Clock word problem

In this problem we will ﬁgure out all the times when the two hands of a

clock are pointing in exactly the same direction and all of the times when

the two hands are exactly 180

◦

apart.

a) According to a certain clock it is currently 2:16. At 2:16, the hour hand

of the clock is not pointing exactly at the 2, rather it is pointing at part way

between the 2 and the 3. Write decimals between 0 and 12 to express in hour

units exactly where each of the hands of the clock is pointing.

b) Suppose the current time is h:m o’clock. That is, it is m minutes after the

hour of h. So h is a whole number between 1 and 12, and m is a real number

between 0 and 60. Write decimals between 0 and 12 to express in hour units

exactly where each of the hands of the clock is pointing.

c) For particular values of h and m, when the time is h:m o’clock, the two

hands of the clock are pointing in exactly same direction. Write an equation

expressing m as a function of h.

d) Use each value of h between 1 and 12 to determine all of the times (up

to 2 decimal places) when both hands of a clock are pointing in exactly the

same direction.

Please answer part d

### 2 Answers | Add Yours

The two hands are exactly in the same direction for a 12 hour clock only 12-1=11 times, hence interval between each such time = 12/11 hours = 1h+(60/11)m or 1h 05.4545m

**d) All such times are:**

1*(1h 05.4545m) = 01h 05.45m = **01:05.45**

2*(1h 05.4545m) = 02h 10.91m = **02:10.91**

3*(1h 05.4545m) = 03h 16.36m = **03:16.36**

4*(1h 05.4545m) = 04h 21.82m = **04:21.82**

5*(1h 05.4545m) = 05h 27.27m = **05:27.27**

6*(1h 05.4545m) = 06h 32.73m = **06:32.73**

7*(1h 05.4545m) = 07h 38.18m = **07:38.18**

8*(1h 05.4545m) = 08h 43.64m = **08:43.64**

9*(1h 05.4545m) = 09h 49.09m = **09:49.09**

10*(1h 05.4545m) = 10h 54.55m = **10:54.55**

11*(1h 05.4545m) = 11h 60.00m = **12:00.00**

On finding part c, we can express m as a function of h, by the equation,

**m = h x 5.4545 **or** m = h x (60/11)**

So, the times at which the two hands of the clock coincide and point towards the same direction are,

time = h + m , where m = h x (60/11)

or **time = h + [h x (60/11)]**

Therefore, the 12 times with value of h from 1 to 12 are,

h = 1 ; time = 1 h + [1 x (60/11)] m ; time = 01 h : 5.45 min

h = 2 ; time = 2 h + [2 x (60/11)] m ; time = 02 h : 10.91 min

h = 3 ; time = 3 h + [3 x (60/11)] m ; time = 03 h : 16.36 min

h = 4 ; time = 4 h + [4 x (60/11)] m ; time = 04 h : 21.82 min

h = 5 ; time = 5 h + [5 x (60/11)] m ; time = 05 h : 27.27 min

h = 6 ; time = 6 h + [6 x (60/11)] m ; time = 06 h : 32.73 min

h = 7 ; time = 7 h + [7 x (60/11)] m ; time = 07 h : 38.18 min

h = 8 ; time = 8 h + [8 x (60/11)] m ; time = 08 h : 43.64 min

h = 9 ; time = 9 h + [9 x (60/11)] m ; time = 09 h : 49.09 min

h = 10 ; time = 10 h + [10 x (60/11)] m ; time = 10 h : 54.55 min

h = 11 ; time = 11 h + [11 x (60/11)] m ; time = 12 h : 0.00 min

h = 12 ; time = 12 h + [12 x (60/11)] m ; time = 1 h : 5.45 min (the value is same as when h = 1 and the cycle continues)

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