March 7, 2005
Problem of the Month

The Balancing Act

A bottle and drinking glass on the left side of a scale balance with a jug on the right side.

The bottle alone balances the drinking glass and a plate.

Three of these plates balance 2 of the jugs.

How many of the glasses will balance a bottle?

Solution to the Problem:

Let B represent a bottle, G a drinking glass, J a jug and P a plate.

Let "=" represent being in balance.
We are given the following:

BG = J

B = GP

PPP = JJ

Given: BG = J, add J to both sides to get: JBG = JJ

Then: PPP = JBG

Add 3 G's to both sides and replace J with BG to get: GPGPGP = GGGBGBG

Now replace each GP with B to get: BBB = GGGGGBB

Lastly, remove two B's from both sides to get: B = GGGGG

Therefore five drinking glasses balance one bottle.

---

Here is a more algebraic approach sent in by Ann Norman:

let b stand for bottle, g stand for glass,and j stand for jug.

b+g=j
b=g+p
3p=2j

b=j-g
j-g=g+p
j=2g+p
3p=2(2g+p)
3p=4g+2p
p=4g
p=b-g
b-g=4g
b=5g

Correctly solved by:

1. Ann Norman	Columbus, Georgia
2. Van Pham	Harrisonburg, Virginia
3. Larry Schwartz	Trumbull, Connecticut
4. John Funk	Ventura, California
5. Tarpley Ashworth	Harrisonburg, Virginia
6. Cameron Burkholder	Winchester, Virginia
7. Jim Arrison	Norristown, Pennsylvania
8. Andrea Eberhard	----------
9. António Baptista	Lisboa, Portugal
10. Charles Washington	Winchester, Virginia
11. Molly Simkins	Columbus, Georgia
12. Ben Swartz	Winchester, Virginia
13. Thomas Woodward	Columbus, Georgia
14. Tom Robb	Winchester, Virginia
15. Weimers Winners	Lordstown, Ohio
16. James Alarie	University of Michigan -- Flint, Flint, Michigan
17. Angelo Leone	Middletown, Connecticut
18. Rich Murray	Ridgetown, Ontario, Canada
19. Cougar5447	Columbus, Georgia
20. Kevin Jacobson	Columbus, Georgia
21. Sarah Watterson	Middletown, Connecticut
22. Sheila Shah	Arlington, Virginia
23. Emily Auerbach	Columbus, Georgia
24. Amanda Auerbach	Columbus, Georgia
25. Jeffrey Gaither	Winchester, Virginia
26. Irene Mathieu	Winchester, Virginia
27. Tristan Collins	Winchester, Virginia
28. David & Judy Dixon	Bennettsville, South Carolina
29. Chelsea Smith	Columbus, Georgia
30. Alex McIntire	Columbus, Georgia
31. Viktor Logrell, Jonathan Jansson, Bahadér Güngör, Ako Saleh	Tullängensskolan, Örebro, Sweden
32. Rob Adams	Winchester, Virginia
33. Gilbert Melanson	Sierra Vista, Arizona
34. David Amos	Winchester, Virginia
35. Jason LaRusso	Winchester, Virginia
36. William Proffitt	Winchester, Virginia
37. Mike Hudson	Winchester, Virginia
38. Wajih Ansari	Harrisonburg, Virginia
39. Okechi "pithy" Egekwu	Harrisonburg, Virginia
40. Arin Smith	Winchester, Virginia
41. Chris Hernley	Winchester, Virginia
42. Kelly Johnson	Columbus, Georgia
43. James Simmons	Harrisonburg, Virginia
44. John Harcourt	Columbus, Georgia
45. Arsalan Heydarian	Harrisonburg, Virginia

Send any comments or questions to: David Pleacher