What is the Rule of Three?

The Rule of Three is a mathematical method to calculate an unknown value when three values of a ratio are known. It is one of the most important tools in everyday mathematics, used in many areas such as shopping, cooking, finance, and technology.

How do you calculate the Rule of Three?

For the proportional Rule of Three: 1) Write down the known values (a units = b value). 2) Divide by a to get the value for 1 unit. 3) Multiply by the desired quantity c. The formula is: x = (b × c) / a.

What is the difference between proportional and inverse?

In proportional Rule of Three, both quantities increase together (twice as much → twice the price). In inverse Rule of Three, one quantity increases while the other decreases (twice as many workers → half the time). The inverse formula is: x = (a × b) / c.

When do you need the inverse Rule of Three?

You need the inverse Rule of Three when one value increases and the other decreases as a result. Typical examples: more workers → less time, higher speed → shorter travel time, more machines → less production time per unit.

What is a compound Rule of Three?

A compound Rule of Three is used when more than two quantities are involved. It is solved step by step: first adjust the first quantity (proportional or inverse), then the second. The result from step 1 becomes the starting value for step 2.

Can you solve percentages with the Rule of Three?

Yes! Percentage calculation is a special case of the Rule of Three. Example: 100% = €500, find: 15%. Solution: €500 ÷ 100 × 15 = €75. The Rule of Three is often simpler because you don't need to distinguish between base value, percentage value, and rate.

In which grade do you learn the Rule of Three?

The Rule of Three is typically introduced in grade 6 or 7. The proportional version comes first, the inverse version usually follows one school year later. The compound Rule of Three is often covered in grade 8 or 9.

How do I tell if it's proportional or inverse?

Ask yourself: 'If one quantity increases, does the other increase or decrease?' If it increases → proportional. If it decreases → inverse. Example: buying more apples → higher price (proportional). More people share a pizza → less per person (inverse).

What is the difference between the Rule of Three and a ratio equation?

Mathematically, both are equivalent. The Rule of Three solves the problem step by step (calculate for 1, then scale up), the ratio equation sets a:b = c:x and solves for x. The Rule of Three is more intuitive, while the ratio equation is more concise.

How do you verify a Rule of Three calculation?

For proportional Rule of Three, check that the ratio stays the same: a/b must equal c/x. For inverse Rule of Three, check that the products are equal: a × b must equal c × x. If the check passes, your result is correct.

Compound Rule of Three Guide | Rule of Three Calculator

Compound Rule of Three: Guide for Multiple Variables

The compound Rule of Three extends the basic principle to problems with more than two quantities. This article shows you how to solve such problems systematically.

What is the Compound Rule of Three?

In the simple Rule of Three, you have two related quantities. In the compound version, three or more quantities are involved, all interconnected.

**Example:** The number of produced parts depends on both the number of machines AND the working hours.

The Method: Solve Step by Step

Break the compound Rule of Three into multiple simple ones:

1. **Hold all variables constant except one** and calculate the intermediate value

2. **Adjust the next variable** using the intermediate value as the new starting point

3. Repeat for additional variables

At each step, decide: Is this variable proportional or inverse to the result?

Detailed Example

**Problem:** 5 workers produce 200 parts in 8 hours. How many parts do 8 workers produce in 6 hours?

**Analysis:** Workers -> proportional (more workers = more parts). Hours -> proportional (more hours = more parts).

**Step 1:** Adjust workers (8 instead of 5, proportional): 200 x 8/5 = 320 parts

**Step 2:** Adjust hours (6 instead of 8, proportional): 320 x 6/8 = 240 parts

**Result:** 8 workers produce 240 parts in 6 hours.

Example with Mixed Relations

**Problem:** 6 pumps fill a pool in 4 hours. How many pumps to fill 3 pools in 2 hours?

**Analysis:** Pools -> proportional (more pools = more pumps needed). Hours -> inverse (less time = more pumps needed).

**Step 1:** Adjust pools (3 instead of 1, proportional): 6 x 3/1 = 18 pumps

**Step 2:** Adjust hours (2 instead of 4, inverse): 18 x 4/2 = 36 pumps

**Result:** You need 36 pumps.

Table Method for Overview

For complex problems, use a table:

|---|---|---|---|

| Workers | 5 | 8 | proportional |

| Hours | 8 | 6 | proportional |

| Parts | 200 | x | result |

This helps you track which variable needs which adjustment.

Typical Problem Types

- **Production:** Machines x Hours -> Output

- **Transport:** Vehicles x Trips -> Freight

- **Agriculture:** Workers x Days -> Harvest

- **Cooking:** Servings x Portion size -> Ingredients

Exam Tips

1. Read carefully and identify all quantities

2. Decide for each quantity: proportional or inverse?

3. Solve one step at a time — never everything at once

4. Verify the result with common sense

Our Calculator Helps

In the 'Compound' tab, you can calculate compound Rule of Three problems directly. Choose the relation for each variable and get the complete solution path.