Ratios and proportions show up everywhere: recipes, maps, scale drawings, unit prices, similar figures, and percent problems. The core skill is keeping the order consistent, then using cross multiplication to solve for the missing value.
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Ratio: compares two quantities, like 3:2 or 3/2.
Proportion: an equation that says two ratios are equal, like a/b = c/d.
Most “proportion problems” are just scaling problems: if something gets multiplied by k, the matching quantity also gets multiplied by k.
This works because you can multiply both sides of an equation by b·d to clear denominators. It’s not “magic,” just algebra.
Problem: 3/5 = x/20
Check: 3/5 = 0.6 and 12/20 = 0.6.
Problem: If 8 apples cost $6, how much do 20 apples cost at the same rate?
Find the unit rate (cost per apple): 6/8 = 0.75 dollars per apple. Then multiply by 20.
Answer: $15.
Problem: A map uses a scale of 1 inch = 25 miles. Two cities are 3.6 inches apart on the map. What is the real distance?
Scaling is direct multiplication:
Answer: 90 miles.
Two ratios are equivalent if you can multiply or divide both terms by the same nonzero number to get from one to the other. This is the same idea as simplifying a fraction.
Example: Simplify the ratio 18:24.
Simplifying early makes proportions easier because you work with smaller numbers.
A percent is “per 100.” That means many percent questions can be solved by writing a proportion.
Example: What is 35% of 80?
You can also do x = 0.35·80, but the proportion method generalizes to many word problems.
Most beginner problems are direct: if one quantity goes up by a factor, the other also goes up by the same factor. But sometimes the relationship is inverse: when one goes up, the other goes down.
Work-rate problems like “more workers finish faster” are often inverse relationships.
One of the most common sources of errors is using the wrong “whole.” A ratio like 3:2 can mean part-to-part(3 red, 2 blue), but many word problems ask for a part-to-whole fraction or percent.
Example: The ratio of red to blue marbles is 3:2 and there are 25 marbles total.
Notice this wasn’t cross multiplication at all—it’s a “parts” method. If you do set up a proportion here, the key is matching the same unit (marbles) and the same order (red/total, blue/total, or red/blue).
Recipes, paint mixes, and “solution concentration” questions are usually direct proportions: if you multiply the batch size by k, you multiply each ingredient by k.
Example: A smoothie uses 3 cups of yogurt for every 2 cups of fruit. If you use 7.5 cups of yogurt, how much fruit should you use to keep the same taste?
Answer: 5 cups of fruit.
If two figures are similar, corresponding lengths have a constant ratio (the scale factor). Perimeters scale like lengths, areas scale by the square of the scale factor.
Example: Two similar triangles have side lengths 6 and 9 for corresponding sides. If the smaller triangle’s perimeter is 24, what is the larger perimeter?
Scale factor k = 9/6 = 3/2.
This is a quick sanity check too: the larger triangle is 50% bigger in length, so the perimeter should also be 50% bigger.
Problem: 4 workers can paint a fence in 6 hours. If the workers all paint at the same rate, how long would it take 8 workers?
This is inverse because more workers means less time. One common setup is workers·time = constant.
Answer: 3 hours. A quick check: doubling workers should cut the time in half.
If you set up apples/bananas in one ratio, keep apples/bananas in the other. Order must match.
Multiply diagonals: a·d and b·c. Don’t multiply across the top or bottom.
Ratios compare units. If the units don’t match, the ratio setup is wrong.
Plug back in or compare decimals. A quick check catches most setup errors.
If you want a compact explanation of proportions and scaling, the overview lessons in OpenStax ratios and ratescan help reinforce why cross multiplication works.
Paste your ratio problem and get a clean proportion + final answer with a check.
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