Things you might have thought about

Real-world problems can be very hard to model in a meaningful way. We will always need to make assumptions, and we need to be aware of how these assumptions might affect our answers.

There are two aspects to these problems. On the one hand, there is the algebraic side: how do we perform a correct calculation with the data we are given? And on the other hand, they require us to make assumptions to be able to answer them in a realistic way. For question 1, we will comment on the algebraic aspect, while for subsequent questions, we will note some of the assumptions we considered ourselves when working on them.

  1. I first travel \(\dots\) miles at \(\quantity{x}{mph}\) and then travel the next \(\dots\) miles at \(\quantity{\dots}{mph}\). If my average speed is \(\quantity{\dots x}{mph}\), what is \(x\)?
  • If I travel the first \(10\) miles at \(\quantity{x}{mph}\) and then travel the next \(10\) miles at \(\quantity{20}{mph}\), with an overall average speed of \(\quantity{x}{mph}\), I know that \(x\) must be …. I could use this to check my answer. If the average speed were \(\quantity{2x}{mph}\) instead, what would \(x\) have to be this time?
  1. I have an alloy made of two different metals. The first metal has density \(\quantity{\dots}{kg\,m^{-3}}\), while the second has density \(\quantity{\dots}{kg\,m^{-3}}\). The resulting alloy has density \(\quantity{\dots}{kg\,m^{-3}}\). What percentage of the alloy is the first metal?
  • If I melt together \(\quantity{1}{cm^3}\) of the first metal and \(\quantity{1}{cm^3}\) of the second one, does the resulting alloy have volume \(\quantity{2}{cm^3}\)?
  1. I need to refill my empty paper stock cupboard. I can order two types of paper: regular paper is \(\quantity{x}{mm}\) thick per sheet, while SuperGrade paper is \(\quantity{\dots x}{mm}\) thick per sheet. I need to have \(\dots\) times as many regular sheets in my stock cupboard as SuperGrade ones, and the stock cupboard is \(\dots\) metres high. How many sheets of each will I be able to store?
  • I’m assuming that I can only put in one stack of paper. If the cupboard is wide enough, then I could put two stacks side-by-side.

  • Do I need to take into account the gaps between individual sheets?

  • Do I need to take into account the boxes or paper wrapping that the paper comes in?

  • Do I need to take into account the thickness of the shelves in the cupboard?

  • Do I need to take into account that I can probably only buy paper in certain quantities, say 500 sheets at a time?

  1. A fish called Wanda and a shark called Jaws live peacefully together in a fish pond. Wanda could drink all of the water in the pond on her own in \(\dots\) hours, while Jaws would only take \(\dots\) hours to do the same. When they are together in the pond, how long would they take to drain it?
  • Perhaps this is not the most realistic of questions to begin with…
  1. I have two chemicals, A and B, which both burn in oxygen. A glass chamber contains a certain amount of oxygen. There is just enough oxygen to completely burn either \(\dots\) grams of chemical A or a mixture of \(x\) grams of chemical A and \(\dots x\) grams of chemical B. How many grams of chemical B would completely burn in the chamber?
  • Do chemicals A and B react with each other when they are both in the glass chamber? Does that affect anything?
  1. Countries A and B are adjacent. This table shows the birth rates, death rates and net migration rates (all given per 1000 population per year) for the two countries:
Country A Country B
Birth rate \(\dots\) \(\dots\)
Death rate \(\dots\) \(\dots\)
Net migration rate \(\dots\) \(\dots\)

The two countries agreed to unify into a single country, after which the net population growth rate of the unified country was \(\dots\) per 1000 population per year. What was the ratio of the populations of the two countries prior to the merger?

  • Will the birth rate or death rate among the former inhabitants of country A remain the same following the unification, and similarly for country B?

    What factors might influence these rates?

  • What will the new net migration rate be for the merged country?

    In particular, will the internal migration between the two formerly separate countries simply cancel each other out?

    And will the unification have an impact on rates of migration out of or into the now-unified country?