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this is something that’s stumped me for a while once I realized that the way I was doing it was giving me the wrong answer and, in hindsight, didn’t make sense.

Let me give you a scenario: In a random country X (let’s call it Mathland), Group A is 50% of the total population of the country, Groups B & C are each 20% and Group D is 10%. The entire population of Mathland is 100,000,000.

In a specific city within the country (let’s called it Mathville), Groups A and B are each 30% of the city’s total population and Groups C and D are each 20%. The population of Mathville is 50,000.

Now, in an alternate world, let’s say that Group A is 70% of the population of Mathland and Groups B, C, and D are each 10%. The population of the country is still 100,000,000 in this new world.

Assuming everything changes proportionally, how would you calculate what percentage of the population each group makes up in the Mathville of the new world, assuming the city’s population is still 50,000?

I assume that in the new Mathville, Group A would be more than 30% of the population and, at the very least, Groups B and C would now be less than 30% and 20% of the city’s population, respectively. But how do you find the exact percentages of the population that each group makes in the new world’s Mathville, based off of the previously given information for the original Mathworld, the original Mathville, and the new Mathworld?

I hope I made sense, please let me know if you have any questions. I look forward to hearing from you all, and thank you in advance! :)

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If by "everything proportional" you mean that the county to country ratio of group A/B/C people are the same in the two countries. Then, for example, if $n_{A,new}$ is the number of group A people in the specified county in the new country and $N_{A,new}$ is the total number of group $A$ people in the new country, then $n_{A,new}=N_{A,new}\frac{n_{A,old}}{N_{A,old}}=(0.7\times100,000,000)\frac{0.3\times50,000}{0.5\times 100,000,000}=0.42\times50,000$ So group $A$ is 42 percent of the population in the new country's county. Others can be constructed similarly.

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  • $\begingroup$ When you do it your way, and solve for each group, the $n_{total,new}$ is not 50,000, it’s 43,500. How do you compensate for this difference and have the $n_{total,new}$ still be at 50,000 like the $n_{total,old}$, yet still have the percentages change to reflect the changed demographics of the new Mathland? $\endgroup$
    – Fahim Azaz
    Sep 15, 2019 at 6:04
  • $\begingroup$ How can that be? We calculated the percentage assuming the two populations were the same both in the county and in the country. To see this, notice that $$\frac{n_{g, new}}{N_{g, new}}=\frac{n_{g,old}}{N_{g,old}}\Rightarrow \frac{n_{g, new}/n_{g, total}}{N_{g, new}/N_{g,total}}=\frac{n_{g, old}/n_{g, total}}{N_{g, old}/N_{g,total}}\Rightarrow \frac{p_{g, new}}{P_{g, new}}=\frac{p_{g, old}}{P_{g, old}}$$ where we have divided both sides by the of the county to country ratio and $p$'s are percentages. $\endgroup$
    – Hassan A
    Sep 15, 2019 at 6:31
  • $\begingroup$ Group A: ${n_{A,new}\over 70,000,000} = {15,000\over 50,000,000}$ Group B: ${n_{A,new}\over 10,000,000} = {15,000\over 20,000,000}$ Group C: ${n_{A,new}\over 10,000,000} = {10,000\over 20,000,000}$ Group D: ${n_{A,new}\over 10,000,000} = {10,000\over 10,000,000}$ $21,000+10,000+7,500+5,000=43,500$ $\endgroup$
    – Fahim Azaz
    Sep 15, 2019 at 15:01

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