Clock

A broken clock is set corectly at 12:00 noon. However, it registers only 20 minutes for each hour. In how many hours will it again register the correct time  ?

A)12

B)18  analog clock(12h)

C)24 

D)30

E)36  digital clock (12h)

1. Analog Clock:

 $\begin{array}{rcll} \text{angular velocity real time: } \omega_1 &=& \frac{2\pi}{12\ h} \\ \text{angular velocity display time: } \omega_2 &=& \frac{2\pi}{36\ h} \\ w_2\cdot t - w_1\cdot t &=& n\cdot 2\pi \\ \frac{2\pi}{12\ h}\cdot t - \frac{2\pi}{36\ h}\cdot t &=& n\cdot 2\pi \qquad &| \qquad :2\pi \\ \frac{1}{12\ h}\cdot t - \frac{1}{36\ h}\cdot t &=& n \\ t\cdot (\frac{1}{12\ h} - \frac{1}{36\ h} ) &=& n \\ t\cdot ( \frac{36-12}{12\cdot 36} ) &=& n \\ t\cdot ( \frac{24}{12\cdot 36} ) &=& n \\ t\cdot ( \frac{2}{36} ) &=& n \\ t\cdot ( \frac{1}{18} ) &=& n \\ t &=& 18\cdot n \\ \end{array}$

next correct time, if n = 1, then t = 18 hours.

2. Digital Clock(24h):

$\begin{array}{rcll} \text{angular velocity real time: } \omega_1 &=& \frac{2\pi}{24\ h} \\ \text{angular velocity display time: } \omega_2 &=& \frac{2\pi}{72\ h} \\ w_2\cdot t - w_1\cdot t &=& n\cdot 2\pi \\ \frac{2\pi}{24 h}\cdot t - \frac{2\pi}{72\ h}\cdot t &=& n\cdot 2\pi \qquad &| \qquad :2\pi \\ \frac{1}{24\ h}\cdot t - \frac{1}{72\ h}\cdot t &=& n \\ t\cdot (\frac{1}{24\ h} - \frac{1}{72\ h} ) &=& n \\ t\cdot ( \frac{72-24}{24\cdot 72} ) &=& n \\ t\cdot ( \frac{48}{24\cdot 72} ) &=& n \\ t\cdot ( \frac{2}{72} ) &=& n \\ t\cdot ( \frac{1}{36} ) &=& n \\ t &=& 36\cdot n \\ \end{array}$

next correct time, if n = 1, then t = 36 hours.

I depends upon the type of clock that we are talking about.....

Note, Solveit.....every 3 hrs, it will  fall 2 hours behind....

In 6 hrs......it will be 4 hours behind

In 12 hrs.......it will be 8 hours behind

So....in 18 hours, it will be 12 hours behind.......and, as long as it's an analog clock....the clock's hands will appear to show the correct time, then !!!!

However.....if it's a digital clock......36 hours would have to elapse to show the correct hour......[ the digital clock would show AM or PM ]

There are 24 hours in a day and the clock is only going 1/3 the speed that it should

So in 24*3=72 hours

So we know that in 72 hours the clockes will be the same but will they be the same before that?

Ok so it is 36 hours.     Now I am trying to put this into a formula............

it is an analog clock 

Hi Solveit,

If you do not differentiate between AM and PM then obviously your answer of 18 hours is correct.

I am not sure about the formula either. :/

Ok Melody thank you and CPhill ! You can put a check mark :)

Thanks Heureka ! :)

Thanks, heureka   !!!!