If consonants are free, determine the cost of the five vowels A, E, I, O, and U from the following clues: 


        Audacious     costs  $260
        Equivocation  costs  $340
        Inimitable    costs  $255
        Onomatopoeia  costs  $435
        Unambiguous   costs  $225


Solution:

The answer is:
a = $60
e = $75
i = $40
o = $50
u = $25 

 

To solve it, I set up 5 simultaneous linear equations:  
(I used brackets to indicate each equation)

                2a      +  i +  o + 2u = 260   [1]
                 a +  e + 2i + 2o +  u = 340   [2]
                 a +  e + 3i +         = 255   [3]
                2a +  e +  i + 4o      = 435   [4]
                 a      +  i +  o + 3u = 225   [5]
       

3*[2] - [5]:    2a + 3e + 5i + 5o      = 795   [6]
-2*[2] + [1]:      - 2e - 3i - 3o      = -420  [7]

[4] - 2*[3]:       -  e - 5i + 4o      = -75   [8]
[6] - 2*[3]:          e -  i + 5o      = 285   [9]
               
[8] + [9]:              - 6i + 9o      = 210   [10]
[7] + 2*[9]:            - 5i + 7o      = 150   [11]

5*[10]:                 -30i + 45o     = 1050  [12]
-6*[11]:                 30i - 42o     = -900  [13]

[12] + [13]:                    3o     = 150

                                 o     = 50
then substitute back to find i = 40, e = 75, 
a = 60, and u = 25.

You could also solve this with Cramer's Rule or with a computer program:
Click here to see Keith Mealy's computer program



Correctly solved by:

1. Rick Jones Kennett Square, Pennsylvania
2. Richard Johnson * La Jolla, California
3. Keith Mealy * Cincinnati, Ohio
4. Joe Heintz Manchester, Tennessee
5. John C. Funk Ventura, California
6. David Dixon Bennettsville, South Carolina
7. Bill Hall Wellington, Florida
8. George Gaither Winchester, Virginia
9. ---------- Feltham, Middx, United Kingdom
10. Chip Schweikarth Winchester, Virginia
11. Bob Hearn Winchester, Virginia
12. Erin McGinnis Winchester, Virginia
* Wrote computer programs to solve the problem