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: Express how much of each candle burns per hour, as a fraction of the whole.
Since the thick candle fully burns out in 6 hours, each hour it burns exactly 1/6 of itself. Since the thin candle fully burns out in 4 hours (6 hours minus 2), each hour it burns exactly 1/4 of itself.
Step 2: Write the fraction of each candle that is left after t hours.
Fraction of thick candle left = 1 - t/6. Fraction of thin candle left = 1 - t/4. Since both candles started at the same length, these fractions can be compared directly without needing to know the actual length.
Step 3: Translate "thick candle is twice as long as thin candle" into an equation using these fractions.
Twice as long means: (fraction of thick left) = 2 x (fraction of thin left). Substituting: 1 - t/6 = 2(1 - t/4).
Step 4: Expand and simplify.
Expanding the right-hand side: 2(1 - t/4) = 2 - t/2. So the equation becomes 1 - t/6 = 2 - t/2. Rearranging: t/2 - t/6 = 2 - 1 = 1.
Step 5: Solve for t using a common denominator.
t/2 = 3t/6, so 3t/6 - t/6 = 2t/6 = t/3. Setting t/3 = 1 gives t = 3 hours.
Step 6: Sanity-check the answer.
After 3 hours: thick candle left = 1 - 3/6 = 1/2 of its length; thin candle left = 1 - 3/4 = 1/4 of its length. Since 1/2 is exactly twice 1/4, this confirms t = 3 is correct.
Ramaswami studied by candlelight for 3 hours. \[\boxed{3 \text{ hours}}\]