How to Simplify Ratios and Solve Proportions
Ratios compare quantities, and proportions express equality between two ratios. This guide explains how to find the GCD to simplify ratios, and how to use cross-multiplication to solve for unknown values in proportions.
What You Need to Know About GCD
The greatest common divisor, or GCD, is the largest number that divides evenly into two or more numbers. Simplifying a ratio means dividing both parts of the ratio by their GCD until no common factor remains. A ratio simplified this way is in its lowest terms, which makes it easier to interpret and compare.
For example, the ratio 12:8 can be simplified. To find the GCD, list the factors of each number. Factors of 12 are 1, 2, 3, 4, 6, and 12. Factors of 8 are 1, 2, 4, and 8. The largest factor both numbers share is 4, so the GCD is 4. Dividing both parts of 12:8 by 4 gives the simplified ratio 3:2.
If you cannot easily list factors, another approach is to repeatedly divide both numbers by any common factor until no further simplification is possible. Start with small primes like 2, 3, and 5. This method is especially useful for larger numbers where listing all factors would take too long.
Step 1 – Find the GCD of Both Numbers
The most systematic way to find the GCD is using the Euclidean algorithm. Divide the larger number by the smaller, then replace the larger number with the remainder. Repeat this process until the remainder is zero. The last non-zero remainder is the GCD.
For the ratio 36:24, divide 36 by 24. The quotient is 1 and the remainder is 12. Now divide 24 by 12. The quotient is 2 and the remainder is 0. Since the remainder reached zero, the GCD is 12. This algorithm works for any pair of positive integers, no matter how large.
For a ratio with three or more parts, such as 12:18:24, find the GCD of the first two values and then find the GCD of that result and the third value. GCD of 12 and 18 is 6. GCD of 6 and 24 is 6. Dividing all three parts by 6 gives the simplified ratio 2:3:4.
Step 2 – Divide Both Numbers by the GCD
Once you have the GCD, divide each part of the ratio by it. For 36:24 with a GCD of 12, divide 36 by 12 to get 3, and divide 24 by 12 to get 2. The simplified ratio is 3:2. This ratio preserves the original relationship between the two quantities while expressing it in the smallest possible whole numbers.
Always verify your result by checking that the two simplified numbers share no common factors other than 1. If they still share a common factor, you used a number smaller than the true GCD and need to simplify further. For 3 and 2, the only common factor is 1, confirming the ratio is fully simplified.
Ratios can also be expressed as fractions. The ratio 3:2 is equivalent to the fraction 3/2. This means for every 3 units of the first quantity, there are 2 units of the second. Expressing ratios as fractions makes them easier to compare and use in further calculations.
Solving Missing Values in Proportions
A proportion states that two ratios are equal, such as 3/4 equals 9/12. When one of the four values is unknown, you can solve for it using cross-multiplication. Multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first by the numerator of the second, then set the two products equal.
For example, if the proportion is 3/4 equals x/20, cross-multiply to get 3 times 20 equals 4 times x, which simplifies to 60 equals 4x. Dividing both sides by 4 gives x equals 15. You can verify the answer by checking that 3/4 and 15/20 are equivalent fractions, which they are since both simplify to 3/4.
Cross-multiplication works because it is algebraically equivalent to multiplying both sides of the equation by both denominators simultaneously. Understanding this makes the technique less mysterious and helps you apply it correctly even when the proportion is presented in a non-standard form.
Real-World Applications
Ratios and proportions appear throughout cooking, construction, science, and finance. When scaling a recipe from 4 servings to 10, you set up a proportion: if 4 servings require 2 cups of flour, how much flour is needed for 10 servings? Solving 4/2 equals 10/x gives x equals 5 cups.
Map reading and scale models rely on ratios. If a map scale is 1:50,000, then 1 centimeter on the map represents 50,000 centimeters, or 500 meters, in the real world. Multiplying the measured distance on the map by 50,000 gives the actual distance.
In finance, ratios like price-to-earnings compare a company's stock price to its annual earnings per share. Simplifying these ratios makes it easier to compare companies of different sizes. If one company has a P/E ratio of 45:3 and another has 20:2, simplifying both to 15:1 and 10:1 reveals that the second company is cheaper relative to its earnings.