Piece of Cake

April 14, 2014

Originally posted 4/18/11

The Cake Place manufactures and sells a cake pan set for creating cakes shaped like any of the digits 0 through 9, as shown.


The set consists of a rectangular, aluminum pan that measures 6” by 10” and some smaller 2” by 2” aluminum inserts. In the bottom of the large, rectangular pan is a grid with grooves used to secure the 2 x 2 inserts in place to cover up the area of the larger pan in which no cake batter will be poured. For example, to create a cake shaped like the number zero, an insert will be placed in the second, third and fourth rows of the second column as illustrated.

Initially, The Cake Place included fifteen 2 x 2 in inserts with each cake pan set. The owner would like to minimize costs by reducing the number of inserts included with each set. What is the minimum number of inserts that should be included in each set to ensure that each of the digits 0 through 9 can be created using the cake pan set, as shown?

 

It costs The Cake Place 10 cents to manufacture each of the 2 x 2 aluminum inserts. How much will they save on the next batch of 1000 cake pan sets produced by reducing the number of inserts included with each set from 15 to the minimum number of inserts calculated in the previous problem?

 

Kori has purchased one of the cake pan sets to make a cake for her daughter’s sweet sixteen party. Kori will make one cake in the shape of the number 1 and another cake in the shape of the number 6. Kori doesn’t want to just divide the batter into two equal parts for the two cakes because she wants the cakes to be proportional in thickness. What is the ratio of the amount of batter that will be used to create the number 1 cake to the amount of batter that will be used to create the number 6 cake? Express you answer as a common fraction.