You may also like

problem icon

Square Mean

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

problem icon

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?

problem icon

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:3 Challenge Level:3 Challenge Level:3

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?