A nine-meter long piece of wood is cut so that one piece is 50% longer than the other. Exactly how long is the longer piece?

This puzzle often trips people up because our first instinct is to divide the total length in half. However, the key is that the two resulting pieces are not based on the halfway point, but on each other. One piece (Review) is defined by the length of the other. If we simply took half of nine (4.5 meters) and added 50%, we would get 6.75 meters, and the other piece would be 2.25 meters. This doesn't work because 6.75 is 200% longer than 2.25, not 50%.

The correct way to think about it is in terms of units. Let's call the shorter piece one unit. Since the longer piece is 50% longer, it is equal to 1.5 of those same units. When you place them end-to-end, their combined length is 2.5 units (1 + 1.5). We know this total length is nine meters. To find the size of one unit, we divide 9 meters by 2.5, which gives us 3.6 meters. This is the length of the shorter piece.

Now that we know the shorter piece is 3.6 meters, we can easily find the length of the longer one. We can either calculate 50% of 3.6 (which is 1.8) and add it on, or more simply, multiply 3.6 by 1.5. Both methods give us the final length of 5.4 meters. This kind of ratio-based thinking is fundamental in many fields, including engineering, cooking, and art, where proportions are everything.