Vayikra- Understanding Fractions through Perspective

"Hashem spoke to Moshe, saying: If a person will commit a misuse, and sins unintentionally against Hashem's holies, he shall bring his guilt-offering to Hashem, an unblemished ram from the flock, with a value of silver shekels, according to the sacred shekel, for a guilt-offering. For what he has deprived the Sanctuary he shall make restitution, and add a fifth to it, and he shall give it to the Kohen; then the Kohen shall provide him atonement with the ram of the guilt-offering and it shall be forgiven him." ~Vayikra 5:14-16

In this section of this week's parsha, we learn about a person who is required to bring a monetary restitution for an unintentional sin of misusing an item in the Tabernacle. His restitution, in addition to bringing a sacrifice of a certain value, is to also repay the value of the damages done, plus 1/5 of the value.

In the Sapirstein Edition of the Artscroll Vayikra (Torah with Rashi's Commentary), there is a footnote on Rashi 5:16 ("For what he has deprived the sanctuary he shall make restitution") that explains that when the Torah speaks of 1/5, it is really referring to 1/5 of the final total. In other words, the amount that was added is 1/5 of the final total after adding in that additional piece. 

What does this mean when thinking of fractions as we know them? If we want to have a section be 1/5 of the total, it means that before that piece was added, there were only 4 of those identically sized pieces. In other words, the original whole was cut into 4 quarters. Then, another 1 piece of equal size to the other 4 pieces is added on. Now, the final added piece is 1/5 of the entire new whole.

Why might this be so confusing to understand?
Think of an optical illusion type of artwork that you may have seen- maybe this well known image where, from one angle, you see an old woman, and from a different angle you see a young woman. It is all a matter of perspective. If you look at one set of features in a certain way, then you can see the old woman; if you focus on a different set of features slightly differently, then you see the young woman. 

The fraction situation that we have in our parsha this week can also be confusing to learners nowadays because of perspective. When we teach and think about fractions nowadays, in most cases, we think about our original value as one whole unit. If someone told us to add 1/5, we would assume that we needed to break our original unit into fifths (ie 5 even pieces) and then add on 1 additional piece of the same size. Alternatively, in the Torah and rabbinical writings through the Talmudic period, the entire unit was determined after the final additional amount was added on. Therefore, the 1/5 that was added on was 1/5 of the final unit, and in order to determine the size of that unit, the smaller segment actually needed to be broken into quarters.

Let's look at an example to see how this would differ, practically:

For ease of calculation, let's take a starting value of $100 restitution that's owed. I will break down the case, for the same starting value, first with our current day perspective, and then with the Torah perspective.

Current Day Perspective:
Starting value owed: $100
We need to add 1/5 of the starting value: 1/5 of $100 is $20 (100 ÷ 5 = 20).
The total value to be paid is the original value + 1/5, or $100 + $20 = $120

Torah Perspective:
Starting value owed: $100
We need to add an amount in order to have that amount be 1/5 of the final value. Remember, I noted above that if you have 4 pieces of equal size, and add one more that is the same size, now you will have 5 pieces of equal size and that last piece that you added will be 1/5 of the total final value. 
So, we need to first figure out 1/4 of the starting value: 1/4 of $100 is $25 (100 ÷ 4 = 25).
When we add $100 + $25 = $125. We can check to confirm that $25 is 1/5 of $125 (125 ÷ 5 = 25).
Since this calculation does work out, we have confirmed that the total value to be paid is the $125.

If we compare these two sample calculations, we can see that, according to our calculations, the Torah Perspective actually turns out to be more costly for the sinner than the Current Day Perspective does, and the calculation process is slightly more involved (although maybe not if that's how you're used to calculating!).

Some questions to keep you thinking about fractions:
Does this perspective difference in the understanding of 1/5 always result in the Torah calculation being more costly than the current day calculation? Does it change if you use a smaller original value? a larger original value? What if you used different fractions and compared them in current day vs. Torah perspective? 

[Note that with an understanding of what a fraction is and represents, these questions are developmentally appropriate for students in younger elementary grades and upwards. The approach to finding the answers would vary based on other skill levels and developmental understanding of the concepts needed to think about the questions and make sense of them.]