Welcome to the World of Proportion!

Hello! Today we are diving into the concept of Proportion. This is a vital part of your AS Level Mathematics B (MEI) course because it helps us describe exactly how two things change in relation to each other. Whether it’s how the area of a circle grows as its radius increases, or how the intensity of light fades as you move away from a lamp, proportion is the mathematical tool we use to model the real world.

Don't worry if this seems a bit abstract at first. We will break it down into simple steps, use plenty of examples, and look at the graphs that bring these relationships to life.

Quick Review: Before we start, remember that in algebra, we often use a constant (usually the letter \(k\)) to represent a fixed value that doesn't change, even when our variables \(x\) and \(y\) do.

1. Direct Proportion

Direct proportion occurs when two quantities increase or decrease at the same rate. If you double one, the other doubles too!

The Basics

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 a math equation we can actually solve, we replace the \(\propto\) with \(= k\):
\(y = kx\)

Here, \(k\) is called the constant of proportionality.

Proportion to Powers and Roots

The syllabus (Ref a14) specifically mentions that you need to understand proportion to a power or root. This just means \(x\) might be squared, cubed, or under a square root.

  • Directly proportional to the square: \(y \propto x^2 \implies y = kx^2\)
  • Directly proportional to the square root: \(y \propto \sqrt{x} \implies y = k\sqrt{x}\)

Example: The area of a square (\(A\)) is directly proportional to the square of its side length (\(s\)). If the side length doubles, the area actually quadruples because \(2^2 = 4\)!

Key Takeaway: In direct proportion, as \(x\) gets bigger, \(y\) gets bigger. The formula always looks like \(y = k \times (\text{something})\).

2. Inverse Proportion

Inverse proportion is the opposite. As one value goes up, the other goes down. Think of it like a seesaw!

The Basics

If \(y\) is inversely proportional to \(x\), we say it is proportional to \(1/x\):
\(y \propto \frac{1}{x}\)

As an equation, this becomes:
\(y = \frac{k}{x}\)

Inverse Proportion to Powers

Just like direct proportion, this can involve powers (Ref C6):
\(y = \frac{k}{x^2}\) (Inverse square law)

Analogy: Imagine you have a fixed amount of pizza to share. The more friends that arrive (increase in \(x\)), the less pizza each person gets (decrease in \(y\)). That is inverse proportion!

Key Takeaway: In inverse proportion, as \(x\) gets bigger, \(y\) gets smaller. The formula always looks like \(y = \frac{k}{\text{something}}\).

3. How to Solve Proportion Problems (Step-by-Step)

Most exam questions will give you a pair of values for \(x\) and \(y\) and ask you to find the formula. Here is the "Golden Path" to solving them:

Step 1: Write the relationship. Use the \(\propto\) symbol based on the wording (e.g., "y is inverse to \(x^2\)").
Step 2: Write the equation. Change \(\propto\) to \(= k\).
Step 3: Find \(k\). Plug in the numbers the question gave you and solve for \(k\).
Step 4: Write the final formula. Put your value of \(k\) back into the equation from Step 2.
Step 5: Solve. Use your new formula to find any other values the question asks for.

Common Mistake: Many students forget to square or square root \(x\) in Step 3 even if the question mentions it. Always double-check the powers!

4. Graphs of Proportional Relationships

Being able to recognize these relationships visually is a key skill (Ref C6).

Direct Proportion Graphs

  • \(y = kx\): A straight line passing through the origin \((0,0)\). The gradient of the line is \(k\).
  • \(y = kx^2\): A parabola starting at the origin. It gets steeper much faster than a straight line.

Inverse Proportion Graphs

Graphs like \(y = \frac{k}{x}\) and \(y = \frac{k}{x^2}\) look very different. They are curves that never actually touch the axes.

Asymptotes: These are lines that the graph gets closer and closer to but never touches.
For \(y = \frac{a}{x}\):
- The vertical asymptote is the y-axis (\(x = 0\)).
- The horizontal asymptote is the x-axis (\(y = 0\)).

Did you know? You can't have \(x = 0\) in an inverse proportion because you can't divide by zero! That is why the graph never touches the vertical y-axis.

Key Takeaway: Direct proportion graphs usually pass through \((0,0)\). Inverse proportion graphs are curves that "avoid" the axes using asymptotes.

Summary and Quick Review

Memory Aid:
Direct = Dultiply (Multiply \(k\) by \(x\))
Inverse = In the denominator (Divide \(k\) by \(x\))

Quick Review Box:

1. Direct: \(y = kx^n\). Line or curve going up.
2. Inverse: \(y = \frac{k}{x^n}\). Curve going down with asymptotes.
3. The Constant: Always find \(k\) first using the given values.
4. The Origin: Direct proportion always goes through \((0,0)\). Inverse proportion never does.

Don't worry if sketching the inverse graphs feels tricky at first. Just remember they are "smooth curves" that get very close to the axes but stay slightly away!