A ratio compares two quantities, a fraction expresses a quotient, often a part of the total, and a proportion says that two ratios are equal. The same 11 marbles give 4/11 if you compare the red ones with the total and 4/7 if you compare them with the blue ones. Here you learn how to choose each one; to calculate, use the ratio to fraction calculator or the proportions calculator.
| Idea | What it does | In the bag |
|---|---|---|
| Ratio | Compares two quantities in an order | 4 red for every 7 blue: 4:7 |
| Fraction | Expresses a quotient; often, a part of the total | Red is 4 of 11 marbles: 4/11 |
| Proportion | States that two ratios are equal | 4:7 = 8:14 |
All three rows use the same numbers, but they answer different questions: the ratio and the proportion compare groups with each other, and the fraction measures a part of the whole set.
The second bag has twice as many marbles, yet neither the ratio nor the fraction changes. That is the idea that ties the three together: comparing without depending on the absolute size.
When a ratio can be turned into a fraction of the total
A ratio between two groups becomes a fraction of the total with a/(a + b), and only if those two groups are everything you are counting. In the bag, 4 and 7 add up to 11, so blue is 7 of 11. To read a part of the total as a fraction, see what fraction a quantity represents.
If you add 2 green marbles, the red to blue ratio is still 4:7, but now there are 13 marbles and red is 4 of 13. The ratio does not depend on the green ones; the fraction of the total does. Before switching from one to the other, define what the total includes.
The ratio can also be written as a quotient to compare red with blue: 4/7 of a red marble for each blue marble. That number is not the share of the marbles that are red. To convert to a percent, use the fraction of the total and the percentage calculator.
Worked example: 4 red marbles and 7 blue
One bag has 4 red marbles and 7 blue. Another has 8 red and 14 blue. Which ratio, which fraction, and which proportion describe the marbles?
- Write the ratio. Red to blue is 4:7.
- Count the total. 4 + 7 = 11 marbles.
- Write the fraction of the total. Red is 4/11 = 0.3636… ≈ 36.4% (error less than 0.05 percentage points).
- Look at the second bag. It has 8:14 and, dividing by 2, it becomes 4:7. The two ratios are equal, so they form a proportion.
- Look at its fraction. 8 of 22 marbles are red, and 8/22 = 4/11.
Result: ratio 4:7, red is 4/11 of the total, and proportion 4:7 = 8:14.
Check: 4 × 14 = 56 and 7 × 8 = 56, so the cross products match.
How to check a proportion and find the missing term
Two ratios form a proportion if their cross products are equal: a/b = c/d when a × d = b × c, with b and d not zero. With 2/5 and 3/7 it fails, because 2 × 7 = 14 and 5 × 3 = 15. The two ratios look alike and still are not equal.
To find a term, keep the order of the quantities. If you have 12 red marbles and want to keep the ratio, set up 4/7 = 12/x. The cross products give 4x = 84, so x = 21 blue. Check by dividing: 12:21 is 4:7 after dividing by 3.
This calculation assumes the situation keeps the ratio. A recipe or a scale map does, but a price with a flat fee does not. The rule of three uses the same equation applied to a problem with two quantities; the guide to map scales uses it with units of length.
Rates: ratios with different units
If the two terms measure different quantities, the ratio is called a rate and carries units. A car that travels 150 miles in 2 hours has the ratio 150:2, which gives 75 miles per hour. It is not a fraction of a set, because miles and hours do not make up a total that can be divided. The guide to speed, distance, and time works through that case with fractions of an hour.
Where to go next
- To reduce a ratio to its smallest terms, use simplify ratios.
- To see the other uses of a fraction (quotient, measure, operator), read meanings of a fraction.
- To split an amount using weights, see sharing with fractions.
- If you are starting from scratch, what are fractions explains the idea of part and whole.
Sources
The Wikipedia article Ratio (accessed October 17, 2026) states that an equality of two ratios A:B and C:D is called a proportion, and illustrates the difference between comparing parts and comparing a part with the whole using oranges and apples: the ratio of oranges to apples is 2:3, and the ratio of oranges to all the fruit is 2:5.