Comparing Fractions by Cross-Multiplying

Mental Math · Axiom Academy

LESSON Comparing Fractions by Cross-Multiplying Which is bigger, or ? Two multiplications settle it exactly — no common denominator, no division. 1. Recut Both to the Same Grid Two fractions are hard to compare because they are cut into different numbers of pieces. Fix that: cut each of the five parts of into eight, and each of the eight parts of into five. Both bars now carry forty identical slices , and counting the shaded ones settles the question. Watch what the two counts turn out to be. The two shaded counts are the two cross products 2. Which Product Belongs to Which Side In practice you skip the bars. Multiply each numerator by the other fraction's denominator, and write each product above the fraction whose numerator you used . That last part is where nearly every mistake happens, so the animation carries each denominator physically across and drops the product on the correct side. Numerator times the other denominator. The answer stays on the side it started from. Multiplying an inequality by a negative number flips it. With a negative denominator, tidy the sign first. against gives 18 and 18 — the same fraction written two ways. against is 55 against 56 — a clean verdict from two small products. Because each comparison costs only two products, you can sort a list of fractions by comparing neighbours. Here are four that decimals handle badly: , , and . Three of them round to 0.6 . Cross-multiplying separates all four with certainty.

This is the written version of the interactive lesson above. See the full Mental Math course.