Averages

Averages

​1. Arithmetic Mean

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​Example 1. A football team has played 4 games so far this season. The amount of points they scored in those games is 4, 5, 7, and 8. What is their average score for a game?

​Example 1. A football team has played 4 games so far this season. The amount of points they scored in those games is 4, 5, 7, and 8. What is their average score for a game?

2. Weighted Average

Weighted average is an arithmetic mean of a set of numbers in which some elements of the set carry more importance (weight) than others.

Grades are often computed using a weighted average.

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Example 2. Suppose that homework counts 30%, quizzes 20%, and tests 50%.

If Pam has a homework grade of 90, a quiz grade of 85, and a test grade of 98,

find her overall grade.

Example 2. Suppose that homework counts 30%, quizzes 20%, and tests 50%. If Pam has a homework grade of 90, a quiz grade of 85, and a test grade of 98, find her overall grade.

Solution: 93.

Pam's overall grade equals (0.3)(90) + (0.2)(85) + (0.5)(98) = 27 + 17 + 49 = 93.

3. Harmonic Mean

The harmonic mean of n numbers is expressed as the reciprocal of the arithmetic mean of the reciprocals of the numbers. The reciprocal of a number, x, is 1/x.

Open Ended

Example 3. Find the harmonic mean of 3 and 4.

Example 3. Find the harmonic mean of 3 and 4.

PROBLEM SOLVING SKILLS

1. Arithmetic Mean

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Example 4. The mean weight of a group of six boys is 56 kg. The individual

weights (in kg) of five of them are 53, 57, 55, 60 and 55. Find the weight of the

sixth boy.

​Example 4. The mean weight of a group of six boys is 56 kg. The individual

weights (in kg) of five of them are 53, 57, 55, 60 and 55. Find the weight of the sixth boy.

​Solution: 56 kg.

Since the mean, or average, weight of the group is 56, the total sum of the weights of six boys is 56 x 6 = 336.

The sum of the given weights of five boys is 53 + 57 + 55 + 60 + 55 = 280.

The weight of the sixth boy is 336 – 280 = 56.

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Example 5. Alex has a mean score of 68 in his 4 math tests. What score does he

need to get in the fifth test to raise his mean score to 70?

​Example 5. Alex has a mean score of 68 in his 4 math tests. What score does he need to get in the fifth test to raise his mean score to 70?

​Solution: 78.

In order to achieve a mean score of 70, the total sum of his scores after the fifth test should be 70 × 5 = 350. The sum of the scores for Alex’s first 4 tests is 68 × 4 = 272. The score he needs to get on the fifth test is 350 - 272 = 78.

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Example 6. Betty played four games. Her average score was 368 for the first

three games. If her score for the 4th game is 128, what is her average score for all four games?

Example 6. Betty played four games. Her average score was 368 for the first three games. If her score for the 4th game is 128, what is her average score for all four games?

2. Average of Base Numbers

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Example 7. The individual heights of a group of ten boys are 145 cm, 142 cm,

138 cm, 136 cm, 143 cm, 146 cm, 138 cm, 144 cm, 137 cm, and 141 cm. Find the

mean height of the group.

Example 7. The individual heights of a group of ten boys are 145 cm, 142 cm,

138 cm, 136 cm, 143 cm, 146 cm, 138 cm, 144 cm, 137 cm, and 141 cm. Find the

mean height of the group.

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Example 8. The daily production of a car company was 46 cars during the first 3

days, 53 cars during the next 4 days. Find the average daily production of the

company.

Example 8. The daily production of a car company was 46 cars during the first 3 days, 53 cars during the next 4 days. Find the average daily production of the company.

3. Average of Consecutive Numbers

​The middle number method

​Given the sum of a number of consecutive integers, the middle number is the arithmetic mean of these integers. If we have an odd number of consecutive integers, the middle number will be an integer. If we have an even number of consecutive integers, the middle number will not be an integer.

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Example 9. The sum of three consecutive integers is 6. Find the middle number.

Example 9. The sum of three consecutive integers is 6. Find the middle number.

Solution:

The middle number is 6/3 = 2.

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Example 10. The sum of five consecutive integers is 30. What is the largest of

these integers?

Example 10. The sum of five consecutive integers is 30. What is the largest of these integers?

Solution:

The third term is 30 / 5 = 6, and the fourth term is 7. The largest term, the 5th term, is 8.

Multiple Choice

Example 11. The sum of 101 consecutive integers is 101. What is the largest

integer in the sequence?

(A) 101 (B) 102 (C) 51 (D) 52 (E) None of them

• A

• B

• C

• D

• E

Example 11. The sum of 101 consecutive integers is 101. What is the largest

integer in the sequence?

(A) 101 (B) 102 (C) 51 (D) 52 (E) None of them

Solution: (C) 51

The middle term, 51^th term, is the average value of the 101 integers: 101/101 = 1.

The largest integer, 101^th term, is 1 + 50 = 51.

Multiple Choice

Example 12. The middle number of a set of 11 consecutive integers is 98. What

is the largest of these 11 integers?

(A) 94 (B) 95 (C) 103 (D) 102 (E) 104

• A

• B

• C

• D

• E

Example 12. The middle number of a set of 11 consecutive integers is 98. What is the largest of these 11 integers?

(A) 94 (B) 95 (C) 103 (D) 102 (E) 104

Solution: C.

The middle number amongst the 11 integers is the 6^th number. We are given that the 6^th number is 98, so the largest of the 11 integers will be the 11th number, or

103.

6^th 7^th 8^th 9^th 10^th 11^th

98 99 100 101 102 103

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Example 13. The sum of 8 consecutive odd integers is 144. What is the value of

the largest of these integers?

Example 13. The sum of 8 consecutive odd integers is 144. What is the value of the largest of these integers?

Solution: 25.

The average of the 8 numbers is 144/8 = 18, which is also the average of the fourth and fifth numbers. It follows that the fourth number is 17 and the fifth number is 19.

6^th 7^th 8^th

21 23 25

The largest of the 8 integers will be the 8^th number, or 25.

4. Weighted Average

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Example 14. A candy store makes a hard candy mix using 10 kilograms apple

hard candy, which costs $4 per kilogram, and 4 kilograms banana hard candy,

which costs $6 per kilogram, and 10 kilograms grape hard candy, which costs $8

per kilogram. What is the cost of the mix per kilogram?

Example 14. A candy store makes a hard candy mix using 10 kilograms apple hard candy, which costs $4 per kilogram, and 4 kilograms banana hard candy, which costs $6 per kilogram, and 10 kilograms grape hard candy, which costs $8 per kilogram. What is the cost of the mix per kilogram?

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Example 15. A candy store makes a hard candy mix costing $185 per kilogram

using a specific ratio of peppermint hard candy and cinnamon hard candy. The

peppermint hard candy costs $203 per kilogram and the cinnamon hard candy

costs $170 per kilogram. To make hard candy mix, for every 5 kg peppermint

hard candy used, x kg of the cinnamon hard candy is used. Find x.

Example 15. A candy store makes a hard candy mix costing $185 per kilogram using a specific ratio of peppermint hard candy and cinnamon hard candy. The peppermint hard candy costs $203 per kilogram and the cinnamon hard candy costs $170 per kilogram. To make hard candy mix, for every 5 kg peppermint hard candy used, x kg of the cinnamon hard candy is used. Find x.

Solution: 6 kg.

The peppermint hard candy costs 203 – 185 = $18 more than the hard candy mix per kilogram. 5 kilograms of the peppermint hard candy costs 18× 5 = $90 more than the 5 kilograms of the hard candy mix.

5. Average Speed (Harmonic Mean)

​5. Average Speed (Harmonic Mean)

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Example 16. The distance from town A to town B is 360 miles. Alex drove from

A to B at 40 miles per hour but returns the same distance at 60 miles per hour.

Find the average speed for the round trip.

Example 16. The distance from town A to town B is 360 miles. Alex drove from A to B at 40 miles per hour but returns the same distance at 60 miles per hour. Find the average speed for the round trip.

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Example 17. An ant moves along the sides of an equilateral triangle with the side length of 80 cm. The speeds the ant moves in three sides are 20 cm per minute, 40 cm per minute, and 40 cm per minute. What is the ant’s average speed if the ant moves one complete loop?

Example 17. An ant moves along the sides of an equilateral triangle with the side length of 80 cm. The speeds the ant moves in three sides are 20 cm per minute, 40 cm per minute, and 40 cm per minute. What is the ant’s average speed if the ant moves one complete loop?

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Example 18. The distance from town A to town B is 240 miles. Alex drove from

A to B at 40 miles per hour but returns the same distance at 60 miles per hour.

Find the average speed for the round trip.

Example 18. The distance from town A to town B is 240 miles. Alex drove from A to B at 40 miles per hour but returns the same distance at 60 miles per hour. Find the average speed for the round trip.

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Example 19. A train traveled at an average speed of 45 miles per hour for 4 hours

and 30 miles for 6 hours. What was the train’s average speed for the round trip?

Example 19. A train traveled at an average speed of 45 miles per hour for 4 hours and 30 miles for 6 hours. What was the train’s average speed for the round trip?

Solution: 36 mph.

Method 1:

The distance traveled at an average speed of 45 miles per hour for 4 hours is 45 × 4 = 180 miles.

The distance traveled at an average speed of 30 miles per hour for 6 hours is 30 × 5 = 180 miles.

So distances travelled in these two sections are the same. Therefore, the average speed is the Harmonic Mean of two speeds of 40 and 60.

6. Average of Average

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Example 20. The average age of thirty people at a nursing home is 73. The

average age of 18 females is 75. What is the average age of the remaining 12

males?

Example 20. The average age of thirty people at a nursing home is 73. The

average age of 18 females is 75. What is the average age of the remaining 12 males?

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Example 21. A, B, and C are three positive integers. The sum of A and B is 90,

the sum of A and C is 82, and the sum of B and C is 86. Find the average of A, B,

and C.

Example 21. A, B, and C are three positive integers. The sum of A and B is 90, the sum of A and C is 82, and the sum of B and C is 86. Find the average of A, B, and C.

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Example 22. The mean of four numbers is 30. If one of the numbers is changed to

50, the mean becomes 40. Find the original value of the changed number.

Example 22. The mean of four numbers is 30. If one of the numbers is changed to 50, the mean becomes 40. Find the original value of the changed number.

Solution: 10.

Method 1:

Before the change, the sum of the four numbers is 30 × 4 = 120.

After the change, sum of four numbers becomes 40 × 4 = 160.

The difference is 160 – 120 = 40, so the original number was increased by 40 to become 50. The original number was 50 – 40 = 10.

Method 2:

The mean increases by 40 – 30 = 10. This indicates that the original number is increased by 10 × 4 = 40. It follows that the original number is 50 – 40 = 10.

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Example 23. The average of three numbers is 85. If one more number is included,

the average gets increased by 2. Find the included number.

Example 23. The average of three numbers is 85. If one more number is included, the average gets increased by 2. Find the included number.

Solution: 93.

Method 1:

Before the number is added, the sum of the original three numbers is 85 × 3 = 255.

Once the number is added, the sum of the four numbers is (85 + 2) × 4 = 348.

The difference between the two sums, or the included number, is 348 – 255 = 93.

​Method 2:

Assume that the three numbers are 85, 85, and 85. Since the average becomes 87

when the new number is added, the new number should be equal to 87 + (87 – 85) + (87 – 85) + (87 – 85) = 93.

7. Tire Problems

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Example 24. Two children at a time can play pairball. In the game of pairball,

only two players can play at a time. For 120 minutes, with only two children

playing at one time, six children take turns so that each one plays the same

amount of time. Find the number of minutes each child plays.

Example 24. Two children at a time can play pairball. In the game of pairball, only two players can play at a time. For 120 minutes, with only two children playing at one time, six children take turns so that each one plays the same amount of time. Find the number of minutes each child plays.

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Example 25. Eight people want to play a 48-minute game as a team but only a

team of exactly five is allowed to play. Between each game, another player can

replace a current player, and suppose each person plays in the game for the same amount of time. How many minutes will each person play?

Example 25. Eight people want to play a 48-minute game as a team but only a team of exactly five is allowed to play. Between each game, another player can replace a current player, and suppose each person plays in the game for the same amount of time. How many minutes will each person play?

PROBLEMS

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Problem 1. The heights of five players of a football team are 147 cm, 149 cm,

150 cm, 151 cm, and 153 cm. What is their average height of the players of the

football team?

Problem 1. The heights of five players of a football team are 147 cm, 149 cm, 150 cm, 151 cm, and 153 cm. What is their average height of the players of the football team?

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Problem 2. Tom wants to sell a package containing store bought candy. The

package contains 15 kg of Banana Candy at $3 per kilogram, and 30 kilograms of

Cherry Candy at $6 per kilogram. How many dollars per kg should he price the

mixture?

Problem 2. Tom wants to sell a package containing store bought candy. The package contains 15 kg of Banana Candy at $3 per kilogram, and 30 kilograms of Cherry Candy at $6 per kilogram. How many dollars per kg should he price the mixture?

Open Ended

Problem 3. Find the harmonic mean of 2 and 4.

Problem 3. Find the harmonic mean of 2 and 4.

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Problem 4. The mean weight of a group of seven boys is 55 kg. The individual

weights (in kg) of six of them are 53, 57, 55, 60, 59 and 55. Find the weight of the seventh boy?

Problem 4. The mean weight of a group of seven boys is 55 kg. The individual weights (in kg) of six of them are 53, 57, 55, 60, 59 and 55. Find the weight of the seventh boy?

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Problem 5. Bob received a grade of 89 for his English test. His math test was 5

more points than his English test. Find out the grade he needs to get on his science test to raise the mean score to 93.

Problem 5. Bob received a grade of 89 for his English test. His math test was 5 more points than his English test. Find out the grade he needs to get on his science test to raise the mean score to 93.

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Problem 6. Sam receives the following scores on his first three English tests: 92,

80, 96. What average score does he need on the last two tests in order to maintain a 93 average?

Problem 6. Sam receives the following scores on his first three English tests: 92, 80, 96. What average score does he need on the last two tests in order to maintain a 93 average?

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Problem 7. The individual scores of a group of six students are 97, 81, 84, 86, 90,

and 96. Find the mean score of the group.

Problem 7. The individual scores of a group of six students are 97, 81, 84, 86, 90, and 96. Find the mean score of the group.

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Problem 8. The mean height of first six students is 123 cm and the mean height

of next four students is 128 cm. Find the mean height of the group.

Problem 8. The mean height of first six students is 123 cm and the mean height of next four students is 128 cm. Find the mean height of the group.

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Problem 9. The sum of four consecutive integers is 10. Find the average value of

the second and the third terms.

Problem 9. The sum of four consecutive integers is 10. Find the average value of the second and the third terms.

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Problem 10. The sum of four consecutive integers is 22. What is the smallest

term of these integers?

Problem 10. The sum of four consecutive integers is 22. What is the smallest

term of these integers?

Multiple Choice

Problem 11. The sum of 5 consecutive integers is 9, 000. What is the value of the

largest of these integers?

(A) 8995 (B) 1801 (C) 1800 (D) 1802 (E) 1798

• A

• B

• C

• D

• E

Problem 11. The sum of 5 consecutive integers is 9, 000. What is the value of the largest of these integers?

(A) 8995 (B) 1801 (C) 1800 (D) 1802 (E) 1798

Multiple Choice

Problem 12. What is the smallest of 9 consecutive integers if the sum of these

integers equals 495?

(A) 56 (B) 55 (C) 51 (D) 49 (E) 486

• A

• B

• C

• D

• E

Problem 12. What is the smallest of 9 consecutive integers if the sum of these integers equals 495?

(A) 56 (B) 55 (C) 51 (D) 49 (E) 486

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Problem 13. The sum of 8 consecutive even integers is 136. What is the value of

the largest of these integers?

Problem 13. The sum of 8 consecutive even integers is 136. What is the value of the largest of these integers?

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Problem 14. A candy store makes a hard candy mix using a specific ratio of

peppermint hard candy, which costs $350 per kilogram, and cinnamon hard

candy, which costs $600 per kilogram. Overall, the mix ends up costing $575 per

kilogram. If the candy store has 16 kilograms of peppermint hard candy, how

many kilograms of cinnamon hard candy will be needed to make the mix?

Problem 14. A candy store makes a hard candy mix using a specific ratio of

peppermint hard candy, which costs $350 per kilogram, and cinnamon hard

candy, which costs $600 per kilogram. Overall, the mix ends up costing $575 per kilogram. If the candy store has 16 kilograms of peppermint hard candy, how many kilograms of cinnamon hard candy will be needed to make the mix?

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Problem 15. The sum of the monthly restaurant spending of a family was $324

during the first 3 months, $278 during the next 4 months and $123 during the last

5 months of a year. Find the average monthly restaurant spending of the family.

Problem 15. The sum of the monthly restaurant spending of a family was $324 during the first 3 months, $278 during the next 4 months and $123 during the last 5 months of a year. Find the average monthly restaurant spending of the family.

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Problem 16. Kali left home and traveled toward her friend's house 60 kilometers

from her home at an average speed of 30 km/h. When she came back, her speed

was 20 km/h. Find the average speed for the round trip.

Problem 16. Kali left home and traveled toward her friend's house 60 kilometers from her home at an average speed of 30 km/h. When she came back, her speed was 20 km/h. Find the average speed for the round trip.

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Problem 17. The distance from town A to town B is 18 miles. Alex walked from

A to B at 3 miles per hour but returns the same distance at 6 miles per hour. Find

the average speed for the round trip.

Problem 17. The distance from town A to town B is 18 miles. Alex walked from A to B at 3 miles per hour but returns the same distance at 6 miles per hour. Find the average speed for the round trip.

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Problem 18. The distance from town A to town B is 3240 miles. A hovercraft

flew from A to B at 810 miles per hour but returns the same distance at 540 miles

per hour. What was the plane’s average speed for the round trip?

Problem 18. The distance from town A to town B is 3240 miles. A hovercraft

flew from A to B at 810 miles per hour but returns the same distance at 540 miles per hour. What was the plane’s average speed for the round trip?

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Problem 19. A round-trip car ride took 12 hour. The first half of the trip took 7

hours at an average speed of 45 miles per hour. What was the average speed on

the return trip?

Problem 19. A round-trip car ride took 12 hour. The first half of the trip took 7 hours at an average speed of 45 miles per hour. What was the average speed on the return trip?

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Problem 20. The average height of 42 boys and girls was calculated to be 131 cm.

The average height of all 24 boys was calculated to be 128 cm. What is the

average height of girls?

Problem 20. The average height of 42 boys and girls was calculated to be 131 cm. The average height of all 24 boys was calculated to be 128 cm. What is the average height of girls?

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Problem 21. A, B, and C are three positive integers. The average of A and B is 42,

the average of A and C is 46, and the average of B and C is 47. Find the value of A.

Problem 21. A, B, and C are three positive integers. The average of A and B is 42, the average of A and C is 46, and the average of B and C is 47. Find the value of A.

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Problem 22. The mean of five numbers is 70. If one of the numbers is changed to

90, the mean becomes 80. Find the original value of the changed number.

Problem 22. The mean of five numbers is 70. If one of the numbers is changed to 90, the mean becomes 80. Find the original value of the changed number.

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Problem 23. The average of four numbers is 88. If one number is excluded, the

average becomes reduced by 3. Find the excluded number.

Problem 23. The average of four numbers is 88. If one number is excluded, the average becomes reduced by 3. Find the excluded number.

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Problem 24. One child carries a bag at a time. For 15 kilometers, five children

take turns to carry two bags so that each child carries the bag for the same amount of distance. Find the number of kilometers each child carries the bag.

Problem 24. One child carries a bag at a time. For 15 kilometers, five children take turns to carry two bags so that each child carries the bag for the same amount of distance. Find the number of kilometers each child carries the bag.

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Problem 25. A bicyclist wants to make a 600-mile trip on his two-wheel bicycle.

He has a spare wheel, which is used to replace either of the other two wheels.

Suppose that each of the three wheels is to have the same mileage for the trip,

how many miles should each wheel travel?

Problem 25. A bicyclist wants to make a 600-mile trip on his two-wheel bicycle. He has a spare wheel, which is used to replace either of the other two wheels. Suppose that each of the three wheels is to have the same mileage for the trip, how many miles should each wheel travel?