Candlestic!

From the time the two candles are at the same length, it takes the initially shorter candle three times as long as the initially longer one to burn out. So if the burn rate of the shorter candle is $y$ cm/hr then the burn rate of the longer candle is $3y$ cm/hr.

Now as the the longer candle takes $7$ hours to burn out and the shorter one $5$ hours, and since their initial difference in length is $32$ cm., we have that

$7*3y - 5*y = 32 \Longrightarrow 16*y = 32 \Longrightarrow y = 2$ cm/hr.

Thus the original length of the longer candle was $3*2*7 = 42$ cm and that of the shorter candle was $2*5 = 10$ cm, giving a sum of $\boxed{52}$ cm.

Let the length of the smaller candle be $x$ cm. The length of the longer candle becomes $(x+32)$ cm.

Now, the longer candle took $10-3=7$ hours to burn. Hence, its rate of burning is $\frac {x+32}{7}$ cm per hour.

Similarly, the smaller candle took $12-7=5$ hours to burn. Hence, its rate of burning is $\frac {x}{5}$ cm per hour.

At $9$ P.M., the lengths of the candles were the same.

At $9$ P.M., for the longer candle, $9-3=6$ hours had passed. This means that $6 \times \frac {x+32}{7} = \frac {6(x+32)}{7}$ cm of the candle was burnt, so the remnant portion of the candle was of length $(x+32)-\frac {6(x+32)}{7} = \frac {x+32}{7}$ cm.

Similarly, at $9$ P.M., for the shorter candle, $9-7=2$ hours had passed. This means that $2 \times \frac {x}{5} = \frac {2x}{5}$ of the candle was burnt, so the remnant portion of the candle was of length $x-\frac {2x}{5} = \frac {3x}{5}$ cm.

Hence, $\frac {x+32}{7} = \frac {3x}{5}$ , which when we solve gives us $x=10$ .

Thus, the sum of the lengths of the two candles is $x+x+32 = \boxed{52}$ cm.

Let La=length of the short candle

Lb=length of long candle =La+32

Ra= burning rate of candleA

Rb=burning rate of candleB

Y=the same length of the two candles @ 9pm

Ra=La /(12-7) = La / 5

Rb=La+32 /(10-3)=La+32 /7

Y=Ra×(12-9)=3Ra

And Y is also=Rb×(10-9)=Rb×1

3Ra=Rb

3(La / 5) =(La+ 32) /7

21La=5 La+ 160

La=10 cm. and Lb=10+32=42cm

La+Lb=10+42 = 52 cm.