Introduction to Rates of Change and Proportionality
Welcome! In this chapter, we are going to explore how quantities change in relation to one another. Whether it's how the cost of petrol increases as you fill your tank, or how the speed of a car affects the time it takes to arrive, we are looking at relationships. Understanding these patterns is a superpower in maths because it allows you to predict what will happen next!
Direct Proportion
When two quantities are in direct proportion, they increase or decrease at the same rate. If you double one, the other doubles too. If you halve one, the other halves.
The Algebraic Rule
We use the symbol \(\propto\) to mean "is proportional to". If \(y\) is directly proportional to \(x\), we write:
\(y \propto x\)
To turn this into an equation, we introduce a constant called \(k\) (the constant of proportionality):
\(y = kx\)
How to solve direct proportion problems:
- Write the equation \(y = kx\).
- Substitute the values of \(x\) and \(y\) given in the question.
- Solve for \(k\).
- Rewrite the equation with your new value of \(k\).
- Use this equation to find any other missing values.
Example: If \(y\) is proportional to \(x\), and \(y = 10\) when \(x = 2\):
\(10 = k \times 2\)
\(k = 5\)
The equation is \(y = 5x\).
If \(x = 4\), then \(y = 5 \times 4 = 20\).
Direct Proportion Graphs
The graph of a direct proportion relationship is always a straight line that passes through the origin \((0,0)\). The gradient of this line is the value of \(k\).
Key Takeaway: If a graph is a straight line but doesn't go through \((0,0)\), it is linear but not directly proportional.
Inverse Proportion
In inverse proportion, as one quantity goes up, the other goes down. For example, the more people you have painting a fence, the less time it takes to finish.
The Algebraic Rule
We say \(y\) is inversely proportional to \(x\), written as:
\(y \propto \frac{1}{x}\)
The equation for inverse proportion is:
\(y = \frac{k}{x}\) (or \(xy = k\))
Higher Tier Only: You may be asked to construct these equations for powers, such as \(y \propto \frac{1}{x^2}\). The steps are exactly the same: find \(k\) first!
Inverse Proportion Graphs
The graph of inverse proportion is a curve (a reciprocal graph) that never touches the \(x\) or \(y\) axes.
Quick Tip: If the question says "one quantity increases as the other decreases," check if they multiply to give the same number (\(k\)). If they do, it’s inverse proportion!
Gradients as Rates of Change
A rate of change tells us how much one variable changes for every unit of another variable. On a graph, the gradient represent this rate.
Straight-Line Graphs
For a straight-line graph, the rate of change is constant. You can find it using:
\(\text{Gradient} = \frac{\text{change in } y}{\text{change in } x}\)
In a real-world context, like a distance-time graph, the gradient represents speed. In a cost-weight graph, the gradient represents price per kg.
Average vs Instantaneous Rates (Higher Tier Only)
When looking at curves (non-linear graphs), the rate of change is always changing. There are two ways to measure it:
- Average Rate of Change: Draw a straight line (called a chord) between two points on the curve. Calculate the gradient of that chord.
- Instantaneous Rate of Change: To find the rate at one specific moment, draw a tangent (a straight line that just touches the curve at that point). Calculate the gradient of the tangent.
Did you know? On a velocity-time graph, the gradient represents acceleration. If the graph is a curve, the tangent gives you the acceleration at that exact second!
Growth and Decay
Growth and decay problems often involve percentages changing over time. A common example is Compound Interest.
The Compound Interest Formula
From your formula sheet:
\(\text{Total accrued} = P(1 + \frac{r}{100})^n\)
Where:
\(P\) = the initial amount (principal)
\(r\) = the percentage rate
\(n\) = the number of time periods (e.g., years)
Example: \(\$200\) invested at \(3\%\) interest for \(4\) years:
\(\text{Total} = 200 \times (1.03)^4\)
Higher Tier Only: You may encounter iterative processes. This is where you use the output of one calculation as the input for the next to show how a value grows or decays step-by-step.
Common Mistakes to Avoid
- Mixing up Direct and Inverse: Always check if the value should get bigger or smaller. If more workers take more time, something is wrong!
- Forgetting the Constant \(k\): You cannot solve these problems without finding \(k\) first.
- Units: Ensure your units are consistent (e.g., don't mix minutes and hours in the same calculation). See the chapter on "Units, scale and compound measures" for more on this.
- Tangent Accuracy: When drawing a tangent, use a sharp pencil and a ruler. Try to make the "angles" between the curve and the ruler look equal on both sides of the point.
Quick Review Box
Direct Proportion: \(y = kx\) (Straight line through origin)
Inverse Proportion: \(y = \frac{k}{x}\) (Curve)
Gradient: Measures the rate of change.
Tangent: Used to find the rate of change at a specific point on a curve (Higher Tier).
Multiplier: For \(5\%\) growth, use \(1.05\). For \(5\%\) decay, use \(0.95\).