# Complicated Math Problem For Solving (:?

I have a math problem called "Hands of Time" and it reads:

Katie is always working on math problems in her spare time. She begins to solve a particularly complicated math problem between 4:00 and 5:00 p.m. when the clock's hour and minute hands are together (on top of each other). She finishes the problem when the minute hand is exactly opposite the hour hand.

Part 1: How many minutes does it take her to solve the problem?

Part 2: At what time does Katie finish the problem?

You must answer both parts to the nearest second.

I could really really use some help from anyone who has time tonight. So thank you so much!! I would really appreciate it. (:

### 4 Answers

- PuzzlingLv 71 decade agoFavorite Answer
Let's start at 4 pm. Here the minute hand is straight up and the hour hand is on the 4. The angle between them is 120° (1/3 of 360°)

Now it takes 60 minutes for the minute hand to make a full revolution. So that is 360/60 = 6° per minute.

It take 12 hours (720 minutes) for the hour hand to make a full revolution. So that is 360/720 = ½° per minute.

In other words, each minute the angle between the two hands changes by a net of 5½°.

Let's first figure out how many minutes it will take from 4pm so that the hands are touching (i.e. 0° apart). That's 120° at a rate of 5½° per minute.

120 / 5.5 = 240/11 = 21 and 9/11 minutes

That's equal to 21 minutes 49 seconds

That's our starting time:

4:21:49

Now how many minutes to go another 180° (so the hands are opposite)?

180 / 5.5 = 360/11 = 32 and 8/11 minutes

That's equal to 32 minutes 44 seconds

Adding that to the time of 4:21:49 we get 4:54:33

Answers:

Part 0 - She starts working at 4:21:49 when the hands are together.

Part 1 - It takes 32 minutes 44 seconds for her to finish the problem.

Part 2 - Her ending time is 4:54:33 pm when the hands are opposite each other.

- 1 decade ago
well, it takes her 30 min, because the hands were together then exactly opposite, in order to do this, half an hour must have past. For the other part, if you think about the tic marks on a a clock, there are five marks between hours (after the four to the five.) So, each mark represents a 12 minute increment. Now, in order to for the hands to start out on top of eachother, she would have needed to start at least around 4: 20 (a little after actually). It can't be at the 36 minute tick mark, b/c then it's too late to complete the half hour, so the start is around 4:24 and the end 4:54 but you'll have to do the finer tuning yourself ^^

- Brian DLv 51 decade ago
the minute hand moves at 360 degrees per hour ..6 degrees per minute

the hour hand at 30 degrees per hour , half a degree per minute

at the time she starts (after 4pm) the hour hand is at 120degrees, second hand at 0degrees

t*(6/min) = 120 + t(.5/min)

t(5.5/min)= 120 .......21.8181...[4hrs,21min, 49.08sec]

finish time is when hands form angle of 180

[t*6/min] - [120+[t*.5/min]] = 180

t* 5.5 = 300 ..

t= 54.545454 ...4hrs,54min32.4 sec

ends ....4:54:32

starts... 4:21:49

32:43