Suppose we have the following data:
start date end date quantity
01/05/2014 07/05/2015 5
07/06/2015 02/06/2016 8
What is the logic of the following expression: $$ \left[\left(\frac{8}{5} \right)^{360/215}-1 \right] \times 100 \% = 1.197 \%$$
Assume that a year is $360$ days. Intuitively I want to find the percentage difference between the two quantities for a year. When I do the following:
$$ \frac{\frac{8-5}{5}}{215/360} \times 100 \% = 1.005 \%$$
I get a percentage that is close to the above but not the same. Maybe I can take the logarithm of the first expression to get something similar to the second expression?
Note that 215 is the number of days between 07/06/2015 and 02/06/2015 .
You have two different models for the growth. The one you ask about assumes the growth is a geometric series, that it starts at $5$ and is multiplied by some number $r$ each day. The second day you would have quantity $5r$, the third $5r^2$ and the $216$th day $5r^{215}$. They are then calculating $r$ and extending the growth to a year. On the first day of the new year the quantity is $5r^{360}=5\left(\frac 85\right)^{(360/215)}\approx 10.984$ which is about $2.197$ times the first day. The calculation of the increase is not correct and should yield $119.7\%$.
Your formula assumes the growth is an arithmetic series, that it grows by a fixed amount each day. The first day you have quantity $5$, the second $5+d$, the third $5+2d$ and the $216$th day $5+215d$. On the first day of the new year year you will have quantity $5+360d$. You can calculate $d=\frac {8-5}{215}\approx 0.014$, giving a quantity on the first day of the new year of about $10.023$, which would be an increase of about $100.5\%$ You should not be dividing the growth in $215$ days by $5$ if you consider it an arithmetic series.
Which model of growth is correct (if either) is not a mathematical question.