Denominated Mischief

I’ve finally got around to reading one of my Christmas presents, Alfred S Posamentier and Ingmar Lehman’s Magnificent Mistakes in Mathematics, which is about ways that mathematical reasoning can be led astray. A lot, at least in the early pages, is about the ways a calculation can be fowled by a bit of carelessness, especially things like dividing by zero, which seems like such an obvious mistake that who could make it once they’ve passed Algebra II?

They got to a most neat little erroneous calculation, though, and I wanted to share it since the flaw is not immediately obvious although the absurdity of the conclusion drives you to look for it. We begin with a straightforward problem that I think of as Algebra I-grade, though I admit my memories of taking Algebra I are pretty vague these days, so maybe I missed the target grade level by a year or two.

\frac{3x - 30}{11 - x} = \frac{x + 2}{x - 7} - 4

Multiply that 4 on the right-hand side by 1 — in this case, by \frac{x - 7}{x - 7} — and combine that into the numerator:

\frac{3x - 30}{11 - x} = \frac{x + 2 - 4(x - 7)}{x - 7}

Expand that parentheses and simplify the numerator on the right-hand side:

\frac{3x - 30}{11 - x} = \frac{3x - 30}{7 - x}

Since the fractions are equal, and the numerators are equal, therefore their denominators must be equal. Thus, 11 - x = 7 - x and therefore, 11 = 7.

Did you spot where the card got palmed there?