Question 145749
Let L=length of each candle
And Let T=time(in hours) that will pass until the slow burning candle is exactly twice as long as the fast burning one

The fast burning candle burns at the rate of (1/6)L per hour

And the slow burning candle burns at the rate of (1/9)L per hour

Now, after T hours the slow burning candle is L-(1/9)L*T long
And the fast burning candle is L-(1/6)L*T long

We are told that the slow burning candle is now twice as long as the fast burning candle, so our equation to solve is:
L-(1/9)L*T=2(L-(1/6)L*T))  simplify

L-(1/9)L*T=2L-(2/6)L*T  divide each term by L
1-(1/9)T=2-(2/6)T  subtract 1 from and add (2/6)T to each side
1-1-(1/9)T+(2/6)T=2-1-(2/6)T+(2/6)T
-(2/18)T+(6/18)T=1
(4/18)T=1 multiply each side by 18

4T=18  divide each side by 4
T=4.5 hrs ----------------------time needed for slow burning candle to be exactly twice as long as the fast burning candle

CK

In 4.5 hours the slow burning candle is (L-(1/9)*4.5L)=(L-0.5L) units long
In 4.5 hours the fast burning candle is (L-(1/6)*4.5L)+(L-0.75L) units long

And we are told that:
(L-0.5L)=2(L-0.75L) or
0.5L=2(0.25)L
0.5L=0.5L


Hope this helps---ptaylor