System of 3 equations APPLIED

A lot of times the applications we come across for algebraic topics feel contrived; here’s one that occurred naturally in my very own home yesterday. 

My wife works in a clinic with two other midwives.  They all do the same amount of “on-call” time, but have different workloads when it comes to clinic hours. And sometimes they’re on call while they’re in the clinic.  She knows their salaries, but didn’t know the hospital’s pay rate for clinic hours, on-call hours, or combo hours – she was interested in finding this out to see if it was worthwhile to pick up some extra shifts at the clinic.

Calling the number of on-call hours X, the number of clinic hours Y, and the number of combo hours Z, she set up an equation for each midwife and was able to figure out the pay rate for each type of work. 

Math to the rescue!

Doomsday 2014

Well, we've evaded the apocalypse for another year.  2014 has its own Doomsday to keep in mind, however - and it's a Friday.

Dr. Conway developed a sweet method (called the Doomsday Algorithm) to determine the day of the week for any day in history.  It takes some practice, but dealing with the current year is pretty simple, provided you know the Doomsday.
Doomsday 2013 is Friday*.

• This means that the following dates are all Thursdays: 4/4, 6/6, 8/8, 10/10, 12/12. 
• If you work nine to five at the local 7-11, you can remember four more Thursdays: 9/5, 7/11, 5/9, 11/7.
• Einstein’s birthday = Pi Day = 3/14 = Thursday.
• The last day of February (2/29 this year) is a Thursday.
• Lastly, 1/3 is Thursday (3 out of 4 years it’s the 3rd… but on leap years it’s the 4th).

What day of the week is your birthday?  Independence Day?  Halloween?  Christmas?

Adding days of the week is easier when you treat them as numbers: mONEday, TWOsday, 3ednesday, FOURsday, 5day, S6turday (Treat Sunday as 0 - "nunday").  What's ten days past Wednesday?  10 + 3 = 13.  Take out any multiples of 7 and you have 6 (Saturday). 

*In case you’re interested, the link up top does a thorough job of teaching the method, but here’s the quick version. Start with the century day.  For 2000 it is Tuesday (2).  Then look at the 2-digit year, 14, and ask how many 12’s are in it. (1). What is the remainder? (2) How many 4’s are in this number? (0). Add up (2)+(1)+(2)+(0) = 5 = 5th day of the week = Friday. 
Craziness? How about one more example from the future: 2055.  Century day is still (2).  How many 12’s in 55? (4). What is the remainder? (7).  How many 4’s in this number? (1).  Add ‘em up: (2)+(4)+(7)+(1) = 14.  Take out multiples of 7 and you get 0.  The 0th day of the week is Sunday. Now you can find the day of the week for any date in 2055.