Pat and Chris are gambling in Atlantic City. When asked to 'put their winnings together,' Pat said that the product was "Nine factorial." Chris added that the sum was "Thirty-eight squared."
How much money did each person win?
Solution to the Problem:
This can be done by looking for two numbers m and n ,
Substituting, we get m(1444 - m) = 362880 ; m2 - 1444 m + 362880 = 0 .
Factor m2 - 1444 m + 362880 = 0
And if you want to do it without algebra, make a table of pairs of numbers
whose PRODUCT (as Pat says) is 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362880 ,
and whose SUM (Chris adds) is 382 = 1444.
So: m * n = 362880 and m + n = 1444, or n = 1444 - m.
Factoring this is the same as looking for two numbers
whose sum is 1444 and product is 362880.
(m-1120)(m-324) = 0
m = 1120 and m = 324
Then n = 324 and 1120
Pat and Chris had winnings of $1120 and $324.
that add up to 1444, with their products:
Num 1 Num 2 Product
-------- -------- -------
1 . . . . 1443 . . . 1443
2 . . . . 1442 . . . 2884
3 . . . . 1441 . . . 4323
. . . . . . . . . . . . . . . . . . . . . . . . . .
323 . . . 1121 . . 362083
324 . . . 1120 . . 362880 . . YES!
Correctly solved by:
1. Jim Arrison | Norristown, Pennsylvania |
2. Jeffrey Gaither | Winchester, Virginia |
3. Richard Johnson | La Jolla, California |
4. James Alarie | University of Michigan -- Flint Flint, Michigan |
5. Saxton Sharad | Melrose, Massachussetts |
6. David & Judy Dixon | Bennettsville, South Carolina |
7. Walt Arrison | Philadelphia, Pennsylvania |
8. Cassie Jordan | Harrisonburg, Virginia |
9. Mikael Wetterholm | Danderyd, Sweden |
10. Josh Feingold | Winchester, Virginia |
11. Celeste Doerwaldt | Winchester, Virginia |
12. Franklin Harcourt | Columbus, Georgia |
13. John C. Funk | Ventura, California |
14. Julie | Dallas, Texas |
15. Larry Schwartz | Trumbull, Connecticut |
16. Mark Mendez | Harrisonburg, Virginia |
17. Chris Maggiolo | Harrisonburg, Virginia |
18. Monica Quach | Harrisonburg, Virginia |
19. Kirby Welsko | Harrisonburg, Virginia |
20. Bella Patel | Harrisonburg, Virginia |
21. Helna Patel | Harrisonburg, Virginia |
22. Patrick Mize | Columbus, Georgia |
23. Kathryn Harris | Winchester, Virginia |
24. Alex Truong | Harrisonburg, Virginia |