Writing up solutions to maths problems is a a tricky business. I was presented the following problem by my colleague Charlie. 

Two candles have different lengths and different thickness. The shorter one would last 11 hours, the longer one would last for 7 hours.

Both candles are lit at the same time, and 3 hours later both have the same length remaining.

What was the ratio of the length of the longer candle to the shorter candle?

You might like to have a go at it first. What different representations might you use?

Often as teachers we show a final complete well rehearsed solution. In reality I attempted a variety of methods using algebra, an area model, a double number line and a graphical method. When we hide these attempts from students they may see the final representation as a bit of a trick. Sometimes they wonder how they are supposed to choose the 'correct' approach. 

Charlie wanted to see whether there was an intuitive way of looking at the problem. In the end we chose to publish the area model.  

Can you make sense of each of the representations? How else could the solution be represented? 
We wrote up our final solution here, it is the first of three different approaches.