Problem Solving

Gertrude and her grandmother have 2 clocks with 12 digits.

the 1st clock loses 3 minutes every hour. It will show that 57 minutes have elapsed when in fact exactly one hour has gone by.

The second clock gains 5 minutes every hour. It will show that 65 minutes have elapsed when in fact one hour has gone by.

Gertrude and grandmother always open a bottle of champain when 2 clocks strike the correct time of 12 at midnight on the same night.

Last night they opened a bottle of champain.

How many full days must go by before they may do so again?

One clock is 3 minute slow and other clock is 5 minutes fast. So if we take the first clock to be correct, then the second one will be 8 minute faster than the first clock.

So in one hour it gains 8 minutes. Now both the clock to strike midnight 12 together, the second clock should have gained one day more that the first clock. i.e 24 hours, which means 24 * 60 minutes.

So how many hours will take to make 24 * 60 minutes gain for the second clock.

= 24 * 60 / 8
= 180 hours
= 7.5 days

Which means that will happen in middle of the day. So for this to happen midnight, the clocks have to go another 7.5 days...totaling to 15 days.

Alara533 wrote:One clock is 3 minute slow and other clock is 5 minutes fast. So if we take the first clock to be correct, then the second one will be 8 minute faster than the first clock.

So in one hour it gains 8 minutes. Now both the clock to strike midnight 12 together, the second clock should have gained one day more that the first clock. i.e 24 hours, which means 24 * 60 minutes.

So how many hours will take to make 24 * 60 minutes gain for the second clock.

= 24 * 60 / 8
= 180 hours
= 7.5 days

Which means that will happen in middle of the day. So for this to happen midnight, the clocks have to go another 7.5 days...totaling to 15 days.

You are changing what the question says by assuming one clock to be correct. The reason the question is good is because it assumes both of them to be wrong.

I also think the question is wrong, in the end it says "that the time should be 12 at midnight in the same night" but how is that possible??

thanks gabriel. I understood my mistake.
But still I think the question is right.

Suppose we have a third clock which is correct.

On day 1, all clock are at 12:00 AM. First clock then starts losing 3 minutes every hours, second clock gains 5 minutes every hour.

For now the question asks, when will the 3 clocks strike 12:00 AM together again.

Now take clock 1, its loses 1:12 hour, each day, so it would take 20 days to lose 24 hours and strike 12:00 AM together with the third clock. This will repeat every 20th day.

For clock 2, which gains 2 hours per day, it will take 12 days to gain 24 hour and then strike 12:00 AM together with third clock. This will repeat for every 12th day.

Now for all the three clock to strike together, 20 th day cycle of first clock, and 12th day cycle of second clock should fall on same day, which would be their LCM... 60 days.