Mathematics (Specification A) | Proportion

Proportion: Understanding Relationships Between Variables

Welcome to the chapter on Proportion! This is a really important area of Mathematics because it helps us define the relationship between two quantities and create a reliable formula (or identity) to predict outcomes.

Don't worry if this seems tricky at first. Proportion is simply about understanding how one thing changes when another thing changes. We use this knowledge every day—whether you're baking a cake (more ingredients, bigger cake) or calculating travel time (faster speed, less time).

Learning Goal: By the end of these notes, you will be able to set up and solve equations involving both direct and inverse proportion, including those involving powers and roots.

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Section 1: The Basics of Proportion

1.1 What is Proportion?

In mathematics, when we say two quantities are in proportion, it means they are linked by a constant multiplier or divisor. We use a special symbol to show that two things are related:

• The symbol \(\propto\) means "is proportional to."

When you see \(\propto\), your immediate goal is to replace it with an equals sign and a Constant of Proportionality, which we call \(k\).

\( \propto \) becomes \( = k \).

1.2 The Constant of Proportionality (\(k\))

The value \(k\) is essential! It is the single number that defines the specific relationship between the two variables. Once you find \(k\), your formula is complete, and you can solve any problem related to those variables.

Think of \(k\) as the "conversion rate" or "recipe ratio" for that specific problem.

Key Takeaway: Proportion defines a relationship. Solving a proportion problem always involves finding the constant \(k\).

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Section 2: Direct Proportion

2.1 Defining Direct Proportion

Two quantities, say \(y\) and \(x\), are in Direct Proportion if, as one increases, the other increases by the same ratio (and vice versa).

• If \(x\) doubles, \(y\) doubles.
• If \(x\) is halved, \(y\) is halved.

Analogy: Imagine photocopying a picture. If you increase the enlargement setting (\(x\)), the resulting picture size (\(y\)) increases directly.

2.2 The Direct Proportion Formula

We write the relationship as:

$$y \propto x$$

To turn this into a usable formula or identity, we introduce the constant \(k\):

$$y = kx$$

If you rearrange this formula, you can see how to calculate \(k\):

$$k = \frac{y}{x}$$

This tells us that in direct proportion, the ratio between the two variables is always constant.

2.3 Step-by-Step Guide to Solving Direct Proportion Problems

Let’s say we are given that \(A\) is directly proportional to \(B\), and when \(A=10\), \(B=5\). We want to find \(A\) when \(B=8\).

Step 1: Write the Proportionality Statement

$$A \propto B$$

Step 2: Change to an Equation using \(k\)

$$A = kB$$

Step 3: Find the value of \(k\)
Use the initial pair of values (\(A=10\), \(B=5\)) to solve for \(k\).

\(10 = k(5)\)
\(k = \frac{10}{5}\)
\(k = 2\)

Step 4: Write the Final Formula and Solve
Now we have the specific formula for this problem: \(A = 2B\). Use this formula to find the missing value.

Find \(A\) when \(B=8\):
\(A = 2(8)\)
\(A = 16\)

Did you know? Establishing this identity (\(A=2B\)) means we have a complete relationship; we can now calculate \(A\) or \(B\) for any value!

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Section 3: Inverse Proportion

3.1 Defining Inverse Proportion

Two quantities, \(y\) and \(x\), are in Inverse Proportion (sometimes called indirect proportion) if, as one increases, the other decreases by the same ratio.

• If \(x\) doubles, \(y\) is halved.
• If \(x\) is quartered, \(y\) quadruples.

Analogy: The speed of a car and the time taken for a fixed journey. If you increase your speed (\(x\)), the time taken (\(y\)) decreases.

3.2 The Inverse Proportion Formula

Because the relationship is opposite (inverse), we must use the reciprocal of \(x\).

We write the relationship as:

$$y \propto \frac{1}{x}$$

To turn this into a usable formula, we introduce the constant \(k\):

$$y = \frac{k}{x}$$

If you rearrange this formula, you can see how to calculate \(k\):

$$k = xy$$

This tells us that in inverse proportion, the product of the two variables is always constant.

3.3 Common Mistake Alert!

DO NOT use \(y = kx\) for inverse proportion. Remember, Inverse means Invert the variable (\(1/x\)).

3.4 Step-by-Step Guide to Solving Inverse Proportion Problems

Let’s say the Time taken, \(T\), is inversely proportional to the Speed, \(S\). If it takes \(T=4\) hours when \(S=60\) mph, find \(T\) when \(S=80\) mph.

Step 1: Write the Proportionality Statement

$$T \propto \frac{1}{S}$$

Step 2: Change to an Equation using \(k\)

$$T = \frac{k}{S}$$

Step 3: Find the value of \(k\)
Use the initial values (\(T=4\), \(S=60\)).

\(4 = \frac{k}{60}\)
\(k = 4 \times 60\)
\(k = 240\)

Step 4: Write the Final Formula and Solve
The specific formula is: \(T = \frac{240}{S}\). Use this to find \(T\) when \(S=80\).

\(T = \frac{240}{80}\)
\(T = 3\) hours

(Notice: The speed increased, and the time decreased—this confirms it is an inverse relationship.)

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Section 4: Proportion Involving Powers and Roots

Proportional relationships don't always have to be between \(y\) and \(x\). Sometimes \(y\) is related to the square of \(x\), the cube of \(x\), or the square root of \(x\). This is often required for higher-grade questions.

The good news? The four-step process for solving the problem remains exactly the same!

4.1 Direct Proportion Examples

If \(y\) is directly proportional to the square of \(x\):

$$y \propto x^2 \quad \implies \quad y = kx^2$$

If \(y\) is directly proportional to the square root of \(x\):

$$y \propto \sqrt{x} \quad \implies \quad y = k\sqrt{x}$$

Example Application: The energy (\(E\)) in a moving object is proportional to the square of its velocity (\(v\)). So, \(E = kv^2\). This means doubling the speed quadruples the energy!

4.2 Inverse Proportion Examples

If \(y\) is inversely proportional to the cube of \(x\):

$$y \propto \frac{1}{x^3} \quad \implies \quad y = \frac{k}{x^3}$$

If \(y\) is inversely proportional to the square root of \(x\):

$$y \propto \frac{1}{\sqrt{x}} \quad \implies \quad y = \frac{k}{\sqrt{x}}$$

Example: Solving a Power Proportion Problem

\(P\) is inversely proportional to the square of \(Q\). When \(Q=2\), \(P=50\). Find the equation linking \(P\) and \(Q\).

Step 1: Statement
\(P\) is inversely proportional to the square of \(Q\).

$$P \propto \frac{1}{Q^2}$$

Step 2: Equation
$$P = \frac{k}{Q^2}$$

Step 3: Find \(k\)
Substitute \(P=50\) and \(Q=2\):
$$50 = \frac{k}{2^2}$$ $$50 = \frac{k}{4}$$ $$k = 50 \times 4$$ \(k = 200\)

Step 4: Final Formula
The equation linking \(P\) and \(Q\) is \(P = \frac{200}{Q^2}\).

Key Takeaway: Always read the question carefully to see if it mentions "square," "cube," or "root." Incorporate this into your initial proportionality statement (Step 1).

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Section 5: Final Review and Success Tips

5.1 The Golden Rule of Proportion

Every proportionality question requires you to define and calculate \(k\) first. This is the most crucial step!

5.2 Summary of Formulas

• Direct: \(y = kx\) (k is \(y\) divided by \(x\))
• Inverse: \(y = \frac{k}{x}\) (k is \(y\) multiplied by \(x\))
• Direct Square: \(y = kx^2\)
• Inverse Square Root: \(y = \frac{k}{\sqrt{x}}\)

5.3 Encouragement

You have now successfully navigated a key part of the "Equations, formulae and identities" section! By using the constant of proportionality, you are creating specific mathematical identities that govern real-world relationships. Practice these four steps diligently, and you will master proportion!