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.

Correctly solved by:

1. K. Sengupta Calcutta, INDIA
2. James Alarie Flint, Michigan
3. Les Walker Ventura, California
4. Tom Robb
    Olga Bushey
    John Crocket 		John Handley High School
Winchester, Virginia
5. David & Judy Dixon Bennettsville, South Carolina
6. Richard Johnson La Jolla, California