Question

Ramaswami was studying for his examinations and the lights went off. It was around 1:00 a.m. He lighted two uniform candles of equal length but one thicker than the other. The thick candle is supposed to last six hours and the thin one two hours less. When he finally went to sleep, the thick candle was twice as long as the thin one. For how long did Ramaswami study in candle light?

• A. 2 hours
• B. 3 hours
• C. 2 hours 45 minutes
• D. 4 hours

Hint:

Write the remaining length of each candle as a fraction of the original length after t hours, then use the "twice as long" condition to solve for t.

Answer 1

The Correct Option is B

Step 1: Note the burn rates of the two candles.
Both candles start out the same length, call it L. The thick candle takes 6 hours to burn completely, so it burns at a rate of L/6 of its length per hour. The thin candle takes two hours less, that is 6 - 2 = 4 hours, to burn completely, so it burns at a rate of L/4 of its length per hour.
Step 2: Write expressions for the remaining length of each candle after t hours.
After burning for t hours, the thick candle has burnt away (L/6)t of its length, so its remaining length is L - (L/6)t = L(1 - t/6). Similarly, the thin candle's remaining length is L - (L/4)t = L(1 - t/4).
Step 3: Use the condition that the thick candle is twice as long as the thin candle when he stops.
At the time Ramaswami stopped studying, after t hours, remaining thick length = 2 x remaining thin length: L(1 - t/6) = 2 x L(1 - t/4). Since L is common and nonzero, it cancels out: 1 - t/6 = 2(1 - t/4) = 2 - t/2.
Step 4: Solve the equation for t.
1 - t/6 = 2 - t/2. Bring the t terms to one side and the constants to the other: t/2 - t/6 = 2 - 1 = 1. Using a common denominator of 6: 3t/6 - t/6 = 1, so 2t/6 = 1, which simplifies to t/3 = 1, giving t = 3.
Step 5: Interpret the result.
Ramaswami studied by candlelight for 3 hours, starting around 1:00 a.m., meaning he stopped studying and went to sleep at around 4:00 a.m. This matches option (2).