Chapter 6.3, Problem 23P

Bob wants to estimate $1893÷275.$ He decides to round 1893 up to 2000. Since he rounded 1893 up, Bob says that he'll get a better estimate if he rounds 275 in the opposite direction namely, down to 250 rather than up to 300. Therefore, Bob says that $1893÷275$ is closer to 8 (which is $2000÷250$ ) than to 7, because $2000÷300$ is a little less than 7. This problem will help you investigate whether or not Bob's reasoning is correct.

1. Which of $2000÷250$ and $2000÷300$ gives a better estimate to $1893÷275?$ Can Bob's reasoning (just described) be correct?
2. Use the meaning of division (either of the two main interpretations) to explain the following: When estimating the answer to a division problem $A÷B,$ if you round A up, you will generally get a better estimate if you also round B up rather than down. Draw diagrams to aid your explanation.
3. Now consider the division problem $1978÷205.$ Suppose you round 1978 up to 2000. In this case, will you get a better estimate to $1978÷205$ if you round 205 up to 250 (and compute 2000 + 250) or if you round 205 down to 200 (and compute $2000÷200)?$ Reconcile this with your findings in part (b).