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Rule of Three Calculator Direct & Inverse Step by Step

Solve mathematical proportions instantly for free with detailed step-by-step solutions, variable substitution, and clear formulas for school homework, cooking recipes, percentages, and everyday math problems.

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Calculate Your Rule of Three

Proportion Type:
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X = 240

Step-by-Step Mathematical Procedure

Applied formula: X = (B · C) / A
Data substitution: X = (100 · 12) / 5
Calculation result: X = 1200 / 5 = 240
Mathematical reasoning:

Al ser una proporción directa, a mayor valor en la primera magnitud (de 5 sube a 12), mayor será el resultado correspondiente de X (sube de 100 a 240).

Quick Preset Scenarios:
Mathematical Concept

What is the Rule of Three and Why is it Useful?

The rule of three is one of the most widely applied and practical mathematical tools in daily life and academic education. It enables you to determine an unknown fourth quantity (commonly labeled as X) based on the proportional relationship established between three known quantities.

The mathematical foundation rests on proportion theory: whenever two quantities are related and we observe a variation in one of them, we can deduce the corresponding value of the other. Depending on whether the quantities increase together or in opposite directions, the method is classified into a direct rule of three or an inverse rule of three.

Direct Proportion

1. Direct Rule of Three

Applied when two quantities are directly proportional. When one quantity increases, the other quantity increases proportionally; when one decreases, the other decreases in the exact same proportion.

X = (B · C) / A

Cross-multiply the opposite known values (B and C) and divide the product by the initial value paired diagonally with the unknown (A).

Inverse Proportion

2. Inverse Rule of Three

Used when two quantities are inversely proportional. In this scenario, when one quantity increases, the other quantity decreases proportionally (for instance, hiring more workers decreases the total days needed to finish a project).

X = (A · B) / C

Multiply the two known values in the reference row horizontally (A and B) and divide by the third known quantity (C).

Common Mistakes

Common Mistakes to Avoid in Rule of Three Calculations

  • Confusing direct and inverse relationships: Applying cross multiplication to labor, workforce, or vehicle speed scenarios where more resources inherently reduce completion time.
  • Mixing incompatible measurement units: Multiplying hours by minutes or meters by kilometers without converting them to matching baseline units first.
  • Misaligning data in proportion columns: Swapping values between columns (for example, placing currency and weight in the same vertical category).
  • Neglecting logic and reasonableness checks: Failing to verify whether the final computed number should logically be greater or smaller than the baseline reference value.
Solution Methodology

Practical Guide: How to Set Up and Solve Any Rule of Three Problem

To solve any proportional mathematics exercise accurately with the rule of three, always follow these structured steps:

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Step 1: Identify the quantities and units. Clearly define what measurements you are comparing (such as miles and hours, or dollars and pounds). Never combine different units without converting them first.

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Step 2: Organize the numbers into a two-row table. Place the two related known quantities (A and B) in the first row. In the second row, place the third known value (C) under its matching column and place X under the unknown column.

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Step 3: Analyze the direction of proportionality. Ask yourself whether an increase in the first quantity will cause the second quantity to grow or shrink. If both grow, it is direct; if one grows while the other drops, it is inverse.

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Step 4: Execute the proper arithmetic formula. For direct proportions, multiply diagonally (B times C) and divide by A. For inverse proportions, multiply across (A times B) and divide by C.

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Step 5: Verify the answer with common sense. Check whether the resulting numeric answer makes practical sense in the real-world context before accepting it as the final solution.

Real-World Examples

Everyday Applications and Real-World Examples

Calculating Discounts and Percentages

If an item priced at $80 represents 100% of its retail price, how much money do you save with a 25% discount? Using the direct rule of three: X = (25 · 80) / 100 = $20 savings. The final price to pay is $60.

Cooking Recipes and Ingredient Scaling

If baking a cake for 4 guests requires 200 grams of flour, how much flour is needed for 10 guests? Direct proportion: X = (200 · 10) / 4 = 500 grams of flour.

Travel Speed, Distance, and Driving Time

A vehicle traveling at a constant speed of 100 km/h completes a highway journey in 3 hours. How long will the trip take at 120 km/h? Since speed and time are inversely proportional: X = (100 · 3) / 120 = 2.5 hours (2 hours and 30 minutes).

Workforce Labor and Construction Deadlines

If a construction crew of 6 workers builds a masonry wall in 12 days, how many days will 9 workers require at the same pace? Inverse proportion: X = (6 · 12) / 9 = 8 days.

Questions & Answers

Frequently Asked Questions About the Rule of Three

How do you calculate the direct rule of three step by step?

First, align the two related quantities in a table: 'If A corresponds to B, then C corresponds to X'. For direct proportion, cross-multiply B by C, then divide that product by A. The standard formula is X = (B · C) / A.

How do you know when a rule of three problem is inverse?

Ask yourself: 'If the first quantity doubles, does the second quantity double or halve?'. If increasing one quantity causes the other to decrease proportionally (such as higher speed requiring less travel time, or more laborers completing a job in fewer days), it is an inverse rule of three.

How can you use the rule of three to calculate percentages?

Assign the whole total to 100%. Set up: 'Total -> 100%' and 'Part -> X%'. If you want to find what percentage 40 is of 200: X = (40 · 100) / 200 = 20%. To calculate 15% of 800: X = (800 · 15) / 100 = 120.

How is the rule of three used in nursing and medication dosage calculation?

In clinical pharmacology: 'Available dosage (mg) -> Liquid volume (ml)'. On the second line: 'Prescribed dosage -> Volume to administer (X ml)'. You multiply the doctor's prescribed dose by the volume and divide by the stock concentration.

What is the difference between simple and compound rule of three?

The simple rule of three involves only two interconnected variables (such as people and cost). The compound rule of three involves three or more variables simultaneously (such as workers, daily working hours, and square meters built).

How do you solve a rule of three in Microsoft Excel or Google Sheets?

If cell A1 holds A, B1 holds B, and A2 holds C, to solve a direct rule of three in cell B2 enter =(B1*A2)/A1. For an inverse rule of three, enter =(A1*B1)/A2.