16. In the college education problem of Example 4, suppose instead that 80% of children of college educated parents get a college education themselves, and that 50% of children of noncollege educated parents get a college education. Assuming still that presently 35% of all parents are college educated and the rest are not, how many will and how many won't be after four generations? 14. In the city/suburbs problem discussed in this section, suppose instead that 5% of city residents move to the suburbs and 10% of suburban residents move to the city each decade. If initially 50,000 live in the city and another 50,000 live in the suburbs, how many will live in each 5 decades later? For each of the transition diagrams shown in Exercises 10-13: (a) Find the iterative equation for the population of either state, Xn or yn, assuming Xn + yn = 1. (b) Find formulas for the solutions Xn and yn in terms of xo and yo- 10. 12. X X 0.5 0.5 0.9 0 0.5 0.1 0.5 Y YEXAMPLE 4 Suppose that in a certain part of the country 90% of the children of college educated parents get a college education, but 10% do not. Also, among parents who are not college educated, 40% of their children get a college education, but the other 60% do not. Assuming that the current distribution of parents in that region includes 35% who are college educated and 65% who are not, what will that distribution be three generations from now? Solution: A transition diagram for this process is shown in Figure 2.10. Letting C" and N\" be the respective fractions of parents who are college educated and not college educated :1 generations from now, we see that CH1 = 0.96,, + 0.4% Since Cu and NR represent the fractions of the total number of parents involved, then C\" + NR = l or N,I =1- C\The equation CH1 = 0.5Cn + 0.4 has the solution 0.5 " l C" = C0(0.5)" + 0.4 0-l_ = Cg(0.5)" 0.8((0.5)" - 1) Using Co = 0.35 this becomes C\" = 0.35(0.5)" 0.8((0.5)" l) = 0.8 0.4S(0.5)\" So three generations from now the fraction of college educated parents will be :33 = 0.8 0.45(0.5)3 = 0.74375 or about 74%, which implies that approximately 26% will not be college educated