I have two candles, one an inch longer than the other. I light the longer candle at 3 PM and the shorter one at 5 PM. At 9 PM, they are the same length. The longer candle goes out at 11 PM and the shorter one goes out at 10:30 PM. How long were they at the start?

Solution to the Problem: The shorter candle was 11 inches and the longer candle was 12 inches. Use algebra to solve: The candles must have different diameters and therefore, different burn rates, because when they were the same length (at 9 PM), one lasted two hours longer but the other lasted only 1.5 hours longer.   Length (inches) Rate (in/hr) Start Time Goes Out at Length (inches) Longer Candle L + 1 x 3 PM 11 PM 8x Shorter Candle L y 5 PM 10:30 PM 5.5y Now, set up three equations from the table: (1)     (L + 1) - 6x = L - 4y     because they are equal at 9 PM (2)     (L + 1) - 8x = 0     because the longer one burns out at 11 PM (3)     L - 5.5y = 0     because the shorter one burns out at 10:30 PM Solving these three equations, L = 5.5y L = 8x - 1 8x - 1 = 5.5y -6x + 1 = -4y 24x - 3 = 16.5y -24x + 4 = -16y 1 = .5y So, y = 2 in/hr. and the length of the shorter candle is (2 in/hr)(5.5 hr) = 11 inches. 8x - 1 = 11 so x = 3/2 in/hr and the length of the longer candle is (3/2 in/hr)(8 hr) = 12 inches.

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