How Long Are The Candles - Answer

(Solution Hidden)


This post presents the solution to the puzzle "How Long are the Candles?"

The first thing to notice is that, at 9pm, the candles are the same length. At this point, it takes the longer candle 2 hours to burn away, but it only takes the shorter candle 1.5 hours. This means that the candles do not burn at the same rate.

I made myself the following diagram:

y+1 y
6 4

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x x
2 1.5

This diagram shows the two candles, and divides them into two parts.

The bottom part is the length of the two candles at 9pm (when they were equal - call it x inches), as well as the number of hours it took each candle to burn that length. So, the longer candle burned at a rate (r1) equal to x/2, and the shorter candle burned at a rate (r2) of x/1.5.

The top part is the length of the two candles that burned up by 9pm, as well as the number of hours it took each candle to burn that length. So, the longer candle burned at a rate (r1) equal to(y+1)/6, and the shorter candle burned at a rate (r2) of y/4.


We now have two equations with 2 unknowns:

r1 = r1
x/2 = (y+1)/6
2y+2 = 6x

r2 = r2
x/1.5 = y/4
4x = 1.5y
x = 1.5y/4

If we now plug the second equation into the first, we get

2y+2 = 6*1.5y/4
2y+2 = 9y/4
2 = 9y/4 - 2y
2 = 9y/4 - 8y/4
2 = y/4
8 = y

We then get x = 1.5(8)/4 = 3

So, the longer candle is x + y +1 = 12 inches.

The shorter candle is x + y = 11 inches.