The answer is: without the extrinsic reward, the execution of the task relies on internal motivation
Extrinsic reward refers to some sort of gain given by others that you get from doing something. Internal motivation on the other hand is a form of gain that derived from yourself after you do something.
From the case above, Rachel's motivation to feed the cats and do the jobs is given by her mother in the form of allowance (extrinsic reward). After her mother lost the job, rachel no longer able to receive this.
Because of this, feeding the cat would most likely be done because of internal motivation from Rachel's part (such as because Rachel loves the cat)