Direct Proportionality

What does direct proportionality mean?

Two quantities, $ x $ and $ y $, are directly proportional when their ratio stays constant. Mathematically, this is expressed as: $$ \frac{y}{x} = k \quad \text{or} \quad y = kx $$ where $ k $ is a constant known as the proportionality coefficient.

Direct proportionality is one of the most basic and essential mathematical relationships between two quantities.

This type of relationship is widely applied in physics and other fields to describe situations where two variables increase or decrease together while maintaining a fixed ratio.

The key features of direct proportionality are:

Constant ratio: For any values of $ x $ and $ y $, the ratio $ \frac{y}{x} $ remains unchanged.
Proportional growth: If $ x $ doubles, triples, or quadruples, $ y $ will also double, triple, or quadruple.
Linear graph: The graph of this relationship is a straight line passing through the origin ($ 0,0 $), with its slope determined by the value of $ k $.

A Practical Example

Let’s examine the relationship between the volume of gasoline purchased and its cost.

If the price of gasoline is $ 1.50 \, \text{€/L} $, the data might look like this:

Volume ($ x $, in L)	Price ($ y $, in €)
1.00	1.50
2.00	3.00
3.00	4.50
4.00	6.00

By calculating the ratio $ \frac{y}{x} $ for each pair of values, we see:

$$ \frac{1.50}{1.00} = 1.50, \quad \frac{3.00}{2.00} = 1.50, \quad \frac{4.50}{3.00} = 1.50, \quad \frac{6.00}{4.00} = 1.50 $$

Clearly, the ratio is constant and represents the unit price of gasoline ($ k = 1.50 $).

The relationship between $ y $ (price) and $ x $ (volume) can be described by the equation:

$$ y = 1.50x. $$

Let’s plot this function on a Cartesian graph.

graph of the function

The graph depicts a straight line passing through the origin ($ 0,0 $), since no gasoline purchase ($ x = 0 $) corresponds to no cost ($ y = 0 $). The slope of the line is determined by $ k = 1.50 $, which represents the price per liter.

Now, imagine the price of gasoline increases to $ 1.90 \, \text{€/L} $. The new equation becomes:

$$ y = 1.90x $$

On a graph, this line would have a steeper slope, reflecting the higher unit cost.

graph of the function

Any change in $ k $ alters the relationship between $ x $ and $ y $, which is visually represented by the slope of the line.

In general, the slope of the line depends on the value of $ k $:

The larger $ k $, the steeper the line.
The smaller $ k $, the flatter the line.

Apart from everyday examples like gasoline prices, this relationship also explains fundamental physical phenomena, such as constant speed in uniform linear motion ($ s = vt $) and Hooke’s law for springs ($ F = kx $).