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The amount of food needed is VARIABLE: it depends on the number of dogs and the number of days.
Food = dog*days.

But the amount of food PER CAN is CONSTANT: it remains the same value, regardless of the number of dogs or the number of days.
(dogs*days)/cans = k.

Since the amount of food per can must always be the same value, we get the following:
(dogs*days)/cans = (dogs*days)/cans.

Thus, given that 10 dogs over 6 days require 35 cans and that 7 dogs over 12 days require x cans, we get:
(10*6)/35 = (7*12)/x
x = 49.

The correct answer is C.

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x/y = k implies the following: if x increases by a factor, then y increases by the same factor.

In my solution, (dogs*days)/cans = k.
This relationship holds true because:
If the number of dogs doubles, then the number of cans must double -- even if the number of days is unchanged.
If the number of days doubles, then the number of cans must double -- even if the number of dogs is unchanged.

In your solution, (cans * days)/no. of dogs = k.
This relationship is not valid: the number of days is NOT directly proportional to the number of dogs.
Quite the opposite.
If the number of cans remains constant, then, as the number of dogs INCREASES, the number of days must DECREASE.
TWICE AS MANY dogs can be fed for HALF the number of days.

_________________
Mitch Hunt
GMAT Private Tutor and Instructor
GMATGuruNY@gmail.com
If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
Contact me about long distance tutoring!