Question 1146675
Two different candles are lit.
 They burn at different rates and one is 3 cm longer than the other.
 The longer one was lit at 5:30 pm and the shorter one at 7 pm.
 At 9:30 pm they were both the same length.
 The longer one burned out at 11:30 pm and the shorter one burned out at 11 pm.
 How long was each candle originally?
:
let a = the length of the longer candle
let b = the length of the shorter
 :
the longer candle burned from 5:30 to 11:30 or 6 hrs, therefore
{{{a/6}}} cm/hr is the rate of burn of the longer
and
the shorter candle burned from 7:00 to 11:00 or 4 hrs, therefore
{{{b/4}}} cm/hr is the rate of burn for the shorter candle
:
at 9:30 the longer candle has burned for 4 hrs and the shorter for 2.5 hrs
a - 4*{{{a/6}}} = b - 2.5*{{{b/4}}}
a - {{{4a/6}}} = b - {{{2.5b/4}}}
multiply both sides by 12, get rid of the denominator
12a - 8a = 12b - 7.5b
4a = 4.5b
a is 3 cm longer than b, therefore
replace b with (a-3)
4a = 4.5(a-3)
4a = 4.5a - 13.5
4a - 4.5a = -13.5
-.5a = -13.5
a = -13.5/-.5
a = 27 cm is the longer candle
and obviously
24 cm is the shorter

:
see of this works out
27/6 = 4.5 cm/hr burn rate of the longer
24/4 = 6 cm/hr is the rate of the shorter
:
27 - 4(4.5) = 
27 - 18 = 9 cm remain
and
24 - 2.5(6) = 
24 - 15 = 9 cm remain