Interesting question. I am no expert, but I wonder if the "piece of the pie" expression is related to the use of "pie charts" in finance and investing, for example. Of course, that leads to the question of, why "pie charts" and not "cake charts"?

I'm pretty sure Thomas Marcyes is right. It's related to the term "pie chart." This article has a picture of a "pie chart:" https://en.wikipedia.org/wiki/Pie_chart A "pie chart" is a visual illustration of a set of percentages that always add up to 100%. The reason why we call it a "pie chart" rather than a "cake chart" is that a pie is fairly flat. A pie chart looks more like a pie than a cake. "Pie chart" has become the standard name for kind of chart. You can't call it anything else; it's just the name for that kind of chart. If you're not sure how pies and cakes look, try a Google Image search on "pie" and then on "cake." Since we call it a "pie chart" it is natural to talk of the percentages as "pieces of the pie" or "slices of the pie." When discussing economic tradeoffs, a set of percentages always adds up to 100%. If my share gets larger, your share gets smaller. But if we are talking about the economy, or income, or something like that, the actual total can get bigger. Since perhaps the 1970s, a common piece of rhetoric has involved the phrase "growing the pie." The idea is that instead of fighting over how to share a limited pie, we can make the pie bigger and then everyone will get more. For example, if workers and management are arguing about how to share profits, workers might say "Our piece of the pie is shrinking." Management might reply "Let's not argue about how to divide up the pie, let's focus on growing the pie." P.S. Former U.S. President George W. Bush once said "You bet I cut the taxes at the top. That encourages entrepreneurship. What we Republicans should stand for is growth in the economy. We ought to make the pie higher." People made fun of him for this, because in U.S. culture the idea of making a pie "higher" is ridiculous. You make a pie bigger by increasing its diameter.

The other answers are right. I just want to add something. In situations where we say, "piece of cake," we can also say, "easy as pie."

Hi Sharpay You said that you understand the literal meaning of pie and cake, and a number of people have given you a possible reason for why the expression "expand the pie of interest" might relate to pie charts. However, when we say something is 'a piece of cake' we are using an idiom, and as such we can't always explain the reasons why these idioms use the words they do. Actually we don't need to make a literal translation of every word, just understand what the whole phrase means. I do understand that it is interesting to speculate on the origin of these idioms. Why does this phrase include the idea of 'cake'? Perhaps it's connected to the idea that cake is a luxury, a pleasure, something nice to eat, something that is not difficult to eat, therefore something that is easy? Sometimes we have to accept we don't know all the idioms. If I said "You need to pay cash on the nail." I think that most people who live near the area where I was born probably know and use this, but don't know the origin of the phrase (I do but it's a long story and not really important to explain now). I'm not sure that people from other places would know the meaning of this idiom. I've never heard 'expand the pie of something' before. Also there are idioms where everybody knows the meaning, but nobody can explain why we use those particular words. This website has some nice examples including an explanation of the 'piece of cake' idiom https://globalgraduates.com/articles/20-bizarre-english-idioms-and-how-to-explain-them

Fun side note: there are serious debates in the US as to which is better: pie or cake. The correct answer is pie.