Understanding Patterns and Relationships in Mathematics

If you were to graph this, you would see a horizontal line because the $y$-value does not move up or down as you move along the $x$-axis. The formula for such a relationship contains no variable cost, looking like this: $\text{Cost of the bus} = R450,00$

Linear Patterns: Constant Difference

A constant difference relationship occurs when the change between two consecutive values of the dependent variable is always the same. This is a linear relationship, meaning that if you plot these points on a graph, they will always form a straight line.

Think about buying produce at a market. If tomatoes cost $R4,95$ per kilogram, every additional kilogram you buy adds exactly the same amount to the total price. We can express this relationship with a formula: $\text{Cost of tomatoes} = \text{Weight of tomatoes in kg} \cdot R4,95$

Proportionality: Inverse and Ratio

Patterns can also be defined by how they multiply or divide.

Indirect/Inverse Proportion

In an indirect or inverse proportion, the two variables move in opposite directions. Specifically, if one variable increases by a certain factor, the other decreases by that same factor. A key mathematical property of this relationship is that the product of the two variables remains constant.

For example, if you have 48 balloons to share among children at a party, the more children that attend, the fewer balloons each child receives. The relationship can be written as: $\text{Number of balloons} = \frac{48}{\text{number of children}}$

Constant Ratio Relationship

Unlike the constant difference (which adds/subtracts), a constant ratio relationship involves multiplying by a factor or a percentage. In this pattern, the value is increased by a specific factor, and that new value is then increased again by that same factor, repeatedly.

Because the amount being added grows larger each time, the graph is not a straight line; instead, it curves upward, increasing faster and faster as time goes on. For instance, if a household's food costs increase by $7,4%$ every year, the difference in cost between Year 1 and Year 2 will be much smaller than the difference between Year 10 and Year 11.

Complex Patterns: Combination of Relationships

In the real world, relationships are rarely "pure." Often, multiple types of patterns are combined into a single scenario.

A common example is a cellphone contract. You might pay a fixed monthly fee (a constant relationship) that includes a set amount of free minutes. Once those minutes are exhausted, you are charged a specific rate per minute (a constant difference relationship).

Consider a contract with a fixed cost of $R169,00$ that includes 100 free minutes. If you use more than 100 minutes, you pay an additional $R0,95$ per minute. The total cost can be calculated using a general equation: $\text{Total monthly cost} = \text{constant value} + (\text{independent variable change})$ $\text{Total monthly cost} = 169 + (\text{no. of minutes after } 100 \cdot 0.95)$

Summary Table for Quick Revision