Judging an Answer with Benchmarks

Every fraction is near 0, near 1/2, or near 1. That is often enough to judge an answer.

How to tell

7/8 is near 1, because the top is nearly the bottom. 1/9 is near 0.

Estimating a sum

7/8 + 1/9 is about 1 + 0, so about 1. An answer of 8/17 would be far too small.

The classic wrong answer

8/17 comes from adding the tops and the bottoms. A benchmark check catches it instantly.

Three benchmarks are enough

Judge every fraction against 0, 1/2 and 1. Is 5/12 near zero, near a half, or near one? Half of 12 is 6, and 5 is just under, so 5/12 is a little under a half. Three reference points give you a usable sense of size for any fraction.

The half test

A fraction is more than half when the numerator is more than half the denominator. For 7/12, half of 12 is 6 and 7 exceeds it, so 7/12 is more than a half. This one comparison resolves a great many questions in a second.

Estimating a sum

5/12 + 7/8 is a bit under a half plus nearly one, so expect a bit under 1 1/2. Now if your calculation produces 12/20 or 3, you know to look again. Benchmark estimation is the fraction equivalent of rounding.

What this catches

The add-the-denominators error always produces an answer smaller than the larger fraction you started with, which a benchmark estimate exposes at once. Almost every serious fraction error fails a benchmark check, which is what makes the habit worth building.