Setting Up Proportions

Pre-Algebra · Axiom Academy

Line the units up the same way on both sides, then cross-multiply — and any proportion becomes one easy equation to solve. Suppose a car goes 120 miles in 2 hours , and at the same speed covers some distance in 5 hours. Before any algebra, set up the ratios so miles sit over hours on both sides . Match the units and the proportion is correct; flip one side and it's wrong. Correct — miles over hours on both sides Wrong — the right side is flipped (hours over miles) Once the proportion is set up, multiply along each diagonal . The two diagonals form an X : one links a with d , the other links b with c . The rule says those two products are always equal. Multiply each numerator by the other fraction's denominator. The two cross products match — that single equation has no fractions left. Why it works (clearing the fractions) Cross-multiplying isn't a magic trick — it's what you get when you multiply both sides of by b and by d . Both denominators cancel, leaving . Now use the X to find a missing value. Back to the car: it goes 120 miles in 2 hours , so how far does it travel in 5 hours? Set up miles-over-hours, cross-multiply, and solve the one-step equation. Step 1 — Cross-multiply. The diagonals give . Step 2 — Multiply out. The left diagonal is a number: 600 = 2x . Step 3 — Divide both sides by 2. That isolates x . 300 miles in 5 hours is 60 mph — the same speed as 120 miles in 2 hours. The answer fits.

This is the written version of the interactive lesson above. See the full Pre-Algebra course.