The general rule when dividing two numbers is that the answer should be given to the minimum number of significant figures of the numerator and denominator. So, $\frac{0.43}{2.546}= 0.168892…$ but can only be reliably accurate to 2 s.f. (from the numerator) so ${0.17}$.
My question is: what is the rule when dividing large numbers when they have been pre-rounded to a particular number of significant figures?
Google says that there are $1.252\times10^9\text{ people}$ in India with an area of $3.288\times10^6 \text{ km}^2$. Directly calculating the population density without taking into account any rounding of those figures gives $381\text{ people per km}^2$ to the nearest whole number.
Let’s imagine that these figures were only provided to 2 s.f. so there are approximately $1.3\times10^9\text{ people}$ in India and the area of India is approximately $3.3\times10^6 \text{ km}^2$. So, the population density is $\frac{1.3\times10^9}{3.3\times10^6}\text{ people per km}^2$. This equates to $393.\overline{93}$. Rounding to the nearest “whole person per square km” would give $394$.
However, as the inputs have only been given to 2 s.f., should I round the answer to 2 s.f., i.e the answer would be $390$ to 2 s.f.?
But the range of numbers that the population could be would be from $1,250,000,000$ to $1,349,999,999$ and the area would be $3,250,000$ to $3,349,999$. Using those extremes would give us a minimum population density of $\frac{1,250,000,000}{3,349,999} = 373.13$ and a maximum of $\frac{1,349,999,999}{3,250,000} = 415.38$.
So, my answer of $390$ isn’t correct either.
What would be the correct answer to “what is the population density of India if there are $1.3\times10^9\text{ people}$ in India and the area of India is approximately $3.3\times10^6 \text{ km}^2$?
Many thanks.
