Watch problems gaining and losing time

[Natasha1]

1. My watch (which is a 12 hour watch) gains 3 minutes every 2 hours.

a) I set my watch to the correct time at noon on 1st January. If I don't reset it, when will it next show the correct time?

I got 48 hours after, so that's at noon on the 3rd January. as if my watch gains 3 minutes every two hours then over 24 lots of 2 hours, it would have gained an extra 2 hours. So that is 24 lots of 2 hours = 24 x 2 = 48 hours added to the 'correct' starting time of noon. Hence, my answer of noon on the 3rd January.2. Mrs Varma's watch (also a 12 hour watch) loses 5 minutes every 2 hours. She also sets her watch to the correct time at noon on 1st January.

b) When will our two watches next show the same time?

I am completely stuck here :(... please help to explain to me how to do this...3. When will our watches next show the same, CORRECT time?

Again, I can't do this... Any explanation would be greatly appreciated. Thank you.

Nat.

 

[Samy_A]

1. My watch (which is a 12 hour watch) gains 3 minutes every 2 hours.

a) I set my watch to the correct time at noon on 1st January. If I don't reset it, when will it next show the correct time?

I got 48 hours after, so that's at noon on the 3rd January. as if my watch gains 3 minutes every two hours then over 24 lots of 2 hours, it would have gained an extra 2 hours. So that is 24 lots of 2 hours = 24 x 2 = 48 hours added to the 'correct' starting time of noon. Hence, my answer of noon on the 3rd January.

If you watch gains 3 minutes in 2 hours, it gains 3*12=36 minutes in a day. So it can't show the correct time after only two days.
Try to write the time on your watch as a function of the correct time.
Say $W(t)=$ some function of the correct time $t$ (expressed in minutes)
Then for your clock to again show the correct time, you should have $W(t)=t+720$ (because there are 720 minutes in 12 hours, and your clock has to gain 12 hours to again show the correct time). Solve that equation for $t$.

Use another similar function to represent the time on Mrs Varma's watch to solve 2) and 3).

 

[Natasha1]

Ok, I got January 4th at 8pm. Is this correct?

Cannot do 2b nor 3. Can anyone help me?

 

[Samy_A]

Ok, I got January 4th at 8pm. Is this correct?

It would help if you showed how you got that result.
I find something different. Note that your watch only gains 36 minutes a day, so it can't catch up in 3 days and 8 hours.

The time on your watch can be expressed as $W(t)=t+t*3/120$, where $t$ is the correct time in minutes, and $t=0$ represents noon on 1st January.
Your watch will again show the correct time when $W(t)=t+720$.

So we look for the solution of $$t+t*3/120=t+720$$

Cannot do 2b nor 3. Can anyone help me?

Express the time on Mrs Varma's watch by a function $V(t)$, similar to the function $W(t)$ expressing the time on your watch.

 

[PeroK]

1. My watch (which is a 12 hour watch) gains 3 minutes every 2 hours.

a) I set my watch to the correct time at noon on 1st January. If I don't reset it, when will it next show the correct time?

I got 48 hours after, so that's at noon on the 3rd January. as if my watch gains 3 minutes every two hours then over 24 lots of 2 hours, it would have gained an extra 2 hours. So that is 24 lots of 2 hours = 24 x 2 = 48 hours added to the 'correct' starting time of noon. Hence, my answer of noon on the 3rd January.2. Mrs Varma's watch (also a 12 hour watch) loses 5 minutes every 2 hours. She also sets her watch to the correct time at noon on 1st January.

b) When will our two watches next show the same time?

I am completely stuck here :(... please help to explain to me how to do this...3. When will our watches next show the same, CORRECT time?

Again, I can't do this... Any explanation would be greatly appreciated. Thank you.

Nat.

Why not try this to get you started and let you see what's going on. Keep two lists: one with the correct time and one with the time your watch shows. For 1a:

Correct Time / My Watch

Jan 1st noon / noon
Jan 1st 2.00 (pm) / 2.03
Jan 1st 4.00 (pm) / 4.06
Jan 1st 6.00 (pm) / 6.09

You might first ask yourself: when is your watch an hour ahead?

 

[Natasha1]

Ok,

a) I set my watch to the correct time at noon on 1st January. If I don't reset it, when will it next show the correct time?

I got 8pm (or 20.00) on 4th January. Can someone tell me if I am right?2. Mrs Varma's watch (also a 12 hour watch) loses 5 minutes every 2 hours. She also sets her watch to the correct time at noon on 1st January.

b) When will our two watches next show the same time?

I got 25th January at noon (lunchtime). Am I right?3. When will our watches next show the same, CORRECT time?

I got 19th February at noon (lunchtime). Am I right?

 

[Samy_A]

Ok,

a) I set my watch to the correct time at noon on 1st January. If I don't reset it, when will it next show the correct time?

I got 8pm (or 20.00) on 4th January. Can someone tell me if I am right?

No, not correct.

2. Mrs Varma's watch (also a 12 hour watch) loses 5 minutes every 2 hours. She also sets her watch to the correct time at noon on 1st January.

b) When will our two watches next show the same time?

I got 25th January at noon (lunchtime). Am I right?

No, not correct.

3. When will our watches next show the same, CORRECT time?

I got 19th February at noon (lunchtime). Am I right?

No, not correct.If you find my approach with the function too difficult, why don't you try what @PeroK suggested for 1a)?

 

[PeroK]

Ok,

a) I set my watch to the correct time at noon on 1st January. If I don't reset it, when will it next show the correct time?

I got 8pm (or 20.00) on 4th January. Can someone tell me if I am right?2. Mrs Varma's watch (also a 12 hour watch) loses 5 minutes every 2 hours. She also sets her watch to the correct time at noon on 1st January.

b) When will our two watches next show the same time?

I got 25th January at noon (lunchtime). Am I right?3. When will our watches next show the same, CORRECT time?

I got 19th February at noon (lunchtime). Am I right?

Given your answer to 1a, I don't think you've understood the problem. Imagine your watch really was 3 mins fast every 2 hours, that's only 36 minutes a day. It will, therefore, take weeks until it shows the correct time again.

I wouldn't look at questions 2 and 3 until you've understood problem 1. Go back to my suggestion in post #5 to give yourself an idea of what's happening.

 

[Natasha1]

if it gains 36mins in every 24 hours then for my watch to catch up time it will have to be 1440/36 = 40 so forty days from 1st of January which would be 10th February at noon.

Is this correct?

 

[haruspex]

if it gains 36mins in every 24 hours then for my watch to catch up time it will have to be 1440/36 = 40 so forty days from 1st of January which would be 10th February at noon.

Is this correct?

Much better, but remember it is a 12 hour watch. How fast will it be when it first shows the correct time?

 

[PeroK]

if it gains 36mins in every 24 hours then for my watch to catch up time it will have to be 1440/36 = 40 so forty days from 1st of January which would be 10th February at noon.

Is this correct?

That's much closer.

How many hours (on a 12-hour watch) does it need to gain to catch up? The question assumes you know that on a 12-hour watch, no distinction is made between 12 noon and 12 midnight. Perhaps you're one of those people who never wears a watch?

 

[Natasha1]

Oh! I did not know about this! It's so confusing... No I do not have a watch, and only really get 24hrs clocks.

is it 20 days later? Is it 720/36 = 20 so on 21st of January at noon. If so, why?

 

[PeroK]

I think you've got the idea. Here's the sort of thing they're talking about:

 

[Samy_A]

Oh! I did not know about this! It's so confusing... No I do not have a watch, and only really get 24hrs clocks.

is it 20 days later? Is it 720/36 = 20 so on 21st of January at noon. If so, why?

Yes.

Your watch has to gain 12 hours, that's 12*60=720 minutes. As it gains 36 minutes a day, it shows the correct time again after 720/36=20 days.

 

[Natasha1]

I see! Thanks... So for:

2)

I think you've got the idea. Here's the sort of thing they're talking about:

Thanks :)... Much clearer!

 

[Natasha1]

Not sure how to go about 2 and 3

 

[haruspex]

Not sure how to go about 2 and 3

How far apart are the two watches after one hour? How far apart do they need to be to be showing the same time?

 

[Natasha1]

they are 8 mins apart... then 16 then 24.. Ahhh so when does 8 fit into multiples of 60, right?

 

[Natasha1]

8
16
24
32
40
48
56
64
72
80
88
96
104
112
120

So that's 30 hours after... So that would be 2nd January at 6pm? Is this right?

 

[haruspex]

they are 8 mins apart... then 16 then 24.. Ahhh so when does 8 fit into multiples of 60, right?

I said "after one hour".
And you did not answer my second question: how far apart will they be when they first show the same time again?

 

[Natasha1]

in one hour that would be 2.5 mins for one watch and 1.5 mins for the other so that's 4 mins difference in one hour or 8 mins in 2hrs.

 

[haruspex]

in one hour that would be 2.5 mins for one watch and 1.5 mins for the other so that's 4 mins difference in one hour or 8 mins in 2hrs.

Right, but please answer my other question: how far apart will the two watches be when when they next show the same time?

 

[Natasha1]

After (in hours) Mrs Varma's watch My watch
1 2.5 1.5
2 5 3
3 7.5 4.5
4 10 6
5 12.5 7.5
6 15 9
7 17.5 10.5
8 20 12
9 22.5 13.5
10 25 15
11 27.5 16.5
12 30 18
13 32.5 19.5
14 35 21
15 37.5 22.5So after 15hrs or 3am on 2nd January will our two watches show the same time.

 

[Natasha1]

How on Earth do I work out 3?

 

[Samy_A]

After (in hours) Mrs Varma's watch My watch
1 2.5 1.5
2 5 3
3 7.5 4.5
4 10 6
5 12.5 7.5
6 15 9
7 17.5 10.5
8 20 12
9 22.5 13.5
10 25 15
11 27.5 16.5
12 30 18
13 32.5 19.5
14 35 21
15 37.5 22.5So after 15hrs or 3am on 2nd January will our two watches show the same time.

Not correct. As you said, the difference between the two watches grows by 8 minutes every 2 hours. After 15 hours the difference will only be 15*4=60 minutes.

So:

how far apart will the two watches be when when they next show the same time?

 

[haruspex]

So after 15hrs or 3am on 2nd January will our two watches show the same time.

You have not understood my question. To answer it, you don't need to know how much each watch is gaining or losing. All you need to know is that they are 12 hour watches running at different speeds. If they show the same time now, how far apart will they actually be when they next display the same time?

 

[Natasha1]

Ok for 3? it will be 720/4 = 180 hrs or 180/12 = 15 days or 16th January at noon

 

[Samy_A]

Ok for 3? it will be 720/4 = 180 hrs or 180/12 = 15 days or 16th January at noon

Both wrong, but let's tackle 2 first.

 

[Natasha1]

I don't know

 

[Natasha1]

it loses 5x12 = 60 mins in a 12 hour day

 

[Natasha1]

what do I do from here?

 

[Samy_A]

it loses 5x12 = 60 mins in a 12 hour day

No, the difference grows by 8 minutes every 2 hours, so that makes 8*12=96 minutes in a day (a real 24 hours day).
But a difference of 96 minutes doesn't make the watches show the same time.
As the difference continues to grow, how big has it to become for the watches to again show the same time?

EDIT: my 24-hour watch says 23.39pm, good night.

 

[Natasha1]

This is too hard for me... I'm lost.

720 (not sure why but there you go) / 96 = 7.5 days so the answer would be 9th January at midnight

 

[SammyS]

How far apart time-wise do the two watches have to in order that they show the same time as each other?

Did haruspex ask that a few times already ?

 

[Natasha1]

Yes and this is what I answered...

After (in hours) Mrs Varma's watch My watch
1 2.5 1.5
2 5 3
3 7.5 4.5
4 10 6
5 12.5 7.5
6 15 9
7 17.5 10.5
8 20 12
9 22.5 13.5
10 25 15
11 27.5 16.5
12 30 18
13 32.5 19.5
14 35 21
15 37.5 22.5So after 15hrs or 3am on 2nd January will our two watches show the same time.

 

[Natasha1]

Can anyone just explain to me by doing it? As I have tried time and time again (proof in my threads)... But simply don't get it?

 

[SammyS]

Can anyone just explain to me by doing it? As I have tried time and time again (proof in my threads)... But simply don't get it?

He did NOT ask how much time must elapse so that the watches read the same time again. He asked how far apart they must be relative to each other when they display the same thing.

 

[Samy_A]

Yes and this is what I answered...

After (in hours) Mrs Varma's watch My watch
1 2.5 1.5
2 5 3
3 7.5 4.5
4 10 6
5 12.5 7.5
6 15 9
7 17.5 10.5
8 20 12
9 22.5 13.5
10 25 15
11 27.5 16.5
12 30 18
13 32.5 19.5
14 35 21
15 37.5 22.5So after 15hrs or 3am on 2nd January will our two watches show the same time.

After 15 hours the difference is 60 minutes, or 1 hour. But these are 12 hour watches, not 1 hour watches. That's why they don't show the same time after 15 hours.

This is too hard for me... I'm lost.

720 (not sure why but there you go) / 96 = 7.5 days so the answer would be 9th January at midnight

Yes, the 7.5 days is correct.

Why?

Well, you know that the difference between the two watches grows by 96 minutes a day.
What does it mean that the difference between the two watches is 96 minutes (1 hour and 36 minutes)? It means that if your watch is (hypothetically) at 12.00 noon, Mr Varma's watch will be 96 minutes behind, that is 10.24 am.
After two days the difference between the two watches is 192 minutes (3 hours and 12 minutes). It means that if your watch is (hypothetically) at 12.00 noon, Mr Varma's watch will be 192 minutes behind, that is 8.48 am.
And so on.

When will the two watches again show the same time? Well, when the difference between them will have gone full circle, meaning 12 hours.
12 hours = 720 minutes. When the difference is 12 hours, the watches again show the same time, precisely because these are 12 hour watches.
That's why you divided 720 by 96 to get the correct result.

 

[Samy_A]

How on Earth do I work out 3?

You already established in the first question that your watch will again show the correct time after 20 days.
That means it will also show the correct time after 20+20=40 days. And also after 40+20=60 days. And so on.

Now, before solving 3, try to establish after how many days Mrs Varma's watch will again show the correct time. This is similar to the first exercise.

 

[Natasha1]

Does that mean that Mrs Varma's watch which loses 5mins every 2 hours then loses 60 mins in 12 hours(or 720 mins)

So Mrs Varma's watch will show the correct time in 720/60 = 12 days and because my watch in 720/36 = 20 days they will both show the correct time in

60 days as this is the lowest common multiple.

Is this correct?

 

[Samy_A]

Does that mean that Mrs Varma's watch which loses 5mins every 2 hours then loses 60 mins in 12 hours(or 720 mins)

So Mrs Varma's watch will show the correct time in 720/60 = 12 days and because my watch in 720/36 = 20 days they will both show the correct time in

60 days as this is the lowest common multiple.

Is this correct?

Yes.

 

[Natasha1]

Thank you :)

 

[dgsspkumar]

Answers
1. Gained time = 12 hours = 720 min
After t hours
t×3/120 = 12
t=480 hours=20 days
Jan 21 St at noon
2. To show same time, time gained in one watch should be equal to 12 hours minus time lost in other watch. It can even be vice versa.
Time gained = 12 - time lost
t * 3/120 = 12 - t* 5/120
t = 240 hours = 10 days
Jan 11 at noon
3 . the clocks show same time every 10 days
The clock shows right time after 20 days ( from problem 1)
Hence both clocks show correct same time after 20 days
Jan 21 at noon

 

[Samy_A]

Answers
1. Gained time = 12 hours = 720 min
After t hours
t×3/120 = 12
t=480 hours=20 days
Jan 21 St at noon

Correct.

2. To show same time, time gained in one watch should be equal to 12 hours minus time lost in other watch. It can even be vice versa.
Time gained = 12 - time lost
t * 3/120 = 12 - t* 5/120
t = 240 hours = 10 days
Jan 11 at noon

t * 3/120 = 12 - t* 5/120 is correct, but then you made a computation error.

3 . the clocks show same time every 10 days
The clock shows right time after 20 days ( from problem 1)
Hence both clocks show correct same time after 20 days
Jan 21 at noon

Not correct, as based on wrong result for 2).

 

[dgsspkumar]

1. Gained time = 12 hours = 720 min
After t hours
t×3/120 = 12 t=480 hours=20 days
Jan 21 St at noon
2. To show same time, time gained in one watch should be equal to 12 hours minus time lost in other watch. It can even be vice versa.
Time gained = 12 - time lost
t * 3/120 = 12 - t* 5/120
t = 180 hours = 7days 12 hours
Jan 8 mid night
3. The clocks show same time every 7days 12 hours.
The 1 st clock shows right time after 20 days ( from problem 1)
The answer is a common multiple of 7.5 and 20.
60 days
Hence Clocks show correct same time after sixty days
jan 1 to jan 31 - 30 days
feb 1 to feb 28 - 28 days
jan 1 and jan 2
hence answer is Jan 2 noon

Reference https://www.physicsforums.com/threads/watch-problems-gaining-and-losing-time.846247/page-3

 

[dgsspkumar]

Correct.
t * 3/120 = 12 - t* 5/120 is correct, but then you made a computation error.
Not correct, as based on wrong result for 2).

1. Gained time = 12 hours = 720 min
After t hours
t×3/120 = 12 t=480 hours=20 days
Jan 21 St at noon
2. To show same time, time gained in one watch should be equal to 12 hours minus time lost in other watch. It can even be vice versa.
Time gained = 12 - time lost
t * 3/120 = 12 - t* 5/120
t = 180 hours = 7days 12 hours
Jan 8 mid night
3. The clocks show same time every 7days 12 hours.
The 1 st clock shows right time after 20 days ( from problem 1)
The answer is a common multiple of 7.5 and 20.
60 days
Hence Clocks show correct same time after sixty days
jan 1 to jan 31 - 30 days
feb 1 to feb 28 - 28 days
jan 1 and jan 2
hence answer is Jan 2 noon
Is this correct now ?

 

[Samy_A]

1. Gained time = 12 hours = 720 min
After t hours
t×3/120 = 12 t=480 hours=20 days
Jan 21 St at noon
2. To show same time, time gained in one watch should be equal to 12 hours minus time lost in other watch. It can even be vice versa.
Time gained = 12 - time lost
t * 3/120 = 12 - t* 5/120
t = 180 hours = 7days 12 hours
Jan 8 mid night
3. The clocks show same time every 7days 12 hours.
The 1 st clock shows right time after 20 days ( from problem 1)
The answer is a common multiple of 7.5 and 20.
60 days
Hence Clocks show correct same time after sixty days
jan 1 to jan 31 - 30 days
feb 1 to feb 28 - 28 days
jan 1 and jan 2
hence answer is Jan 2 noon
Is this correct now ?

Yes.

(I mean the sixty days for 3. You probably made a typo at the end, where you meant March and not January.)

 

[dgsspkumar]

Yes.

(I mean the sixty days for 3. You probably made a typo at the end, where you meant March and not January.)

Ya it was march 2 noon.