Just a bit of procrastination here but at the bus stop I was wondering how should I study before multiple tests given the percentage of the grade each one was worth, the days until each test and the amount of hours you want to study that day.
Here's my attempt
I have come to the conclusion that it would be an inverse relationship between the percentage of time you should allocate (S) and the time until the test (t). Multiplied this by the grade percentage the test was worth (G).
$$S\propto \frac{G}{t}$$
Given that you have multiple tests and 8 hours (S=8) a day to study I sum the above relationship to and equate the hours to yield the following: (k= 8/sum)
$$8=k \sum^n_{i=1} S_i$$ $$8=k \sum^n_{i=1} \frac{G_i}{t_i}$$
For instance let's say that I have 6 tests: English($G_1=0.25$, $t=58$ days) Maths Methods($G_2=0.5$, $t=60$ days), Specialist Maths($G_3=0.5$, $t=65$ days), Chemistry($G_4=0.5$, $t=72$ days), Modern History($G_5=0.25$, $t=74$ days), Physics($G_6=0.5$, $t=76$ days). Today I have 8 hours to study (more like 7 after I'm done with this), using my formula I can calculate how long I should study for each test today ($S_i$).
$$8=k \sum^n_{i=1} \frac{G_i}{t_i}= k\Big(\frac{G_1}{D_1}+\frac{G_2}{D_2}+\frac{G_3}{D_3}+\frac{G_4}{t_4}+\frac{G_5}{t_5}+\frac{G_6}{t_6}\Big)$$
$$=k\Big(\frac{0.25}{58}+\frac{0.5}{60}+\frac{0.5}{65}+\frac{0.5}{72}+\frac{0.25}{74}+\frac{0.5}{76}\Big)$$
$k\approx 214.84$ Therefore,
- English $S= 0.93$ hours
- Maths Methods $S= 1.79$ hours
- Specialist Maths $S= 1.65$ hours
- Chemistry $S= 1.49$ hours
- Modern History $S= 0.73$ hours
- Physics $S= 1.41$ hours
Now given that the time approaches the day before the first test (English) I want to just study the content of the first test so I am ready for the test. So let $t_1=1$
$$8=k\Big(\frac{0.25}{1}+\frac{0.5}{3}+\frac{0.5}{8}+\frac{0.5}{15}+\frac{0.25}{17}+\frac{0.5}{19}\Big)$$
$k\approx 14.45 $ Therefore
- English $S= 3.61$ hours
- Maths Methods $S= 2.4$ hours
- Specialist Maths $S=0.90 $ hours
- Chemistry $S= 0.48$ hours
- Modern History $S= 0.21$ hours
- Physics $S= 0.38$ hours
This is where my model somewhat fails as I want most of the day to be studying the most urgent subject. If the test dates were more 'spaced out' or English had a 50% of the marks test then I would spend most of the day before the test studying the subject.
Can any of you 'test anxious' mathematicians suggest any improvements of this crude model (I am a high school student)? I was thinking about throwing in some differential equations ($\frac{\partial S}{\partial t_1} )$ but I'm not smart enough for that yet. Thank you
Edit:
I have more of a think about it and have come up with the following conditions. Ignoring $S\propto \frac G t$
$$1=k \sum^n_{i=1} S_i $$
$$0=\sum^n_{i=1} \frac{dS_i}{dt} $$
$$ S_i(t_i=1)=1 $$
$$ S_i(t_i<1)=0$$
$t$ is the days until the exam. I have come up with the following model for 3 different tests with test dates t=10, t=20, t=30 https://www.desmos.com/calculator/jrrawaofki
This application can be extended to all tasks you pursue which have a deadline. This isn't a serious model and there is plenty of space to play with.
How could I extend and improve this model using differential equations?
This is the second bounty