On XS56a, I don't get how they simplified the inequality.
I don't get the question. How does the number of children affect standard deviation?
number of children don't affect standard deviation, but affect standard error of mean and that is what you need and what you see in that inequality. So standard error of mean being,
because of some reasons like, bigger the sample is, better the estimation of true population mean will be(denominator), bigger the variability in the data is, less confidence in that one value (the mean) accurately reflects them we have(numerator). Radical in denominator is there because you really computing variance, which is SD^2.
So and for simplifying that inequality, knowing std=3.4 and under the assumption that population is normally distributed(that 1.96 therm for 95%) you should want to your SE be less or equal(if mean would be 0) than 0.7. You know how many times of SE you need(=1.96 from normality assumption), you also know that std is 3.4 and knowing that your 1.96SE have to be in max. equal to 0.7 all what's left is n, so that's the reason why you should solve that for n.