Right angle = money

Geometry Level 4

It is 1:00 am on April 9th. How much money would you make in the next 48 hours if you made 8 dollars every time the hands of a clock formed a 90 degree angle?

A note on the source

This problem was taken from the Ole Miss Math Contest , which was hosted by Dr. David Rock of The University of Mississippi; the contest has not been so active recently and will probably be on hiatus for the foreseeable future. This problem was taken from the 'Middle School Madness' Challenge Mode section.


The answer is 704.

Dana Chiueh by dana chiueh , 21, Taiwan

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3 solutions

Dana Chiueh
Sep 29, 2014

Right angles occur twice each hour except that the second right angle between 2 o'clock and 3 o'clock occurs exactly at 3 o'clock, which is also the first occurrence between 3 and 4 o'clock. The same happens at 9 o'clock. Therefore, only 22 right angles occur in a 12-hour period (original thought process would lead to 24) or 44 in a 24-hour period or 88 in two days.

Therefore, 88 × 8 = 704 88 \times 8 = \boxed{704}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

There should be 24 right angles in 12 hours but why 22 is taken here, can you explain.

Nashita Rahman - 6 years, 2 months ago
Manas Vema
Dec 26, 2014

the clock makes 90 degree angle 44 times in 24 hours .so it makes 88 90 degree angles in 48 hours .SO ANSWER IS 88*8=704

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Min. hand moves 6X degrees in X minutes, Hr. hand X/2 degrees. However first 90 will take place after 30 more degrees, since it is 1 a.m.
So the first 90 is after X minutes. So The equation 6X - X/2 - 30 = 90.~~~~~X=24/1.1 Minutes ....
Total minutes we have are 48 * 60=2880 . After first, we are left with 2880 - 24/1.1 Minutes
Now after every 180 degrees relative motion there will be a 90 degree.
After the first, we will have 6Y - Y/2=180. So Y=36/1.1 minutes per every 90 degrees.
(2880 - 24/1.1)/(36/1.1) = 87.33 + 1=88 integers. So 88*$8=$ 704.




1 Helpful 0 Interesting 0 Brilliant 0 Confused

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