# 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.
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