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## Example Questions

### Example Question #1 : How To Find Arithmetic Mean

Column A

The mean of the sample of numbers 2, 5, and 10.

Column B

The mean of the sample of numbers 1, 5, and 15.

**Possible Answers:**

The relationship cannot be determined from the information given.

The quantity in Column B is greater.

The quantity in Column A is greater.

The two quantities are equal.

**Correct answer:**

The quantity in Column B is greater.

The arithmetic mean is the average of the sum of a set of numbers divided by the total number of numbers in the set. This is not to be confused with median or mode.

In Column A, the mean of 5.66 is obtained when the sum (17) is divided by the number of values in the set (3).

In Column B, the mean of 7 is obtained when 21 is divided by 3. Because 7 is greater than 5.66, Column B is greater. The answer is Column B.

### Example Question #2 : How To Find Arithmetic Mean

Bill runs for 30 minutes at 8 mph and then runs for 15 minutes at 13mph. What was his average speed during his entire run?

**Possible Answers:**

9^{2}/_{3} mph

10^{1}/_{2} mph

10 mph

11 mph

**Correct answer:**

9^{2}/_{3} mph

Rate = distance/time.

Find the distance for each individual segment of the run (4 miles and 3.25miles). Then add total distance and divide by total time to get the average rate, while making sure the units are compatible (miles per hour not miles per minute), which means the total 45 minute run time needs to be converted to 0.75 of an hour; therefore (4miles + 3.25 miles/0.75 hour) is the final answer.

### Example Question #3 : How To Find Arithmetic Mean

Sample Set A has 25 data points with an arithmetic mean of 50.

Sample Set B has 75 data points with an arithmetic mean of 100.

Quantity A: The arithmetic mean of the 100 data points encompassing A and B

Quantity B: 80

**Possible Answers:**

The two quantities are equal.

Quantity B is greater.

Quantity A is greater.

The relationship cannot be determined from the information given.

**Correct answer:**

Quantity A is greater.

Note that:

The arithmetic mean of the 100 data points encompassing A and B =

(total data of Sample Set A + total data of Sample Set B)/100

We have Mean of Sample Set A = 50, or:

(total of Sample Set A) / 25 = 50

And we have Mean of Sample Set B = 100, or:

(total of Sample Set B) / 75 = 100

We get denominators of 100 by dividing both of the equations:

Divide [(total of Sample Set A) / 25 = 50] by 4:

(total of Sample Set A) / 100 = 50/4 = 25/2

Multiply [(total of Sample Set B)/75 = 100] by 3/4:

(total of Sample Set B)/100 = 75

Now add the two equations together:

(total data of Sample Set A + total data of Sample Set B)/100

### Example Question #4 : How To Find Arithmetic Mean

The average (arithmetic mean) of x, y and z is 15. If w is 10, then what is the average of w, x, y and z?

**Possible Answers:**

12.5

15

10

13.75

12.75

**Correct answer:**

13.75

We can calculate the arithmetic mean by adding up the numbers in a set, and dividing that total by the count of numbers in the set.

Thus, we know that (x + y + z) / 3 = 15. (Multiply both sides by 3.)

x + y + z = 45

We add w = 10 to that, and divide by the new count, 4.

55 / 4 = **13.75**

### Example Question #5 : How To Find Arithmetic Mean

The combined height of John and Sandy is 130 inches. Sandy, John, and Allen together have a combined height of 215 inches. Sandy and Allen have combined height of 137 inches. How tall is John?

**Possible Answers:**

78

67

69

75

62

**Correct answer:**

78

Translate the question into a series of equations:

J + S = 130; J + S + A = 215; S + A = 137

Although there are several ways of approaching this, let us choose the path that is most direct. Given that J, S, A are all involved in the second equation, we can isolate J if we eliminate S and A - which can be done by using the data we have in the third equation. Since S + A = 137, we can rewrite J + S + A = 215 as:

J + 137 = 215.

Now, we only need to solve for J:

J = 215 - 137

J = 78 inches.

### Example Question #6 : How To Find Arithmetic Mean

On a given exam, four students have an average score of 81 points. If another student takes the exam, what must he or she score in order to increase the overall average to at least 83 points?

**Possible Answers:**

97

86

95

84

91

**Correct answer:**

91

Begin by translating the current state of affairs into an equation. We know that for four students, the average is ascertained by taking the sum of the scores (*s*) and dividing that by four:

s / 4 = 81; s = 324

Now, when we add the additional student, that person's score (*x*) will be added to the value for *s*. The average of this new group will be divided among five students. We must solve for the case in which the average is 83 points (thus giving us the case for the minimum score *x* necessary.) This yields the following equation:

(324 + x) / 5 = 83

Solve for x:

324 + x = 415; x = 91

The minimum score necessary is 91 points.

### Example Question #5 : Arithmetic Mean

Find the arithmetic mean of the following series of numbers:

1, 2, 2, 3, 4, 5, 11, 12

**Possible Answers:**

11

5

3.5

2

**Correct answer:**

5

To find the arithmetic mean of a series of numbers, add up all of the numbers and divide by the number of numbers in the series. Adding all the numbers gives us 40 and there are 8 numbers in the series. 40/8 = 5

### Example Question #5 : How To Find Arithmetic Mean

Consider the given set of numbers 70, 81, 91, 83, 88, and 55.

A new number equal to a 12% increase in the median of this set is added to the list. What percentage larger is the mean of the new list in comparison with the original?

**Possible Answers:**

2.441%

1.534%

2.535%

1.71%

4.44%

**Correct answer:**

2.535%

Reorder the set: 55, 70, 81, 83, 88, 91

Since the set is even, we find the median by taking the average of the middle two values: (81 + 83)/2 = 82

The average of the first set = (55 + 70 + 81 + 83 + 88 + 91)/6 = 468/6 = 78

The new value to be added is 82 * 1.12 = 91.84

This means the new mean is 559.84/7 = 79.9771

The increase in mean value is 79.9771 – 78 = 1.9771

This is (1.9771/78) 100, or 2.535% larger than the original mean.

### Example Question #7 : How To Find Arithmetic Mean

A group of ten candy bars has an average cost of $0.89 per candy bar. How many bars must be bought at the cost of $0.72 a piece to bring the average down to $0.80?

**Possible Answers:**

5

11.25

12

8

11

**Correct answer:**

12

We know to begin with that for x candy bards:

x / 10 = 0.89; x = 8.90

We are trying to ascertain how many bars (y) must be added to this to amount to have an average of 0.80. This is represented:

(8.9 + 0.72y) / (10 + y) = 0.8

8.9 + 0.72y = 0.8 (10 + y)

8.9 + 0.72y = 8 + 0.8y

8.9 – 8 = 0.8y - 0.72y

0.9 = 0.08y

y = 0.9 / 0.08 = 11.25

However, since we cannot have a partial amount of candy bars, we will have to buy 12.

### Example Question #5 : Arithmetic Mean

In a given translation project, three translators each took sections of a book to translate. The first translator took 15000 words, which he translated at a rate of 500 words per 20 minutes. The second translator took 200000 words, which he translated at a rate of 1250 words per half hour. The third took 10000 words, which he translated at a rate of 250 words per 15 minutes. In terms of words per hour, what was the overall average translation rate for this project?

**Possible Answers:**

667 words per hour

1575 words per hour

1667 words per hour

2250 words per hour

750 words per hour

**Correct answer:**

2250 words per hour

To find the answer, we need to know the total words and the total number of hours involved.

The first is easy: 15000 + 200000 + 10000 = 225000 words

To ascertain the number of hours, we have to look at each translator separately. Although there are several ways to do this, let's consider it this way:

Translator 1 can translate at 500 words per 20 minutes OR 1500 words per hour.

Translator 2 can translate at 1250 words per half hour OR 2500 words per hour.

Translator 3 can translate at 250 words per 15 minutes OR 1000 words per hour.

Therefore, we know each translator took the following amount of time:

Translator 1: 15000 / 1500 = 10 hours

Translator 2: 200000 / 2500 = 80 hours

Translator 3: 10000 / 1000 = 10 hours

The total number of hours was therefore 10 + 80 + 10 = 100 hours.

The average rate was 225000 words/100 hours, or 2250 words per hour.

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