Suppose that you are concerned about your test average in your biology class. Imagine that you have received scores of 91, 92, and 95. One test remains, and you want to be sure that your overall test average is greater than 90. What must you score on the final test?
This type of question arises with surprising frequency on the ACT, on the SAT, and in life. Fortunately, such questions may be handled methodically through the use of algebra.
For the sake of argument, I’ll assume that the reader can follow the development below without any extra explanation. (If I am mistaken, feel free to comment. I am happy to provide more details.)
unknown fourth score: x
average = (91 + 92 + 95 + x) ÷ 4
average = (278 + x) ÷ 4
average > 90
(278 + x) ÷ 4 > 90
278 + x > 360
x > 82
The fourth score must be greater than 82.
I hope that the reader is able to follow the laconic solution above. I further hope that the reader is able to produce a similar solution for a similar problem. I would like to pause, however, to make a few notes. Perhaps they will seem useless.
- 91 is greater than 90 by 1.
- 92 is greater than 90 by 2.
- 95 is greater than 90 by 5.
- 82 is greater than 90 by –8. (It is less than 90 by +8.)
- 1 + 2 + 5 + (–8) = 0.
After this mysterious detour, the reader is now invited to solve a very, very similar problem: Suppose that you are concerned about your test average in your zoology class. Imagine that you have received scores of 83, 82, and 86. One test remains, and you want to be sure that your overall test average is greater than 80. What must you score on the final test? After completing the algebra, see if the following holds any meaning:
- 3 + 2 + 6 + (–11) = 0.
Could such a random, bullet-pointed sum be of any use?
One Comment
I think what you’ve proven, dear sir, is that reasoning about averages with a basic “mathematical sense” is really the higher order reasoning approach we’d love students to take. For example, when working with a student who scored a 29 on the ACT, and who wants a 30, I’ll say, “Well, you just need to turn your 29 on English into a 33.” [For you laypersons, there are 4 sections on the ACT: English, Math, Reading, Science, all weighted equally towards a composite score] That student will then reach for the calculator — and then I launch into a lesson like your laconically elegant one above….