On Diwali eve, two candles, one of which is 3 cm longer than the other, are lighted. The longer one is lighted at 5.30 p.m. and the shorter at 7 p.m. At 9.30 p.m. they both are of same length. The longer one burns out at 11.30 p.m. and the shorter one at 11 p.m. How long was each candle originally?

Let's assume that the longer candle shorten at rate of x cm/hr when burning and the smaller candle shorter at rate of y cm/hr.

Given, the longer candle burns out completely in 6 hours and the smaller candle in 4 hours,

So, their lengths are 6x cm and 4y cm respectively.

According to first condition,

⇒ 6x = 4y + 3

⇒ 6x - 4y = 3 …..(i)

At 9:30 p.m. length of longer candle = (6x - 4x) = 2x cm.

At 9:30 p.m. the length of smaller candle = $4y - \dfrac{5y}{2} = \dfrac{8y - 5y}{2} = \dfrac{3y}{2}$

Now, according to second condition given in problem,

2x = $\dfrac{3y}{2}$ (As both candles have same length at 9:30 p.m.)

⇒ 4x = 3y

4x - 3y = 0 ……(ii)

Multiplying (i) by 3 and (ii) by 4 we get,

18x - 12y = 9 …….(iii)

16x - 12y = 0 …….(iv)

Subtracting (iv) from (iii) we get,

⇒ 18x - 12y - (16x - 12y) = 9 - 0
⇒ 2x = 9
⇒ x = $\dfrac{9}{2}$ = 4.5 cm/hr.

On substituting value of x in (ii) we get,

4x - 3y = 0

4(4.5) - 3y = 0

18 - 3y = 0

3y = 18

y = 6 cm/hr.

Length of longer candle = 6x = 6(4.5) = 27 cm.

Length of smaller candle = 4y = 4(6) = 24 cm.

Hence, length of longer candle = 27 cm and length of smaller candle = 24 cm.