# How many times is the light bulb turned on?

A light bulb is automatically turned off for 21 secs. Next, it lights up automatically for 15 secs. Once again, it is automatically turned off for the following 21 secs. The process keeps repeating.

After 200 full mins, how many times has the bulb light up after turning off?

My approach was the following:

21 + 15= 36 secs

200 mins x 60 = 12000 secs

12000/36 = 333

so the bulb lit up around 333 times but this isn't the correct answer.

• Can you post the exact wording of the problem? I doubt that the problem asked how many times "has the bulb light up"? Commented Feb 21 at 2:11

We know that...

• The bulb is off for $$21$$ seconds.
• Then the bulb is on for $$15$$ seconds.
• The pattern occurs for $$200$$ full minutes, which is $$12,000$$ seconds.

From this, we see that one full cycle of on/off states of the bulb can be represented by $$(21+15)$$ seconds. Now we need to figure out how many times that cycle occurs in $$12,000$$ seconds.

$$12,000/(21+15)=333\frac13$$

This tells us that the bulb will go through $$333$$ on/off cycles, and only $$\frac13$$ of the $$334$$th cycle (which is $$12$$ seconds). Those $$12$$ extra seconds are not enough for the bulb to turn on again, so the bulb will turn on a total of $$333$$ times.

However, this might be wrong due to the ambiguity of the question. You said that the bulb would first be off for $$21$$ seconds, then turn on for $$15$$ seconds, then turn off for another $$21$$ seconds. If this was the writer's intended pattern, then our answer would change.

$$(21+15+21)$$ seconds = $$1$$ on/off cycle

$$12,000/(21+15+21)\approx210.526$$

This means that the bulb would go through $$210$$ on/off cycles, and about halfway through the $$211$$th one (which is $$\approx30$$ seconds). This extra $$30$$ seconds is enough for the bulb to transition from off to on, so the bulb will turn on a total of $$211$$ times.

• This doesn't seem right though: The number of on-off cycles should be about the number of times the lightbulb goes on, to within $\pm 1$. In particular, if you consider the first cycle to have $21+15+21$ seconds, then wouldn't the second have $15+21+15$ seconds?
– Mike
Commented Feb 20 at 23:15
• @Mike That's if we assume the pattern is a repetition of 21 seconds off and 15 seconds on. But if we take the question exactly as it's worded, then the pattern would be a repetition of 21 seconds off, 15 seconds on, and 21 seconds off. It would be like an ABA pattern, as opposed to an AB pattern. Commented Feb 21 at 0:51
• Yes I do see it now...ABAABAABAABA... [so ABA repeating until the time is up], as opposed to ABAB...
– Mike
Commented Feb 21 at 21:34

Your answer will be right if we assume at the beginning (t=0 second) the bulb turns from light to off. But if for t=0 second, the bulb turns from off to light, the answer is 333+1 since 12000/36=333.33333, the last incomplete loop also has one light up.