Average with the number of values unknown

The candles in a box have an average height of 12 centimetres. One more candle, 40 centimetres tall, is put into the box. The average height of the candles becomes 16 centimetres. How many candles were in the box at first? Try it first. Pause if you like. Here's what's already in the box, drawn as blue bars. We are not told how tall each candle is, only that they level off at 12 centimetres, so 12 is the height we draw them at. How many of them there are is what the question asks for, so the row ends in a question mark. The new candle is 40 centimetres. That's the tall bar beside them. Once it's in the box, every candle levels off at 16 centimetres, so 16 is a line right across the row. Every candle already in the box was 12 centimetres. Now they all have to reach 16. 16 minus 12 is 4, so every one of them is 4 centimetres short. So where does that extra come from? Nothing went into the box except one candle. The new candle came in at 40, but it only has to stand at 16, the same as the rest. 40 minus 16 is 24. Everything above the line is spare. That is the 24 centimetres we colour purple. That purple 24 is the only thing that can fill the gaps, and every gap is 4 centimetres. So we cut the purple piece into pieces of 4 centimetres. 24 divided by 4 is 6. That is 6 pieces, so there were 6 candles below the average. One piece fills each gap. Every candle in the box is level at 16 centimetres now, and the total never changed. So there were 6 candles in the box at first. The extra above the new average is shared out in equal pieces, and the number of pieces is the number of candles.