Answer to December 2016 Problem of the Month

Determine the value of each picture, given the total of each horizontal row in the present below.

You must show your work to get credit.

Solution to the Problem:

Tree = 15 Ornament = 55 Snowman = 35 Snowflake = 35

Let A = tree
Let B = Ornament
Let C = Snowman
Let D = Snowflake

Then write the four equations:
(1) A + C + 2D = 120
(2) A + 2B + C = 160
(3) 2A + B + D = 120
(4) D + 3C = 140

You can solve the system of equations using linear combinations or you can use Cramer's Rule with determinants.

Here is the solution using Cramer's Rule:

James Alarie sent in an excellent algebraic solution:

Ball    = 55
Tree    = 15
Flake   = 35
Snowman = 35

01:  T + F + S + F = 120
02:  T + B + B + S = 160
03:  B + T + T + F = 120
04:  F + S + S + S = 140

Combining items:
05:  T + 2 * F + S = 120
06:  T + 2 * B + S = 160
07:  B + 2 * T + F = 120
08:  F + 3 * S = 140

Solving equation 08 for F and 
substituting in the others:
09:  F + 3 * S = 140
     F = 140 - 3 * S
10:  T + 2 * F + S = 120
     T + 2 * (140 - 3 * S)+S = 120
     T + 280 - 6 * S + S = 120
     T - 5 * S = -160
11:  T + 2 * B + S = 160
12:  B + 2 * T + F = 120
     B + 2 * T + (140 - 3 * S)=120
     B + 2 * T + 140 - 3 * S = 120
     B + 2 * T - 3 * S = -20

Solving #10 for T and substituting
 in #11 and #12:
13:  T - 5 * S = -160
     T = -160 + 5 * S
14:  T + 2 * B + S = 160
     (-160 + 5 * S) + 2 * B+S=160
     -160 + 5 * S + 2 * B +S =160
     6 * S + 2 * B = 320
     3 * S + B = 160
15:  B + 2 * T - 3 * S = -20
     B + 2 *(-160 + 5 * S)-3*S=-20
     B -320 + 10 * S - 3 * S = -20
     B + 7 * S = 300

Subtracting #14 from #15:
16:  (B + 7 * S) - (3*S+B)=300-160
     B + 7 * S -3 * S - B = 140
     4 * S = 140
     S = 35

Put this value for S into #09,
 #13, and #15:
17:  F = 140 - 3 * S
     F = 140 - 3 * 35
     F = 140 - 105
     F = 35
18:  T = -160 + 5 * S
     T = -160 + 5 * 35
     T = -160 + 175
     T = 15
19:  B + 7 * S = 300
     B + 7 * 35 = 300
     B + 245 = 300
     B = 55

Correctly solved by:

1. Kimberly Howe	Vienna, Virginia
2. James Alarie	Flint, Michigan
3. Eliza Sheffield	Tuscaloosa, Alabama
4. Todd Craver	Waukegan High School, Waukegan, Illinois
5. Christian Andersen	Mountain View High School, Mountain View, Wyoming
6. Hope Pfeifer	Mountain View High School, Mountain View, Wyoming
7. Behlee Aimone	Mountain View High School, Mountain View, Wyoming
8. Sreeroopa Sankararaman	Singapore, Singapore
9. Adalene Thomas	Mountain View High School, Mountain View, Wyoming
10. Valerie Alfeo	Waltham Public Schools, Waltham, Massachusetts
11. Jim Hooker	Johnson City, Tennessee