In Time

Algebra Level 2

The great 2-ton Detroit clock was the hit of the 1876 Centennial exhibition in Philadelphia. Not only did it give the time in 13 cities, but it kept track of the seasons and plotted the orbits of the planets around the sun. It also inspired the following problem:

At noon, the big and little hands of the clock meet. How many times do the hands meet between noon and midnight?

12 2 11 13

5 solutions

Hardeep Singh Boparai
May 18, 2014

IN FIRST HOUR THREY DONOT CROSS BUT AFTER THAT THEY CROSS ONCE EVERY HOUR/SO IT IS 12 -1=11

They cross each other again actually after more than one hour. After 12:00, they meet again at 1:05.4545..., then again at 2:10.9090..., and so on. The meeting of the hands will happen after 1pm, 2pm, 3pm, 4pm, 5pm, 6pm, 7pm, 8pm, 9pm and 10pm (a total of 10 meetings). After 11:00pm, they would meet again at exactly 12:00 midnight. So they only cross 10 times between noon and midnight.

Emmanuel Jao - 7 years ago

That is what I say. It doesn't ask how many times will they after noon till midnight

ann law - 6 years, 12 months ago

But it asks how many times they meet. It specifies that they "meet" at noon.

Billybob Jenkins - 7 years ago

hanep na yan ah!!

Ord Sobremonte Jr. - 7 years ago

Vishnu Menon
May 16, 2014

Every hour, they meet about 5 minutes later than the previous hour. Starting at noon, the next times they'll meet are 1:05, 2:10, 3:15, 4:20, and so on...

The 11th time that they would meet will be at 12:00 midnight. That should not count as a time BETWEEN noon and midnight. So the answer should be 10. If you include the 12:00 noon and the 12:00 midnight meetings of the big and little hands as inclusive, then the answer should be 12.

Emmanuel Jao - 7 years ago

BETWEEN = can't include noon or midnight: 1:05, 2:10, 3:15, 4:20, 5:25, 6:30, 7:35, 8:40, 9:45, 10:50, & 11:55 = 11 times

Doug Darfus - 7 years ago

Please see comment of Mr. Prashant Keshvani below, they don't meet exactly at 1:05 but at "1:05+x," etc... The big and little hands will not meet at exactly 11:55 (try it if you have a mechanical watch), but they meet again after 11:00 o'clock at exactly 12:00 o'clock. So the answer should be 10 and not 11.

Emmanuel Jao - 7 years ago

You are absolutely correct. Thanks for the correction. I was thinking the 0.5 degrees the hour hand moves every minute wouldn't amount to enough, but it does on that last hour. Thanks!

Doug Darfus - 7 years ago

To be precise, I guess they meet at 1:05 + x , 2:10 + 2x/12 , 3:15 + 3x/12 and so on

prashant keshvani - 7 years ago

Amogh Agrawal
May 30, 2014

Roughly... 1:05, 2:10, ..., 11:55. Ans = 11

Gopi Haris
May 27, 2014

1.05,2.10,3.15.....11.55.so,totally 11 times between noon and midnight

Krishna Garg
May 21, 2014

Starting from 12.00 Noon,both hands will meet every next hour,so it will meeet 11 times till 11:59 PM,then already it is 12.00 mid night.So answer is 11.

K.K.GARG,India

Beautiful Problem! Guess 11 is the right answer because we should either count 12 noon or 12 midnight. I would however point out another more elegant solution. This is how I learnt it about 22 years back. Look at the relative velocities of the hands as (360-30)=330 degrees per hour. The answer becomes 12*330/360 Why I am suggesting it: For questions of defective watches where one hand moves faster or slower!!! The relative velocity comes in very handy. Happy Mathematicking

Amit Srivastava - 7 years ago

In Time

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