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### Number and algebra

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# About Average

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Age 11 to 14

Challenge Level

Can you find sets of positive integers that satisfy the following?

1. Three numbers with mean $3$ and mode $2$

2. Three numbers with mean $7$ and mode $10$

3. Three numbers with mean $8$, median $10$ and range $8$

4. Four numbers with mean $7.5$, mode $6$ and median $7$

5. Four numbers with mean $6$, median $6.5$ and range $11$

6. Five numbers with mean $4$, mode $3$ and range $9$

7. Five numbers with mean $4$, mode $2$ and range $6$ (two possible solutions)

8. Five numbers with mean $7$, mode $7$ and range $10$ (three possible solutions)

**Extension**

Can you find a set of four numbers with mean $4$, mode $1$, median $2$ and range $10$?

How about a set of five numbers?

Or six numbers?

Or...

Or 100 numbers?

*If you enjoyed this problem, you may also be interested in M, M and M.*

*With thanks to Don Steward, whose ideas formed the basis of this problem.*