# Melting Candles | Puzzle

**Problem Statement:**

- There are two candles of equal lengths but of different thicknesses.
- The thicker candle lasts for 6 hours while the thinner one lasts for 2 hours lesser than the thicker one.
- A person lights the two candles at the same time and goes to play.
- After returning home, the person saw that the thicker candle was twice the length of the thinner one. How long ago did the person light the two candles?

**Solution:**

- According to the problem statement, the thicker candle lasts for 6 hours while the thinner one lasts for 2 hours lesser than the thicker one.
- Therefore, the thinner candle lasts for 4 hours.
- The problem reduces to finding the number of hours it takes for the thicker candle to become twice the length of the thinner candle.
- Analyzing the problem hour by hour, after each hour
**(1/6)**part of the thicker candle is melted while**(1/4)**part of the thinner candle is melted. - After 1 hour,
**(5/6)**part of the thicker candle remains while**(3/4)**part of the thinner candle remains. - After 2 hours,
**(4/6)**part of the thicker candle remains while**(2/4)**part of the thinner candle remains. - After 3 hours,
**(3/6)**part of the thicker candle remains while**(1/4)**part of the thinner candle remains. - On observing carefully, it can be found that at the end of 3 hours the remaining part of the thicker candle is
**(1/2)**while it is**(1/4)**for the thinner candle i.e., the thicker candle is twice the length of the thinner one.

**Another Approach : **

- Assume Candle 1 is thinner and Candle 2 is thicker.
- Suppose Length of Candle 1 = L and Length of Candle 2 = L
- According to the problem statement, the thicker candle lasts for 6 hours while the thinner one lasts for 2 hours lesser than the thicker one. Therefore, the thinner candle lasts for 4 hours.
- After returning home, the person saw that the thicker candle(Candle 2) was twice the length of the thinner(Candle 1) one.
- Suppose remaining length of Candle 1 = x. So, Remaining length of Candle 2 = 2x.
- For Candle 1, Time taken for melting (L – x) part =T1= ((L – x)*4)/L
- Similarly, For Candle 2, Time taken for melting (L – 2x) part = T2 = ((L – 2x)*6)/L
- We know that T1 andT2 are same. So, T1= T2
- So, By solving above two equation, we get the relation L = 4x
- So, Total Length of Candle 1 and Candle 2 are 4x.
- So, For Candle 2, (4x – 2x) = 2x part is melting that means half of the Candle 2 is melting. So Time taken for melting half of the Candle 2 = 6 / 2 = 3 hour.
- So, person light the two Candles 3 hours ago.