What is the area of a rectangle that has a reported height of 7.5 cm and length of 10.5 cm? (There's more to this question than meets the eye!)

Question

Question: Report the area of a rectangle that has a reported height of 7.5 cm and length of 10.5 cm.

According to an HMH Algebra 1 textbook, the product of two [reported] measurements should have no more significant digits than the least precise measurement." This goes against the usual convention of reporting the product using no more significant digits than the factor with the least number of significant digits.

Using the textbook's criteria for comparing precision, the length and width of the rectangle have the same precision, as they are both measured to the nearest tenth of a centimeter.

So, using the textbook's rule, my question is what happens in this case? Since both measurements have the same level of precision, how many significant digits should the reported area have?

(Or maybe yet, and dare I even ask, do you agree with the textbook's wording of this rule?)

Answer 1

7,5 * 10,5 = 78,75

But since the least precise digit is one digit after the comma, the answer must be modified as 78,7cm^2.

My math is not very good, so if there's anything wrong, please don't hesitate to correct me!

Answer 2

Ignore, for a while, the "general rule" or "rule of thumb" quoted.
Let's say that the two dimensions are $H$ and $L$, and the reported measurements mean $$ 7.45 \le H \le 7.55,\qquad 10.45 \le L \le 10.55 . $$ We may then conclude that $$ 77.8525 \le HL \le 79.6525 . \tag1$$ Conclusion: Reporting the answer as "$78.7$" would mean between $78.65$ and $78.75$, which is certainly not justified by the information in $(1)$.

Reporting the answer as "$79$" would mean between $78.5$ and $79.5$, which is better, but still slightly more than what is known in $(1)$.

Reporting the answer "$80$" would mean between $75$ and $85$. This does follow from $(1)$, but has lost a lot of information.

The "Rule of thumb" about significant digits says we should report "$79$". It is a compromise, but I agree it is the best of our choices.