A siren sounds once every 5 seconds. How many times will it sound in three quarters (3/4) of an hour?

Introduction / Context:
This puzzle is about counting events that occur at regular time intervals. The siren sounds once every 5 seconds, and we are asked how many times it will sound in three quarters of an hour. Such problems test your understanding of time conversions and whether you correctly interpret whether the first sounding happens at time zero or after the first interval. These types of questions appear frequently in basic arithmetic and reasoning tests.

Given Data / Assumptions:

- The siren sounds at a regular interval of every 5 seconds. - The total duration considered is three quarters of an hour, that is 3/4 hour. - One hour has 60 minutes, and each minute has 60 seconds. - We assume the siren sounds immediately at the start (time zero) and then at every 5 second interval afterward. - The sounding is treated as an instantaneous event at each interval.

Concept / Approach:
First, convert three quarters of an hour into seconds. Then determine how many 5 second intervals fit into this total duration by dividing the total seconds by 5. If we assume the siren sounds at time zero and at the end of each full interval, the total number of soundings is one more than the number of completed intervals. This interpretation is common in standard exam questions of this form, which is why the answer often differs by one from the simple division.

Step-by-Step Solution:
Step 1: Convert three quarters of an hour into minutes: (3 / 4) * 60 = 45 minutes. Step 2: Convert 45 minutes into seconds: 45 * 60 = 2700 seconds. Step 3: Compute the number of 5 second intervals in 2700 seconds: 2700 / 5 = 540 intervals. Step 4: If the siren sounds first at time zero and then at the end of each 5 second interval, the number of soundings equals the number of intervals plus 1. Step 5: Therefore, total soundings = 540 + 1 = 541. Step 6: Hence the siren will sound 541 times in three quarters of an hour.

Verification / Alternative check:
Consider a smaller example: if the siren sounds every 5 seconds over a period of 10 seconds, it would sound at time 0, time 5, and time 10 seconds. That is 3 soundings because there are two full intervals plus the initial sounding. Similarly, in 2700 seconds there are 540 full 5 second intervals and one sounding at time zero, so 540 plus 1 equals 541. This confirms that including the starting sounding is consistent with the typical interpretation of such questions.

Why Other Options Are Wrong:
The value 540 corresponds to counting only the completed intervals and omitting the initial sounding at time zero, which is not the usual interpretation of sounds occurring every 5 seconds. The value 450 would imply an interval larger than 5 seconds or a shorter overall time, and 275 is even further off. The option 600 does not align with the precise conversion of 45 minutes into 2700 seconds and dividing by 5 seconds per interval. Therefore, only 541 times matches the correct counting logic.

Common Pitfalls:
The most common error is to answer 540 by simply dividing 2700 by 5 and forgetting to consider the first sounding at time zero. Another pitfall is mixing minutes and seconds, for example dividing 45 by 5 instead of converting to seconds first. Carefully converting units and thinking about whether the event occurs at the start of the timing window as well as at the end of each interval will help avoid these traps.

Final Answer:
The siren will sound 541 times in three quarters of an hour.