Question

There are two candles of the same length and same size. Both of them burn at a uniform rate. The first one burns in 5 hours and the second one burns in 3 hours. Both the candles are lit together. After how many minutes the length of the first candle is 3 times that of the other?
1) 90
2) 120
3) 135
4) 150

Answer 1

Hint: Here, we will find the burning rate of candles by dividing the lengths of candles with the time it takes to burn. Then we will find the lengths of the candle after time \[t\] and substitute the obtained values in the given condition.

Complete step-by-step answer:
Let us assume that the length of each candle is \[l\].

First, we will find the burning rate\[l\] of the first candle by dividing the above length with 5 hours.

\[\dfrac{l}{5}\]

Then, we will find the burning rate of the second candle by dividing the above length \[l\] with 3 hours.

\[\dfrac{l}{3}\]

Now, we will find the lengths of the first candle \[{l_1}\] and second candle \[{l_2}\] after time \[t\].
\[{l_1} = l - \dfrac{l}{5}t\]

\[{l_2} = l - \dfrac{l}{3}t\]

Since we are given that after time \[t\], the length of the first candle \[{l_1}\] is 3 times the length of the second candle \[{l_2}\].

So, we will have \[{l_1} = 3{l_2}\] to find the value of \[t\].

Substituting the above values of \[{l_1}\] and \[{l_2}\] in \[{l_1} = 3{l_2}\], we get

\[
   \Rightarrow l - \dfrac{l}{5}t = 2\left( {l - \dfrac{l}{3}t} \right) \\
   \Rightarrow \dfrac{{5l - lt}}{5} = 3\left( {\dfrac{{3l - lt}}{3}} \right) \\
   \Rightarrow \dfrac{{l\left( {5 - t} \right)}}{5} = 3l\left( {\dfrac{{3 - t}}{3}} \right) \\
   \Rightarrow \dfrac{{5 - t}}{5} = 3 - t \\
\]

Cross-multiplying the above equation, we get

\[ \Rightarrow 5 - t = 15 - 5t\]

Combine the like terms in the above equation, we get

\[
   \Rightarrow 5t - t = 15 - 5 \\
   \Rightarrow 4t = 10 \\
\]

Dividing the above equation by 4 into each of the sides, we get

\[
   \Rightarrow \dfrac{{4t}}{4} = \dfrac{{10}}{4} \\
   \Rightarrow t = 2.5{\text{ hrs}} \\
\]

Converting this value in minutes, we get

\[2.5 \times 60 = 150{\text{ min}}\]

Thus, we have found out that it will take 150 minutes to the length of the first candle to be 3 times the length of the other.

Hence, option D is correct.

Note: In this question, we have to first assume the length of the candle and then use this length to find the rate of change of both the candles. We have to use the given condition of that after time \[t\], the length of the first candle is three times the length of the second candle.