Rule of Three (Cross Multiplication)

Calculator for solving proportion problems using the rule of three

Rule of Three Calculator

Rule of Three (Proportion equation)

Mathematical method to compute the unknown fourth value from three known values of a proportion using the cross product

Proportionality

The rule of three solves proportion problems using the equation a/b = c/x, where x is the unknown value.

Variable a
First value of the first ratio
Variable c
First value of the second ratio
Variable b
Second value of the first ratio
Proportion equation
4
2
=
6
x
Formula: x = (b × c) ÷ a = (2 × 6) ÷ 4 = 3
Rule of Three Result
Variable x:
—
Calculation:
—
Enter three known values and compute the fourth

Rule of Three Info

Rule of Three

Proportion equation: a/b = c/x

Proportion Direct Inverse

Direct: More A → More B
Inverse: More A → Less B

Quick examples
5 apples = 2€: 8 apples = ?
3 workers = 12h: 4 workers = ?
100km = 8L: 150km = ?
Basic formula
\[\frac{a}{b} = \frac{c}{x}\]
Rearranged: \[x = \frac{b \cdot c}{a}\]
Types
• Direct: Proportional (more → more)
• Inverse: Antiproportional (more → less)


Rule of Three Methods and Applications

Direct rule of three
Proportional relationship:
\[\text{More A} \rightarrow \text{More B}\]
x = (b × c) ÷ a

Both quantities change in the same direction

Inverse rule of three
Antiproportional relationship:
\[\text{More A} \rightarrow \text{Less B}\]
x = (a × b) ÷ c

Quantities change in opposite directions

Proportion equation
Basic proportion:
\[\frac{a}{b} = \frac{c}{x}\]

Two ratios are equal

Cross product
Rewriting the proportion:
\[a \cdot x = b \cdot c\]

Multiply across and solve for x

Step-by-step examples

Direct rule of three: Buying apples
5 apples = 2 Euro 8 apples = ? Euro
Step 1: 5 apples = 2 Euro
Step 2: 1 apple = 2 ÷ 5 = 0.40 Euro
Step 3: 8 apples = 0.40 × 8 = 3.20 Euro
Answer: 3.20 Euro

More apples → More cost (proportional)

Inverse rule of three: Work time
5 workers = 10 hours 8 workers = ? hours
Step 1: 5 workers = 10 hours
Step 2: 1 worker = 10 × 5 = 50 hours
Step 3: 8 workers = 50 ÷ 8 = 6.25 hours
Answer: 6.25 hours

More workers → Less time (inverse proportional)

Formula application
Direct rule of three
Formula: x = (b × c) ÷ a
Example: x = (2 × 8) ÷ 5 = 3.20
5 apples : 2€ = 8 apples : x€
Inverse rule of three
Formula: x = (a × b) ÷ c
Example: x = (5 × 10) ÷ 8 = 6.25
5 workers : 10h = 8 workers : xh

The choice of formula depends on the type of proportionality

Applications of the rule of three

The rule of three is a universal tool for solving proportional problems:

Trade & Economy
  • Price calculations and discounts
  • Currency conversions
  • Bulk discounts and scaling
  • Cost-benefit analyses
Production & Engineering
  • Material requirement calculations
  • Work time and capacity planning
  • Scale calculations
  • Energy and resource consumption
Everyday & Household
  • Scale recipes
  • Calculate fuel consumption
  • Plan travel time and costs
  • Shopping and budgeting
Education & Science
  • Basic mathematical education
  • Physical calculations
  • Statistical extrapolations
  • Scaling laboratory experiments

The rule of three: Foundation of proportional relationships

The rule of three is one of the most practical techniques in elementary mathematics. This simple technique—solving proportional problems using proportion equations—forms the basis for countless everyday and scientific calculations. From ancient trade arithmetic to industrial production and modern data analysis, the rule of three appears wherever proportional relationships arise. Distinguishing between direct and inverse proportionality helps recognize and describe relationships beyond pure arithmetic.

Summary

The rule of three embodies systematic thinking: from three known values of a proportion the fourth can be reliably determined. The method not only teaches calculation skills but also logical thinking and problem-solving. From practical applications in commerce and household to technical calculations and scientific analysis, the rule of three shows how mathematical principles solve everyday problems and model proportional relationships.