Ratio, Proportion, and Percent

A ratio is just a comparison, like 3 to 5. A proportion says two ratios are equal, like 3/5 = x/20, and you solve it by cross-multiplying: 3 times 20 equals 5 times x, so x = 12. This works because equal ratios describe the same relationship at different scales, so whatever you do to one side's numbers you're really doing to both.

Percent problems are ratios in disguise, with 100 as one side. 'Discount 30 percent' means the new price is 70 percent of the old one, a multiplier of 0.70. Multiple percent changes don't add together, they multiply in sequence, because each change acts on whatever value exists at that moment, not on the original number. That's why a 20 percent raise followed by a 20 percent cut leaves you behind: you take 20 percent of a smaller number on the way back down, so you lose more than you gained.

This chaining idea connects directly to multi-step word problems: discount then tax, population growth then decline, dilution then concentration. Each step is its own multiplier, applied left to right. Ratios also show up in part-to-part comparisons, like mixing paint in a 3 to 5 ratio of blue to yellow, and these behave differently from part-to-whole percents, especially when you add or remove a fixed amount from both parts. Keeping straight what the ratio is being compared to, the whole, the part, or the previous step, is the skill underneath every problem on this page.