Two Candles

Let $a$ and $A$ represent the cross-sectional areas of the long and short candle respectively. Similarly, let $H$ and $h$ represent the heights of the long and short candle respectively. Then the long candle has a volume of $aH$ and short $Ah$ .

$\text{Volume consumption per hour must be equal for both.}$

$\implies~~\dfrac{aH}{3.5}=\dfrac{Ah}{5}.$ $\text{Let x be the height left after 2 hrs.}$
$\therefore \text{volumes ~left~ are} ~~~aH\dfrac{3.5-2} {3.5}=ax~~and~~Ah\dfrac{5-2} {5}=Ax.$
$\therefore~ H\dfrac{3.5-2} {3.5 }=x= h\dfrac{5-2} {5}$
$\implies~\dfrac h H=\dfrac{ \frac {1.5}{3.5}} {\frac{3}{5} } =\Large \color{#D61F06}{\dfrac 5 7}$

LET HEIGHT OF LONGER CANDLE BE "x" AND HEIGHT OF SHORTER ONE BE "y".
We need to find out y/x.
According to question:
$x-(\frac {2}{3.5})x = y-(\frac{2}{5})y$
$=> \frac{1.5}{3.5}x = \frac{3}{5}y$
$=> \frac{1.5×5}{3.5×3} = \frac{y}{x}$
$=> \frac{75}{105} = \frac{y}{x}$
$\boxed{\frac{5}{7}}=\frac{y}{x}$

Suppose volume & length of long candle is 100kg & 100m,
There fore volume of short candle = (100 * 5)/3.5 = 1000/7 kg
After 2 hrs. the length of long candle remain = 100 - 100 * 2/3.5 = 300/7 m & is also equal to length of short candle.
So total length of short candle = (300 * 5)/7 *3 = 500/7 m
Ratio = (500/7)/100 = 5/7

The hint is to notice that the height of both the candles will be the same after burning 2 hours.

Here the trick is to divide the length of the candles proportionate to their total times.

In case of the short candle, after burning 2 hours the remaining height is equal to the height of the tall candle and must be burned in 3 hours because it takes a total of 5 hours to burn.

In case of the tall candle, after burning 2 hours the remaining height must be burned in 1.5 hours since it takes 3.5 hours in total to burn the whole.

So, to stitch the two equality of height we divide the heights proportionally with respect to their time of consumption.

Clearly, we know the small candle is divided into 5 equal parts where each part takes an hour to burn. Therefore, to apply this method to the tall candle, we think of way to divide each section of the tall candle such that the ratio is a natural number because it must fit the description. Which clearly leads to 0.5 hours per section of the tall candle where we can see 7 times 0.5 gives 3.5.

Therefore, with this we can side step the volume calculation, speed ratio calculation etc.. Which gives 5/7 as the ratio and the answer to the question.

First I was going to consider the volumes as well but then I realized it wasn't necessary, and a similar problem could have different candles or objects. For example one candle could be made of different materials that allow it to burn more slowly. So in my opinion the volume and width of the candles are actually irrelevant. But it is also the same as a problem with two objects at two different distances and different speeds, who meet in the middle.

So: Hs = Height of the small candle, Ss = "speed" (or the rate at which it burns down) of the small, Ht = height tall, Ss = "speed" tall, Ha = Height of both after two hours.

$\frac {Hs}{Ss} = 5h$ and $\frac {Ht}{St} = 3.5h$

$\frac {Ha}{Ss} = (5h-2h) = 3h$ and $\frac {Ha}{St} = (3.5h - 2h) = 1.5h$

$Ha = 3Ss$ and $Ha = \frac {3St}{2}$

$6Ss = 3St$ -->> $2Ss = St$

Go back to the start and replace St for 2Ss

$5Ss = Hs$ and $7Ss = Ht$

$\frac {Hs}{Ht} = \frac {5}{7}$

Suppose, The height of the long candle is x meter and the short candle is y meter For the long candle, 3.5 hours is needed to burn x meter so, after 2 hours it will burn 2x/3.5 meter Then the remaining length of the candle is, (x - 2x/3.5) = 1.5x/3.5 meter

For the short candle, 5 hours is needed to burn y meter So, after 2 hours it will burn 2y/5 meter The remaining length of the candle is (y - 2y/5) = 3y/5

Therefore, 1.5x/3.5 = 3y/5 (Since, after 2 hours the length of the both candle is equal)
=> 7.5x = 10.5y => y/x = 7.5/10.5 => y/x = 5/7

           So, the answer is 5/7

Long one height x and its rate b dx Let shott one height y and its rate dy Then x=dx ×3.5 Y=dy x5 Again x-dx×2=y-dy×2 After solving both equation we have 5:7