# Rate of Percentage Charge

So I was wondering how long it took my iPad to go from 60% to $$100\%$$ since I got a new USB port. I found it took $$216$$ minutes, or about $$3.6$$ hours. My goal was to figure out the time it takes to charge $$1\%$$ (namely in minutes), assuming of course, the rate of charge is consistent. I also tried backtracking the math to figure out how long it should take to go from $$0\%$$ to $$100\%$$ (based on the fact that $$\frac{.40}{216}=1\% \text{ per} \ 540 \ \text{minutes}$$), for example, but I realized I can't trust those results if I can't figure out a conventional rate per minute!

So I figured $$\frac{0.40}{216}\propto \frac{0.01}{9}$$

I decided to represent this algebraically where the percentage of charge, $$P$$, is a function of time in minutes, $$t$$

Which should mean

$$P(t)=\frac{0.01}{9}t$$

So I tested this function with the actual data from above $$P(216)=\frac{0.01}{9}(216) \implies P=0.24$$

Clearly, $$24\%\ne40\%$$

I figured no calculus would be needed because I'm assuming rate of charge is consistent/linear. What am I missing here? Why is my function not working?

• I am pretty sure that your assumption that the rate of change is linear is what makes this wrong. Charging is a complicated chemical/physical process depending on the forces between atoms and their electrons and between atoms and molecules od different substances whiich makes this question unlinear. – Vinyl_cape_jawa Feb 23 at 11:33
• I'd suggest testing the charging process from 0 to 100%, check the charge every 10 minutes, make a graph afterwards and see how linear it is. Then you can choose your assumptions. – Hugh Feb 23 at 11:36
• @Vinyl_coat_jawa I figured. I was hoping I could constrain nonlinear factors and use conventional math to get conventional answers. I thought inputting the actual real-life values would, at least, work even if its rate doesn't work for other values. – Lex_i Feb 23 at 11:40
• @Hugh, that's a good idea, lol. I'll try that when I find the time – Lex_i Feb 23 at 11:42
• 0.40/216 = 0.1/54 = 0.01/5.4,: 1% per 5 & a half minutes. Whereforth 0.01/9? – William Elliot Feb 23 at 12:12