Inverse Rule of Three

Calculator for inverse rule of three (inverse proportionality)

Inverse Rule of Three Calculator

What is the inverse rule of three?

The inverse rule of three applies to inverse proportionality: when one quantity increases, the other decreases. The product of both quantities remains constant.

4
×
20
=
5
x
Enter known values
First quantity, first situation
Second quantity, first situation
First quantity, second situation
Calculation result
x =
Calculation: a × b = c × x → x = (a × b) / c

Inverse Proportion Info

Properties

Inverse proportionality: When one quantity rises, the other falls

a × b = constant inverse antiproportional

Rule: More workers → Less time
Formula: a₁ × b₁ = a₂ × b₂

Quick examples
Workers & Time: 4×20 = 5×16 hrs
Speed & Time: 60×2 = 80×1.5 hrs
Illumination & Distance: 100×1 = 25×4 m


Inverse rule of three formulas

Basic formula
\[a_1 \times b_1 = a_2 \times b_2\]

Product remains constant under inverse proportionality

Solve for x
\[x = \frac{a_1 \times b_1}{a_2}\]

Compute unknown via cross product

General form
\[\frac{a_1}{a_2} = \frac{b_2}{b_1}\]

Ratio of first quantities is inverse to ratio of second quantities

Recognition
\[\text{If } a \uparrow \text{ then } b \downarrow\] \[\text{If } a \downarrow \text{ then } b \uparrow\]

One quantity increases, the other decreases

Hyperbola function
\[y = \frac{k}{x} \text{ with } k = \text{const.}\]

Graphical representation as hyperbola

Constant k
\[k = a_1 \times b_1 = a_2 \times b_2\]

Product constant characterizes inverse proportionality

Example calculations for inverse rule of three

Example: Workers and working time
Problem
5 workers → 12 days 8 workers → ? days

More workers need less time for the same work.

Step 1: Identify relation
Inverse proportionality: More workers → Less time
Work volume (workers × time) remains constant
Step 2: Setup equation
5 workers × 12 days = 8 workers × x days
60 = 8x
Step 3: Solve for x
x = 60 ÷ 8 = 7.5 days
Result: 8 workers need 7.5 days
\[5 \times 12 = 8 \times x\] \[60 = 8x \Rightarrow x = \frac{60}{8} = 7.5\]
Example 2: Speed and travel time
60 km/h → 2 hours 80 km/h → ? hours
Inverse proportionality: Higher speed → Shorter time
60 × 2 = 80 × x → 120 = 80x → x = 1.5 h
Result: At 80 km/h the trip takes 1.5 hours
More examples
Illumination & Distance:
100 Lux at 1m → 25 Lux at 2m
Pumps & Time:
2 pumps × 6h = 3 pumps × 4h
Force & Lever arm:
10 N × 30 cm = 15 N × 20 cm
Recognition signs
One quantity rises
Other quantity falls
Product constant
Keywords: "The more... the less"
Difference: Direct vs. Inverse rule of three
Direct rule of three
Both quantities change in the same direction
a/b = c/x → x = (b×c)/a
Inverse rule of three
Quantities change oppositely
a×b = c×x → x = (a×b)/c

The type of proportionality determines the calculation method

Applications of inverse rule of three

The inverse rule of three applies to all inverse proportionalities:

Staff planning & working time
  • Workers and required time
  • Machines and production duration
  • Teams and project time
  • Shift planning and capacity
Speed & time
  • Travel speed and duration
  • Production speed and time
  • Data transfer and transfer time
  • Flow rate and filling time
Physical laws
  • Light intensity and distance (1/r²)
  • Gravitational force and distance
  • Electric field strength
  • Pressure and volume (Boyle's law)
Mechanics & lever
  • Force and lever arm
  • Torque calculations
  • Transmission ratios
  • Mechanical advantage calculations

Inverse rule of three: Understanding inverse proportionality

The inverse rule of three is the mathematical tool for situations where two quantities vary oppositely: when one increases, the other decreases. This inverse proportionality appears daily—from more workers requiring less time to complex physical laws like the inverse-square law. The key feature is the constancy of the product: a₁ × b₁ = a₂ × b₂. This insight enables precise predictions and calculations in engineering, science and everyday life.

Characteristics
  • Quantities vary oppositely
  • Product constant: a × b = constant
  • Hyperbola function y = k/x
  • Cross-product calculation
Recognition
  • "The more... the less"
  • Workers ↔ Time, Speed ↔ Time
  • Force ↔ Lever arm, Illumination ↔ Distance
  • One quantity doubles → other halves
Distinction
  • Not applicable for direct proportionality
  • Be careful with complex dependencies
  • Consider physical laws
  • Perform plausibility checks
Summary

The inverse rule of three reveals the mathematical elegance of inverse proportionalities. While the direct rule of three describes quantities changing in the same direction, the inverse rule captures opposing relationships. From staff planning to physics, wherever "more of one means less of the other" applies, the inverse rule of three delivers precise answers. It converts intuition into calculable relations and makes complex dependencies predictable.