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Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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Cubic Rotations

There are thirteen axes of rotational symmetry of a unit cube. Describe them all. What is the average length of the parts of the axes of symmetry which lie inside the cube?

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Tyneside Average Speed

Weekly Problem 12 - 2006
A van travels from Newcastle to South Shields at 30mph and returns at 40mph. What is its average speed?


Stage: 4 Challenge Level: Challenge Level:1

Diagram for solution.

Choose any two numbers. Call them $a$ and $b$ ($b < a$). Work out the arithmetic mean $(a + b)/2$ and the geometric mean $\sqrt{ab}$. Which is bigger? Repeat for other pairs of numbers. What do you notice?

In the diagram PQRS is a rectangle measuring $a$ units by $b$ units. The green rectangle measures $(a + b)/2$ by $b$ and the orange and blue rectangles both measure $(a - b)/2$ by $b$. By considering the areas of the rectangles explain why this diagram shows that $$ab < ({{a + b}\over 2})^2. $$ What does this tell us about the arithmetic mean and the geometric mean of two numbers?