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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 5-6, Math Kangaroo 7-8, Math Kangaroo 9-10.

The average of a data set is defined as the sum total of all the values in the set divided by the number of values:

If we have the data set:

we notice that there are 5 numbers in the set. Then, the average of value of the data set is:

Common error: is to automatically divide by 2.

Typical problem: When the average is given but one or more of the data in the set are unknown, we have to use the definition in product form:

Example:

Pippi Longstocking opened her treasure chest to count her possessions. She counted 100 silver doubloons worth $5 each in today's money and a few old Turkish golden drachmas worth $14 each in today's money. She calculated the average worth per coin to be $8. How many drachmas did she count?

(character from Astrid Lindgren's "Pippi Longstocking" series)

We have to write an equation that expresses the equality of the total value of the coins when added up individually with the total value of the coins when calculated from their average.

Denote with x the number of drachma coins. The total number of coins (drachmas plus doubloons) is:

Since the average value per coin is $8, the value of the whole bounty when calculated from the average is:

On the other hand, we can calculate this value also by adding up the value of all the doubloons and the value of all the drachmas:

These two values must be equal:

This is an equation for x.

Typical problem: The average of two data sets is not the average of the two separate averages.

Example:

In Nils Holgersson's flock of 23 wild geese, the average weight for a goose is 5.3 kg. When they visit a neighboring flock of 21 geese, the average weight of a goose in this other flock is 5.1 kg. What is the average weight calculated over both flocks?

(characters from Selma Lagerlof's "The Wonderful Adventures of Nils")

We have to apply the definition of the average.

The total weight of Nils' flock is:

The total weight of the neighboring flock is:

The total weight of both flocks is:

The average weight over both flocks is:

Typical problem: Compute the average speed.

In physics (kinematics), the average speed is defined as:

Common error: It is not correct to average speeds in order to calculate the average speed!

To compute the average speed it is important to correctly apply the definition in all the different situations:

Example:

A pilgrim has walked 3 hours in the morning at a speed of 3 mph, has rested for 6 hours during the hot noon hours and has continued for 4 hours in the late after-noon at a speed of 3.5 mph. What has his average speed been?

Apply the definition to calculate the partial distances walked in the morning and in the evening. Compute the total distance by adding up the partial distances.

Compute the total time just by adding up.

Apply the definition to compute the average speed: