Solve compound direct and inverse proportion problems with 2 to 5 quantities, editable factors and a worked solution.
Solve proportions involving several quantities
Enter the known result and how the related quantities change. Mark each one as direct or inverse with respect to the result, then review every multiplier.
Compound proportion result
You changed the inputs. The last result remains visible until you press “Calculate proportion”.
Overall multiplier:
Step-by-step calculation
Quantity
Type
Applied ratio
Multiplier
Running result
The answer is only valid if each relationship is proportional, compared units match and the other relevant factors remain unchanged.
How to solve a compound proportion step by step
This compound proportion calculator combines up to five quantities that affect a known result proportionally. Enter the reference result, the starting and new values of each quantity, and mark whether its relationship to the result is direct or inverse. The tool then shows each multiplier and the complete formula. It assumes all other relevant conditions remain constant and cannot automatically decide proportionality from a word problem.
Example: 4 workers make 300 parts in 12 days. With 6 workers working at the same rate, how many days are needed for 450 parts? Days = 12 × (4 ÷ 6) × (450 ÷ 300) = 12.
Results are estimates and depend on the supplied values.
Compound proportion formula and worked examples
What is a compound proportion?
A compound proportion relates one result to two or more changing quantities. To find the new result, compare each quantity with the result while considering the others fixed.
New result = known result × factor₁ × factor₂ × ...
When to use direct proportion
Use a direct factor when increasing that quantity, all else equal, increases the result in the same proportion. For example, producing more parts takes more days with the same team and productivity.
Direct factor = new value ÷ starting value
When to use inverse proportion
Use an inverse factor when increasing the quantity, all else equal, decreases the result proportionally. For example, more workers need fewer days for the same job if their work rates remain constant.
Inverse factor = starting value ÷ new value
Example: workers, days and output
Four people make 300 parts in 12 days. With six people and an order for 450 parts, assuming identical productivity and shift length, days are inverse to workers and direct to parts.
New days = 12 × 4/6 × 450/300 = 12 days
How to check a problem
Before calculating, ask: if you only increase the number of workers, should the time fall? If you only increase the parts needed, should the time rise? If the selected relationships disagree, review them.
Always compare matching units: hours with hours, parts with parts, days with days.
Important assumptions
Every quantity must be in a purely direct or inverse relationship to the result, with other conditions held fixed. The model does not include diminishing returns, delays, fixed minimum times, changing productivity or nonlinear relationships. The user must decide whether each factor is direct or inverse.
Mathematical basis
Derived by multiplying proportional change factors from the reference case. Khan Academy covers direct/inverse variation and simple and compound proportion in its arithmetic materials.
What is the difference between simple and compound proportion?
Simple proportion compares two quantities; compound proportion relates a result to two or more additional quantities. This tool accepts between two and five factors.
How do I tell direct proportion from inverse proportion?
Keep the other conditions fixed. If the result rises in the same proportion as the quantity, it is direct. If it falls proportionally, it is inverse.
Can one problem contain both direct and inverse relationships?
Yes. Multiply by new/starting for direct factors and by starting/new for inverse factors.
Can I enter decimals and different units?
You may use a comma or a period for decimals. Each quantity’s starting and new measurements must be in the same unit or converted before entry.
Why is zero not accepted as a starting value?
The ratios require nonzero denominators and the model starts from a positive reference result. If zeros occur, use a different mathematical formulation.
Does the result always apply in real situations?
No. It is valid only when the chosen relationships are truly proportional. Worker productivity, for example, may change as a team gets larger.