Introduction to Proportion
In mathematics, proportion describes a specific relationship between two quantities. Whether you are adjusting a recipe, calculating how long a car journey will take, or figuring out how much you’ll get paid for overtime, you are using proportional reasoning. In the Algebra and graphs section of your IGCSE, we move beyond simple ratios and look at how to describe these relationships using algebraic formulas and the special proportionality symbol \( \propto \).
Note for Core students: While the algebraic formulas are primarily for Extended candidates (Syllabus 2.8), understanding the logic of how quantities change together is a vital skill for everyone (Syllabus 1.11).
1. Direct Proportion
When two quantities are in direct proportion, they increase or decrease at the same rate. If you double one, the other doubles. If you halve one, the other halves.
The Proportionality Symbol
We use the symbol \( \propto \) to show that two variables are proportional. If \(y\) is directly proportional to \(x\), we write:
\( y \propto x \)
The Algebraic Formula
To solve math problems, we need to turn that symbol into an equation. We do this by introducing a constant called \(k\) (the constant of proportionality):
\( y = kx \)
This equation tells us that \(y\) is always equal to \(x\) multiplied by some fixed number \(k\).
Direct Proportion with Powers (Extended Only)
Sometimes, a quantity is proportional to the square or cube of another.
- If \(y\) is proportional to the square of \(x\): \( y \propto x^2 \) which becomes \( y = kx^2 \)
- If \(y\) is proportional to the square root of \(x\): \( y \propto \sqrt{x} \) which becomes \( y = k\sqrt{x} \)
Quick Tip: If a graph of \(y\) against \(x\) is a straight line passing through the origin \( (0,0) \), the two quantities are in direct proportion!
2. Inverse Proportion
In inverse proportion, as one quantity goes up, the other goes down. A classic example is speed and time: the faster you drive, the less time the journey takes.
The Formula
If \(y\) is inversely proportional to \(x\), we write:
\( y \propto \frac{1}{x} \)
To turn this into an equation, we use our constant \(k\) again:
\( y = \frac{k}{x} \)
Did you know? In inverse proportion, if you multiply the two variables together (\(x \times y\)), you will always get the same constant value \(k\).
3. The 4-Step Method for Solving Proportion Problems
Don't worry if these formulas look intimidating at first. Almost every proportion question in your exam can be solved using these same four steps:
Step 1: Write the relationship. Use the \( \propto \) symbol based on the wording in the question.
Step 2: Write the equation. Replace \( \propto \) with \( = k \).
Step 3: Find \(k\). Plug in the pair of numbers the question gives you and solve for \(k\).
Step 4: Answer the final question. Rewrite your equation with the value of \(k\) you found, then plug in the new number to find the missing value.
Worked Example (Direct Proportion)
Question: \(y\) is directly proportional to \(x\). When \(x = 5\), \(y = 20\). Find \(y\) when \(x = 8\).
Step 1: \( y \propto x \)
Step 2: \( y = kx \)
Step 3: \( 20 = k \times 5 \implies k = \frac{20}{5} \implies k = 4 \)
Step 4: Our formula is \( y = 4x \). Now, if \(x = 8\), then \( y = 4 \times 8 = 32 \).
Worked Example (Inverse Proportion)
Question: \(P\) is inversely proportional to \(Q\). When \(Q = 10\), \(P = 2\). Find \(P\) when \(Q = 4\).
Step 1: \( P \propto \frac{1}{Q} \)
Step 2: \( P = \frac{k}{Q} \)
Step 3: \( 2 = \frac{k}{10} \implies k = 2 \times 10 \implies k = 20 \)
Step 4: Our formula is \( P = \frac{20}{Q} \). Now, if \(Q = 4\), then \( P = \frac{20}{4} = 5 \).
4. Common Mistakes to Avoid
- Forgetting the Power: If the question says "proportional to the square of \(x\)", make sure you use \(x^2\) in your formula, not just \(x\).
- Mixing up Direct and Inverse: Always read carefully. "Inversely" means the variable goes in the denominator (bottom) of the fraction.
- Stopping at \(k\): Many students find the value of \(k\) and think they are finished. Remember to use \(k\) to find the final value requested!
- Rounding too early: Keep the full value of \(k\) (or use a fraction) until the very end to ensure your final answer is accurate to 3 significant figures.
Key Takeaways
Quick Review Box:
- Direct Proportion: \( y = kx \) (One goes up, other goes up)
- Inverse Proportion: \( y = \frac{k}{x} \) (One goes up, other goes down)
- The Constant (\(k\)): Always find this first using the numbers given.
- Symbol: \( \propto \) means "is proportional to".
For more on how these relationships look on a graph, check out the chapter on Graphs of functions (Syllabus 2.10).