When's Your B-Day?
A mother doesn't remember the birthday of her 4 children. All she knows is that they were all born in the same year, and either in June, March or December. The names of the children are Charlie, Nicolas, Eric, and Joey. Month numbers are the number of the month in a year. April is 4, for example. Day numbers are, well, day numbers. The following are true statements made by the kids about each other's birthdays.
Charlie: "The cube-root of the day number of Joey's birthday is equal to the day number of Nicolas's birthday. Eric's day number is 8 more than the number of days STRICTLY BETWEEN today and March 1." Strictly means excluding "today" and March 1.
Nicolas: "My birthday is tomorrow. Oh Yes!!!! Only 12 more days until Pi Day!!!"
Eric: "Ahhhh!!! March, my favorite month of the whole year. Pi Day, School, NO VACATION!! Luckily my birthday is not in March. That would ruin March... And don't get me talking 'bout my little brother Nicolas, or my older brother Joey. It's good enough to not share a birthday month with either of them…"
Joey: "Fine, y'all wanna know my month? My month number is 300% greater than Nicolas's month number. Happy???"
Mother: "My day number is the number of kids who it is impossible to find the birthday of, added to the product of Eric's day number and Nicolas's month number. My month number is one less than (the sum of twice Eric's month number and Joey's month number, divided by 8% of my day number).
If the mother's day number can be expressed as d and her month number can be expressed as m, calculate (m * n) / (m + n - 1). Express as a decimal, rounded to the nearest thousandth.