definite variables wrote:

I have a question about the problem below:

Consider a city with two kinds of population: the inner city population and the suburb population. We assume that every year 40% of the inner city population moves to the suburbs, while 30% of the suburb population moves to the inner part of the city. Let I (resp. S) be the initial population of the inner city (resp. the suburban area). So after one year, the population of the inner part is

0.6 I + 0.3 S

My questions are as follows:

(1) How did they arrive at 0.6 I + 0.3 S ?

(2) What does "I (resp. S)" and "(resp. the suburban area)" stand for ?

To answer the second part first, "resp." means "respectively". In other words, the sentence containing that abbreviation is a shorthand way of saying "Let I be the initial population of the inner city and let S be the initial population of the suburban area."

Notice that 40% = 0.4, and 30% = 0.3. If, out of an initial population of I, an amount given by 0.4 I leave the inner city then only 0.6 I remain. But we have to add to that the 0.3 S that move in from the suburbs. That's where the expression 0.6 I + 0.3 S comes from.