MATHEMATICAL IDEA
What is a compound rule of three?
A compound rule of three extends the simple rule of three when two or more quantities are connected to an unknown. Instead of comparing one pair of values, you start with a known situation and observe how every condition changes in the new situation. The method is useful for work, production, recipes, consumption, speed, scale, and proportional sharing. This calculator keeps each quantity separate so you can review the setup before accepting the answer.
The key idea is to preserve the same total work or relationship. A direct quantity moves X in the same direction: more ingredients produce more servings, for example. An inverse quantity moves the other way: more workers normally require fewer days to finish the same task. The label changes the ratio: use new ÷ base for a direct relationship and base ÷ new for an inverse one.
Before calculating, define what X means and use a consistent unit in each column. Do not mix hours with minutes or litres with millilitres without converting them first. The tool shows the multiplication of every ratio and rounds only the displayed result. Keep extra precision when the exercise needs it, and always check whether the answer makes sense in context.
METHOD
How to solve a compound rule of three
A careful setup prevents you from reversing a relationship by intuition. Follow this order and use the steps panel as an independent check of your arithmetic.
1
1. Define the unknown
State what you need to find and enter the base value of X. It can be days, consumption, price, output, or any other comparable unit.
2
2. Order each quantity
For every condition, record the reference value and the new value. Keep the same unit in both fields and use names that distinguish workers, hours, kilograms, and pieces.
3
3. Choose the relationship
Ask what happens to X when only that quantity increases. If X increases too, choose Direct; if X decreases to preserve the same task, choose Inverse.
4
4. Multiply the ratios
For Direct use new ÷ base. For Inverse use base ÷ new. Multiply each ratio by the base value of X, in any order.
5
5. Interpret and verify
Read the explanation, round according to the unit, and check the direction of change. If the answer conflicts with the story, review a label or conversion.
FORMULA
Compound rule of three formula
If T is the base value of X, each row adds one ratio. The final expression is X = T × r₁ × r₂ × … × rₙ. Direct and inverse rows can be combined; the important part is explaining every choice from the behavior of the quantities.
| Type | Applied ratio | When to use it |
|---|
| Direct | new ÷ base | X changes in the same direction as the quantity. |
| Inverse | base ÷ new | X changes in the opposite direction. |
| Mixed | T × r₁ × r₂ | Combines direct and inverse rows in one problem. |
EXAMPLES
Where this method is useful
These cases show why each condition deserves its own row instead of one unexplained multiplication.
Workers and days
A job takes 15 days with 10 workers and 8 hours per day. With 15 workers and 6 hours, both quantities are inverse to days: X = 15 × (10 ÷ 15) × (8 ÷ 6) = 13.33 days.
Recipes and servings
If a recipe uses 250 grams for 4 people and you need 10 servings, the number of people is direct. Add rows for ingredients when you want to preserve the full proportion.
Production
A machine makes a known output in one shift. When machines and hours change, machines are direct when they increase output, and hours are direct when they extend working time.
Fuel and distance
Distance is usually direct with fuel used. If you compare speeds and travel time, check whether the rate is per hour or per kilometre before choosing a relationship.
CHECK YOUR WORK
Limits and common mistakes
A calculator cannot decide whether two quantities are truly proportional. The arithmetic is correct for the relationships you select, so interpreting the wording is the most important part.
- Do not choose Inverse just because the final number should be smaller; first explain what remains constant.
- Convert units before entering them: 90 minutes and 1.5 hours match, but 90 and 1.5 do not without conversion.
- Never put zero in a denominator. A zero base for Direct or a zero new value for Inverse does not define a valid ratio.
- Negative values rarely describe physical quantities. Use them only when an algebra exercise clearly requires them.
- The visible result is rounded to four decimals. For money, doses, or critical measurements, keep the problem precision and verify it.
- For medical, industrial, or financial decisions, use the calculation as support and confirm the units and procedure with a qualified professional.
The calculation runs in this browser. The quantities are not saved or sent to a calculation API; read the privacy policy for the site's general services.
FREQUENT QUESTIONS
Compound rule of three questions
How many variables does this calculator support?
You can use between two and six quantities in addition to the base value of X. Add a row only when that condition has a clear proportional relationship with the unknown.
How do I know if a quantity is direct or inverse?
Imagine increasing only that quantity. If X rises as well, it is Direct; if X falls to preserve the same job or amount, it is Inverse. More workers for the same job is the classic inverse example.
What formula does a compound rule of three use?
The tool uses X = T multiplied by every ratio. Each Direct row adds new ÷ base, while each Inverse row adds base ÷ new. The steps panel displays the full substitution.
Can I mix direct and inverse relationships?
Yes. One problem can contain a direct quantity and an inverse quantity. Select the type for each row and read the explanation before copying the solution.
Why does my result contain decimals?
A ratio may not produce a whole number. The decimal is valid when the unit allows it; for people or objects, interpret it as an estimate and decide how to round in practice.
Does the tool save my data?
No. JavaScript performs the operation on your device and the form does not make a calculation request. Avoid entering personal information in quantity names.
Continue with a simple check
If your exercise has one quantity and three known values, the main simple rule of three calculator may be faster. For several changes, return here and keep the formula visible so the method is easy to explain.
Open simple calculator
