There are two candles of same length and same size.both of them burn at uniform rate. The frist one burns in 5 hours and the second one the second one burns in 3 hours. Both the candles are lit together. After many minutes the length of the first candle is 3 times is 3 times that of the other ?

Options

A. 90
B. 120
C. 135
D. 150

Explanation

To solve the problem, we need to determine how long it takes for the length of the first candle to be three times that of the second candle after both are lit together.

### Step-by-Step Solution:

1. **Define the Variables:**
Let the initial length of both candles be \( L \).
- The first candle burns completely in 5 hours.
- The second candle burns completely in 3 hours.

2. **Calculate the Burning Rates:**
- The burning rate of the first candle, \( V_1 \):
\[ V_1 = \frac{L}{5} \text{ (length burned per hour)} \]
- The burning rate of the second candle, \( V_2 \):
\[ V_2 = \frac{L}{3} \text{ (length burned per hour)} \]

3. **Determine the Lengths After Time \( t \):**
After \( t \) hours:
- Length of the first candle remaining:
\[ L_1 = L - V_1 \cdot t = L - \frac{L}{5}t = L \left(1 - \frac{t}{5}\right) \]
- Length of the second candle remaining:
\[ L_2 = L - V_2 \cdot t = L - \frac{L}{3}t = L \left(1 - \frac{t}{3}\right) \]

4. **Set Up the Equation:**
We need to find \( t \) such that:
\[ L_1 = 3L_2 \]
Substituting the expressions for \( L_1 \) and \( L_2 \):
\[ L \left(1 - \frac{t}{5}\right) = 3L \left(1 - \frac{t}{3}\right) \]

5. **Simplify the Equation:**
Dividing both sides by \( L \) (assuming \( L \neq 0 \)):
\[ 1 - \frac{t}{5} = 3 \left(1 - \frac{t}{3}\right) \]
Expanding the right side:
\[ 1 - \frac{t}{5} = 3 - t \]

6. **Rearranging the Equation:**
Bringing all terms involving \( t \) to one side:
\[ 1 - 3 = -t + \frac{t}{5} \]
\[ -2 = -t + \frac{t}{5} \]
Multiplying through by 5 to eliminate the fraction:
\[ -10 = -5t + t \]
\[ -10 = -4t \]
\[ t = \frac{10}{4} = 2.5 \text{ hours} \]

7. **Convert to Minutes:**
Since \( t \) is in hours, converting to minutes:
\[ t = 2.5 \times 60 = 150 \text{ minutes} \]

### Final Answer:
After 150 minutes, the length of the first candle will be three times that of the second candle.

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