A confectioner has 300 pounds of a chocolate that is 1 part cocoa butter to 7 parts caramel. How much chocolate that is 1 part cocoa butter to 15 parts caramel should be combined with the first in order to create a new chocolate that is 1 part cocoa butter to 9 parts caramel?

The answer is 200 pounds

Let x be the number of pounds of the second chocolate that should be added

First chocolate: 300 pounds

cocoa to caramel = 1 to 7

So, there are in total 8 parts (1 + 7 = 8)

There is:

Cocoa: 1/8 * 300 pounds

Caramel: 7/8 * 300 pounds

Second chocolate: x pounds

cocoa to caramel = 1 to 15

So, there are in total 16 parts (1 + 15 = 16)

There is:

Cocoa: 1/16 * x pounds

Caramel: 15/16 * x pounds

Resulting chocolate: 300 + x pounds

cocoa to caramel = 1 to 9

So, there are in total 10 parts (1 + 9 = 10)

There is:

Cocoa: 1/10 * (300 + x) pounds

Caramel: 9/10 * (300 + x) pounds

Since in all chocolate, there is only one part of cocoa, it is easy to add up the amounts of cocoa:

1/8 * 300 + 1/16 * x = 1/10 * (300 + x)

300/8 + x/16 = 300/10 + x/10

75/2 + x/16 = 30 + x/10

75/2 - 30 = x/10 - x/16

75/2 - 60/2 = 8x/80 - 5x/80

(75 - 60) / 2 = (8x - 5x) / 80

15/2 = 3x/80

80 * 15/2 = 3x

40 * 15 = 3x

600 = 3x

x = 600/3

x = 200 pounds