x* + 2 + 15*x* + 10 = 21*x* + 12

The mean is (21*x* + 12)/4 = (21*x*/4) + (12/4) = (21*x*/4) + 3

### Example Question #24 : Statistics

Jack traveled for 3 hours at a rate of 70 miles per hour and for 5 hours at a rate of 55 miles per hour. What was his approximate average speed for the 8-hour period, rounded to the nearest mile per hour?

**Possible Answers:**

65

60

61

70

59

**Correct answer:**

61

You multiply the number of hours that Jack went 70 mph by 3 hours, multiply the hours he went 55 by 5 hours, and add the totals giving you a total of 485 miles. You then divide this by 8 hours, giving you 60.625, rounding to 61 mph.

### Example Question #21 : Statistics

If *i* = 2*j* = 5*k*, what is the average of *i*, *j* and *k* in terms of *i*?

**Possible Answers:**

7*i*

7*i*/10

7*i*/30

17*i*/10

17*i*/30

**Correct answer:**

17*i*/30

You solve each variable in terms of *i*, giving you *j *= *i*/2 and *k *= *i*/5, then add both to *i*, giving you 17*i*/10 and then divide by 3 giving you 17*i*/30.

### Example Question #25 : Statistics

This semester, Mary had five quizzes that were each worth 10% of her grade. She scored 89, 74, 84, 92, and 90 on those five quizzes. Mary also scored a 92 on her midterm that was worth 25% of her grade, and a 91 on her final that was also worth 25% of her class grade. What was Mary's final grade in the class?

**Possible Answers:**

89

87

93

85

91

**Correct answer:**

89

To find her average grade for the class, we need to multiply Mary's test scores by their corresponding weights and then add them up.

The five quizzes were each worth 10%, or 0.1, of her grade, and the midterm and final were both worth 25%, or 0.25.

average = (0.1 * 89) + (0.1 * 74) + (0.1 * 84) + (0.1 * 92) + (0.1 * 90) + (0.25 * 92) + (0.25 * 91) = 88.95 = 89.

Looking at the answer choices, they are all spaced 2 percentage points apart, so clearly the closest answer choice to 88.95 is 89.

### Example Question #26 : Statistics

A function *f*(*x*) = 3*x* + 2 is defined for all values of *x*. What is the average value of the function when *x* can only take the values {–1, 0, 2, 5, 9}, each with equal probability?

**Possible Answers:**

6

13

9

11

3

**Correct answer:**

11

Let's find out what values the function takes for the given *x*-values.

*f*(–1) = 3(–1) + 2 = –1

*f*(0) = 2, *f*(2) = 8, *f*(5) = 17, *f*(9) = 29

We're told that the function takes these different values with equal probability, so to find the average we need to sum up the *f*(*x*) terms and divide by the number of terms, 5.

average = (–1 + 2 + 8 + 17 + 29) / 5 = 11

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