Math · Ratios & measurement
Ratios and Proportions
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Start here: key ideas
- A ratio compares two quantities in a stated order.
- Equivalent ratios multiply or divide both quantities by the same nonzero number.
- A proportion states that two ratios are equal.
- A relationship is proportional only when its unit ratio stays constant.
What you’ll be able to do
- Write ratios in equivalent forms while keeping units in the same order.
- Solve proportions by scaling and verify them with cross-products.
- Recognize when two quantities are not proportional.
A ratio compares two quantities in a stated order. Equivalent ratios multiply or divide both quantities by the same nonzero number.
Complete fractions operations mixed numbers first.
Keep the comparison and units in order
A ratio is a comparison by division. “3 red tubes for every 5 blue tubes” can be written 3 to 5, 3:5, or 3/5. The order matters because 3/5 describes red compared with blue, while 5/3 describes blue compared with red. Include units when they differ: 120 miles/3 hours, not just 120/3.
A part-to-part ratio compares categories. With 2 red and 3 blue tubes, red:blue is 2:3. A part-to-whole ratio compares a category with the total: red:all is 2:(2+3)=2:5.
To simplify a ratio, divide both terms by the same common factor. For 18:24, divide both by 6 to get 3:4. Simplifying does not change the comparison.
Build equivalent ratios
Two equivalent ratios describe the same relationship. Apply one scale factor to both entries. If 2 cups of rice serve 5 people, multiplying both by 3 gives 6 cups for 15 people.
| Cups | People | Scale from 2:5 |
|---|---|---|
| 2 | 5 | ×1 |
| 4 | 10 | ×2 |
| 6 | 15 | ×3 |
Denny Burzynski and Wade Ellis, Jr. / OpenStax CNX, via Mathematics LibreTexts · CC BY · https://math.libretexts.org/Bookshelves/PreAlgebra/Fundamentals_of_Mathematics_(Burzynski_and_Ellis)/04%3A_Introduction_to_Fractions_and_Multiplication_and_Division_of_Fractions/4.03%3A_Equivalent_Fractions_Reducing_Fractions_to_Lowest_Terms_and_Raising_Fractions_to_Higher_Terms · https://creativecommons.org/licenses/by/ · Source image reproduced unchanged.
Solve and verify proportions
A proportion is an equation between two ratios. Worked example 1: Four notebooks cost $10. At the same price, what do 14 cost? Set 4/10 = 14/x with matching notebook and dollar positions. Scaling 4 to 14 uses 3.5, so x = 10 × 3.5 = $35. Verify: 4x = 10(14), so 4(35)=140.
Worked example 2: A mixture uses 3 mL concentrate for 8 mL water. For 20 mL water, 3/8 = c/20. Cross-products give 8c=60, so c=7.5 mL. Cross-products verify equality; the ratio meaning explains why the equation was arranged that way.
Test before using a proportion
A proportional relationship has one constant of proportionality. Divide corresponding output by input in several rows. The pairs (2,6), (4,12), and (5,15) all have y/x=3. The pairs (2,7) and (4,13) do not: 7/2 differs from 13/4. A fixed starting charge often creates a nonproportional relationship because doubling the input does not double the total.
Label corresponding ratio positions before calculating. A sentence such as “cups per serving equals cups per serving” checks that meanings and units stay aligned. A check cannot repair a reversed setup.
Scaling is fastest when one quantity is an obvious multiple; algebra is useful when it is not. Both work because the numerator and denominator are changed by the same nonzero factor.
Test proportionality with several pairs. A $5 starting fee plus $2 per mile is not proportional: one mile costs $7, but two miles cost $9, so total cost per mile is not constant. A proportional graph passes through the origin.
Finally check direction. If servings increase in a proportional recipe, every ingredient must increase. A smaller ingredient result signals an inverted scale factor.
When units differ inside a ratio, do not simplify the numbers until the units are compatible. A comparison of 2 feet to 8 inches becomes 24 inches to 8 inches, or 3:1.
Key terms
- ratio: A comparison of two quantities by division.
- equivalent ratios: Ratios that name the same comparison.
- proportion: An equation stating that two ratios are equal.
- scale factor: The multiplier that changes one equivalent ratio into another.
- cross-product: A product made from diagonally opposite terms of a proportion.
- constant of proportionality: The unchanged unit rate in a proportional relationship.
Summary
- A ratio compares two quantities in a stated order.
- Equivalent ratios multiply or divide both quantities by the same nonzero number.
- A proportion states that two ratios are equal.
- A relationship is not proportional when equivalent input ratios do not produce one constant output ratio.
Ratio order follows the words: “3 nurses to 8 patients” means 3:8, not 8:3.
Sources
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