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At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

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Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?


Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2


Let the extra volume of antifreeze required be $x$ litres. Initially, there are 3.5 litres of antifreeze in the solution. So $3.5+x=0.3(14+x)$, that is, $35+10x=42+3x$, so $x=1$.

This problem is taken from the UKMT Mathematical Challenges.