Calculate the compound ratio of two or more ratios by multiplying the antecedents and consequents, then simplifying the result to lowest terms.
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Compound Ratio Formula
A compound ratio combines two or more ratios into a single ratio by multiplying their antecedents together and their consequents together.
Compound Ratio = (a_1 \times a_2 \times \cdots \times a_n) : (b_1 \times b_2 \times \cdots \times b_n)
Where:
a_1, a_2, ... a_n are the antecedents (the first or left-hand term of each ratio).
b_1, b_2, ... b_n are the consequents (the second or right-hand term of each ratio).
You multiply all the antecedents to get the new antecedent, multiply all the consequents to get the new consequent, then divide both terms by their greatest common divisor to express the result in lowest terms. For example, the compound ratio of 5:4 and 8:5 is (5 x 8):(4 x 5) = 40:20, which simplifies to 2:1.
Reference Values
The first table shows how related ratio terms are defined, and the second gives worked compound ratios you can use to check your inputs.
| Term | Definition | Example |
|---|---|---|
| Compound ratio | Product of antecedents : product of consequents | 2:3 and 4:5 give 8:15 |
| Duplicate ratio | A ratio compounded with itself once (a:b gives a^2:b^2) | 3:4 gives 9:16 |
| Triplicate ratio | A ratio compounded with itself twice (a:b gives a^3:b^3) | 2:3 gives 8:27 |
| Ratios | Product of terms | Compound ratio (simplified) |
|---|---|---|
| 2:3 and 4:5 | 8:15 | 8:15 |
| 5:4 and 8:5 | 40:20 | 2:1 |
| 3:5, 7:9 and 15:28 | 315:1260 | 1:4 |
| 6:7 and 14:9 | 84:63 | 4:3 |
Example Problems
Example 1. Find the compound ratio of 2:3 and 4:5. Multiply the antecedents: 2 x 4 = 8. Multiply the consequents: 3 x 5 = 15. The compound ratio is 8:15, which is already in lowest terms because 8 and 15 share no common factor greater than 1.
Example 2. Find the compound ratio of 3:5, 7:9, and 15:28. Multiply the antecedents: 3 x 7 x 15 = 315. Multiply the consequents: 5 x 9 x 28 = 1260. This gives 315:1260. Dividing both terms by their greatest common divisor of 315 gives 1:4.
FAQ
What is a compound ratio?
A compound ratio is the single ratio you get when you multiply two or more ratios term by term. You multiply all the antecedents to form the new antecedent and all the consequents to form the new consequent.
How is a compound ratio different from a duplicate ratio?
A duplicate ratio is a special case of a compound ratio where a ratio is compounded with itself, so a:b becomes a squared : b squared. A general compound ratio combines different ratios rather than the same one.
Do I need to simplify the result?
Simplifying is optional but standard. The raw product of terms and the simplified form represent the same ratio, so you can report either, though lowest terms is the conventional answer.