(Took this from the Palette of Problems in the August 2013 MTMS journal.) “In the town of Nuptuville, 2/3 of the men are married to 5/7 of the women. Which is greater, the number of men or the number of women in the town? What is the ratio of men to women in the town?”
Immediately, students went for common denominators and insisted their answer was correct.
After a careful analysis of what was represented by each part of the ratios, one student commented, “Oh! We don’t want common denominators. We want common numerators!” There was still quite a bit of arguing for common denominators, but I was able to stand back and let her convince her classmates.
Afterwards, one of the students who took the most convincing commented, “So that’s why you nag us so much about labels!!”