A buzzer sounds every $15$ minutes. If the buzzer sounded at $12:40$PM, which of the following could be a time at which the buzzer sounded?
A. $4:05$
B. $5:30$
C. $6:45$
D. $7:15$
E. $8:10$
I chose E. This is exactly the same as but I don't know why as if the buzzer sounded at $12:30$PM and the answer choices were the following:
A. $3:55$
B. $5:20$
C. $6:35$
D. $8:00$
Now you'd have to have the minute hand be at $00, 15, 30, 45$. E.
Why does this subtraction strategy work?
Of course the problem would be exactly the same according to me if you added five to the $12:40$ and every answer choice by five, but why does this simplification work?
The key is that the buzzer is going off every 15 minutes, when it starts is really immaterial - what affects the possibility of an answer is whether the difference between the given start time and the answer is a multiple of 15 minutes. So long as you add or subtract the same number of minutes from the start time and all of the answers, you haven't changed the length of time between them.
It might be helpful to think about an hourly buzzer to remove any considerations about minutes and modular arithmetic. If an hourly buzzer went off at 3:05, it could only go off next at 4:05, not 4:15. Subtracting 5 minutes from all of those times gives an equivalent problem - an hourly buzzer going off at 3:00 would go off at 4:00, not 4:10.