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Rule of Three Calculator: Solve Proportions Online

Solve proportional equations effortlessly with the classic Rule of Three method. Perfect for scaling culinary recipes, calculating fuel usage, comparing unit prices, and solving math homework.

Rule of Three Calculator

Solve proportional math problems using the classic rule of three method.

Known Ratio
Target Quantity
Rule of Three Result
Step 1: Known Ratio:3 → 15
Step 2: Find Unit Rate:1 → 5
Step 3: Scale to Target:8 → 40
Result (D):0
Direct proportionality: When quantity increases, total yield/cost increases by the exact same ratio.
βš–οΈ Proportions & Mathematical Logic

What is the Rule of Three and How Does It Work?

The Rule of Three (also known in mathematics as cross-multiplication or solving proportions) is an ancient, fundamental arithmetic method that calculates an unknown fourth value from three known, related values. It forms the backbone of commercial math, chemistry stoichiometry, culinary portioning, and school algebra.

The 3 Standard Steps of the Method

The reason this technique is called the "Rule of Three" is because it resolves proportional relationships across three distinct statements:

Statement 1 (Known Relationship): A units correspond to B units (A β†’ B)
Statement 2 (Find the Unit Rate): 1 unit corresponds to B Γ· A (divide both sides by A)
Statement 3 (Scale to Target Quantity): C units correspond to (B Γ· A) Γ— C (multiply by C)

Direct Proportion vs. Inverse (Indirect) Proportion

Before solving any problem, you must determine whether the relationship is direct or inverse:

Direct Proportion ("More yields More")

Both values move in the exact same direction. If you double quantity A, quantity B doubles.

Cross-Multiplication Formula: x = (B Γ— C) Γ· A

Examples: Buying more items costs more money; driving further requires more gasoline; cooking for more guests requires more flour.

Inverse Proportion ("More yields Less")

Values move in opposite directions. If you double quantity A, quantity B is cut in half.

Inverse Formula: x = (A Γ— B) Γ· C

Examples: Adding more workers decreases construction time; driving faster reduces travel duration; sharing a pizza among more people yields smaller slices.

Real-World Worked Examples

πŸ§‘β€πŸ³ Scaling Kitchen Recipes (Direct Proportion)

Problem: A pancake recipe for 4 people requires 350 grams of flour. How much flour do you need for 10 people?

  • Step 1: 4 people = 350g
  • Step 2 (Unit Rate): 1 person = 350g Γ· 4 = 87.5g
  • Step 3 (Target): 10 people = 87.5g Γ— 10 = 875 grams of flour
⏱️ Workforce Scheduling (Inverse Proportion)

Problem: 6 painters can finish repainting a school in 10 days. If the school hires 9 painters working at the same pace, how long will the job take?

  • Total Labor required: 6 painters Γ— 10 days = 60 painter-days
  • Scaling to 9 painters: 60 painter-days Γ· 9 painters = 6.67 days (approx. 6 days and 5 hours)

The Compound (Double) Rule of Three

When a problem involves three or more variables simultaneously (such as workers, hours per day, and output volume), you use the compound rule of three:

Example Scenario:

If 4 machines produce 600 components in 5 hours, how many components will 7 machines produce in 8 hours?

1. Find production per machine per hour: 600 Γ· (4 machines Γ— 5 hours) = 30 components / machine-hour
2. Multiply by new variables: 7 machines Γ— 8 hours Γ— 30 components = 1,680 components

Frequently Asked Questions About the Rule of Three

How is the Rule of Three related to percentage calculations?

Every standard percentage calculation is simply a direct rule of three proportion where the total base quantity is tied to 100%. For example: If $80 equals 100%, then 1% is $80/100 = $0.80, and 15% is $0.80 Γ— 15 = $12.00.

What is cross-multiplication?

Cross-multiplication is the algebraic shortcut for solving equivalent fractions. If A / B = C / x, multiplying diagonally gives A Γ— x = B Γ— C, which quickly simplifies to x = (B Γ— C) Γ· A.