Stewart's birthday is in February and he wants a nine inch by twelve inch rectangular birthday cake
with chocolate icing and yellow icing as shown in the diagram below. There is no icing on the sides.
The yellow icing must be a constant width all around the cake, and the amount of chocolate icing must
equal the amount of yellow icing.
Determine the width of the yellow icing.
You must show your algebra and geometry to get credit for this problem.
Solution to the Problem:
The width of the yellow icing must be 1.5 inches.Let x = width of the yellow icing
Then 9-2x and 12 - 2x would represent the dimensions of the chocolate icing. See diagram below:
Set up expressions for the area of the chocolate icing and the yellow icing and set them equal to each other.
Yellow: A = 12x + 12x + (9 - 2x) x + (9 - 2x) x
Chocolate: A = (9 - 2x) ( 12 - 2x)
Now set these two expressions equal to each other and solve.
Sam Loyd gives an example of a similar problem in Martin Gardner's book, More Mathematical Puzzles of Sam Loyd. Farmers use a simple rule for cutting a swath around a rectangular field until half the crop was gathered: "One quarter the difference between a short cut cross lots and round by the road."
Mathematicians understand it better by saying, "From the sum of the sides subtract the diagonal of the field and divide the remainder by four."
In Stewart's cake problem above, the diagonal of the cake is 15" (it is a 3-4-5 right triangle). So the width of the yellow icing would be (9 + 12 - 15) / 4 = 3/2 or 1.5 inches.
Correctly solved by:
| 1. Ivy Joseph | Pune, Maharashtra, India |
| 2. James Alarie | Flint, Michigan |
| 3. Bryce Villanueva |
University of Central Florida, Orlando, Florida |
| 4. Kathy Mann | Birmingham, Alabama |
| 5. SreeRoopa Sankararaman | Singapore, Singapore |
| 6. Brijesh Dave | Mumbai City, Maharashtra, India |
| 7. Alyssa Marke |
Delta High School, Delta, Colorado |