Imagine a detailed scene. At the center of the scene, there's a large, intricately designed clock. Its hands are pointed to 9, indicating that it's 9 a.m. Behind the clock, you notice a grand factory. The facade of the factory features an assortment of windows and other architectural details. It's a peaceful morning, despite the constant hum of industry coming from the factory. Surrounding the scene are bees, flying back and forth, their wings creating a slight buzz. In the distance, you can hear the consistent ringing of bells. The entire picture feels like a harmonious blend of nature and industry.

In a certain factory a buzzer and bell both sound together at 9 a.m. every day, when the factory opens. The buzzer buzzes every 1 3/5 hours after that, and the bell gongs every 54 minutes. When they both sound together again the factory closes for the night, and the buzzer and bell stop their noise making until until 9 a.m. the next day. How many minutes before midnight does the factory close?

(Please, please, PLEASE answer)

1 3/5 hr = 96 min

LCM(54,96) = 2^5 * 3^3 = 864 min = 14:24
So the factory closes at 23:24, or 36 min before midnight

Yo you from spirit of math?

Why did the factory need a buzzer and a bell in the first place? Did they hire a DJ to make sure everyone knows when it's time to start and end the factory party? 🎉

Anyway, let's get down to business. To find out when the factory closes, we need to find the time when the buzzer and bell sound together again, right before midnight.

The buzzer buzzes every 1 3/5 hours, which is the same as 1 hour and 36 minutes. The bell gongs every 54 minutes. We need to find the least common multiple (LCM) of these two times.

LCM of 96 (1 hour and 36 minutes) and 54 is 432 minutes.

That means, every 432 minutes, the buzzer and bell will sound together.

So, to find out how many minutes before midnight the factory closes, we need to find how many times 432 minutes can fit into the time remaining until midnight.

There are always 24 hours in a day, which is 1440 minutes. If we subtract the current time (9 a.m.) from midnight (12 a.m.), we get 15 hours, which is 900 minutes.

900 minutes divided by 432 minutes gives us a quotient of 2 with a remainder of 36.

Since we can fit the buzzer and bell sounding together twice into the remaining time before midnight, the factory will close 2 times 432 minutes before midnight. That's 864 minutes.

So, the factory will close 864 minutes, or 14 hours and 24 minutes, before midnight. Time to close up shop and get your beauty sleep! 😴

To determine how many minutes before midnight the factory closes, we need to calculate the time interval between the initial opening of the factory at 9 a.m. and when the buzzer and bell sound together again.

First, let's convert 1 3/5 hours into minutes to find the time interval for the buzzer.

1 3/5 hours = 1 hour + 3/5 hour = 60 minutes + 3/5 * 60 minutes = 60 + 36 = 96 minutes.

Now, we know that the bell gongs every 54 minutes. To find the time interval for the bell, we need to find the least common multiple (LCM) of 96 and 54.

The LCM of 96 and 54 is 864. This means that the buzzer and the bell will both sound together every 864 minutes.

To find how many minutes from 9 a.m. until midnight is 12 hours * 60 minutes = 720 minutes.

Now, we divide the total number of minutes until midnight (720) by the time interval for the buzzer and bell to find the number of times both sound together:

720 minutes ÷ 864 minutes = 0.8333.

This means that the buzzer and bell will sound together approximately 0.8333 times before midnight.

Since the buzzer and bell sound together when the factory closes for the night, and they start at 9 a.m., the factory will close approximately at:

9 a.m. + (0.8333 times * 864 minutes) = 9 a.m. + 720 minutes = approximately 9:12 p.m.

Therefore, the factory will close approximately 12 minutes before midnight.

Don't you think we would've answered by now if we knew?

Read the "Bob"!

PLZ PLZ PLZZZZZZ ANSWERRR!!!!!!!!!!!!!