Direct proportionality

Online calculator and formulas for computing direct proportionality

Proportionality calculator

Direct proportionality

In direct proportionality doubling one quantity results in a doubling of the other. The proportionality constant k describes the constant ratio: y = k · x.

y = k · x (basic formula of direct proportionality)
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Independent variable
Proportionality constant
Dependent variable
Calculation result
Variable y =
Calculation: y = k · x (direct proportionality)

Proportionality Info

Properties

Direct proportionality: y = k · x

y = k·x k = y/x linear

Rule: Doubling x causes doubling of y
Graph: Line through the origin

Quick examples
Speed: v = const, s = v·t
Costs: Price = const, Total = Price·Quantity
Area: Rectangle with b = const, A = b·h


Formulas of direct proportionality

Basic formula
\[y = k \cdot x\]

y is directly proportional to x with constant k

Compute constant
\[k = \frac{y}{x}\]

Proportionality constant from given values

Compute x
\[x = \frac{y}{k}\]

Independent variable from y and k

Properties
\[\frac{y_1}{x_1} = \frac{y_2}{x_2} = k\]

Ratio remains constant (proportional equality)

Quotient equality
\[\frac{y_1}{y_2} = \frac{x_1}{x_2}\]

Ratio of y-values equals ratio of x-values

Graphical representation
\[f(x) = kx \text{ (line through origin)}\]

Linear function through the origin

Example calculations for direct proportionality

Example: Speed and distance
Given
x = 4 hours y = 200 km

A car drives for 4 hours and covers 200 km.

Step 1: Compute proportionality constant k
Formula: k = y/x
k = 200 km ÷ 4 h = 50 km/h
k = 50 km/h (speed)
Step 2: Compute distance for 6 hours
Formula: y = k · x
y = 50 km/h × 6 h = 300 km
Result: In 6 hours the car covers 300 km
\[k = \frac{200 \text{ km}}{4 \text{ h}} = 50 \frac{\text{km}}{\text{h}}\] \[y = 50 \cdot 6 = 300 \text{ km}\]
Proportionality check
t = 2h: s = 50 × 2 = 100 km
t = 4h: s = 50 × 4 = 200 km
t = 6h: s = 50 × 6 = 300 km
t = 8h: s = 50 × 8 = 400 km

Doubling time → doubling distance ✓

More examples
Costs: 3€/kg → 5kg cost 15€
Wages: 20€/h → 6h yield 120€
Consumption: 8L/100km → 250km: 20L
Properties of direct proportionality
Linear
Line through origin
Constant k
Slope remains constant
Doubling
x doubled → y doubled
y/x = constant
Quotient unchanged

Direct proportionality describes linear relationships

Applications of direct proportionality

Direct proportionality appears in many everyday contexts:

Commerce & economics
  • Shopping: Price × Quantity = Total cost
  • Wages: Hourly rate × Time = Pay
  • Production: Rate × Time = Total output
  • Currency conversion at fixed rates
Engineering & transport
  • Speed: v × t = distance
  • Fuel consumption per 100 km
  • Material consumption per unit
  • Ohm's law: U = R × I
Natural sciences
  • Density: ρ = m/V (at constant density)
  • Hooke's law: F = k × s
  • Radioactive decay (exponential)
  • Light speed: c × t = distance
Geometry & measurements
  • Circumference: U = π × d (circle)
  • Area: A = a × b (rectangle with const. a)
  • Scale calculations
  • Similarity ratios

Direct proportionality: Foundation of linear relationships

Direct proportionality describes one of the most fundamental relationships in mathematics and natural sciences. Two quantities are directly proportional if their quotient is constant: y/x = k. This simple relation y = k·x defines a linear function through the origin and forms the basis for understanding linear dependencies. From pricing to physics, direct proportionality shows how mathematical models can describe real phenomena.

Characteristics
  • Linear relation: y = kx
  • Graph through the origin (0,0)
  • Constant slope k
  • Quotient y/x remains constant
Identification
  • Doubling → doubling
  • Halving → halving
  • Proportional equality
  • Three-rule problems solvable
Distinctions
  • Not valid if y-intercept ≠ 0
  • Different from inverse proportionality
  • Not for exponential relations
  • Beware of rounding errors
Summary

Direct proportionality is more than a formula – it is a mental model for linear relationships in the real world. The simple relation y = k·x enables predictions, calculations and understanding of cause-effect ratios. Whether shopping, in physics or at work – direct proportionality helps reduce complex situations to simple mathematical relations and makes them calculable and predictable.

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