Ratio and proportion

Chapter N2: Ratio and Proportion

Welcome to one of the most practical chapters in O-Level Mathematics! Whether you are mixing ingredients for a cake, exchanging currency for a holiday, or reading a map to find your way, you are using Ratio and Proportion. In this guide, we will break down these concepts into simple, bite-sized steps to help you master them with confidence.

1. Understanding Ratios

A ratio is a way of comparing two or more quantities of the same kind. It tells us how much of one thing we have compared to another.

Writing Ratios in Simplest Form

Just like fractions, we always prefer to write ratios in their simplest form. This means the numbers in the ratio should be whole numbers with no common factors other than 1.

Steps to simplify:
1. Ensure all quantities are in the same units (e.g., all in cm or all in grams).
2. Divide all parts of the ratio by their Highest Common Factor (HCF).
3. If the ratio contains fractions or decimals, multiply to clear them first.

Example: Simplify the ratio 800 g : 2 kg.
Step 1: Convert to same units → 800 g : 2000 g.
Step 2: Divide by HCF (400) → \( \frac{800}{400} : \frac{2000}{400} = 2 : 5 \).
Final Answer: 2 : 5

Ratios with Rational Numbers (Fractions)

Don't worry if you see fractions in a ratio! To simplify them, find the Lowest Common Multiple (LCM) of the denominators and multiply every term by it.

Example: Simplify \( \frac{1}{2} : \frac{2}{3} \).
The LCM of 2 and 3 is 6.
Multiply both by 6: \( (\frac{1}{2} \times 6) : (\frac{2}{3} \times 6) = 3 : 4 \).
Final Answer: 3 : 4

Quick Review:
- Ratios have no units (because the units cancel out).
- The order matters! \( 2 : 3 \) is not the same as \( 3 : 2 \).

2. Map Scales (Distance and Area)

Maps are "scale models" of the real world. A map scale is usually written as a ratio 1 : n, where 1 unit on the map represents \( n \) units in real life.

Distance on Maps

The standard format is 1 : n.
Note: Both sides must be in the same units (usually cm) for the ratio to work!

Memory Aid: Use the formula \( \text{Map Distance} \times n = \text{Actual Distance} \).

Common Mistake: Forgetting to convert units. If a map says 1 : 50,000, it means 1 cm on the map is 50,000 cm on the ground. You usually need to convert that 50,000 cm into meters or kilometers (50,000 cm = 500 m = 0.5 km).

Area on Maps

This is where many students get tripped up, but here is the secret: If you square the distance scale, you get the area scale.

If the distance scale is \( 1 \text{ cm} : n \text{ km} \),
Then the area scale is \( (1 \text{ cm})^2 : (n \text{ km})^2 \).
Which simplifies to: \( 1 \text{ cm}^2 : n^2 \text{ km}^2 \).

Example: A map has a scale of 1 : 20,000. Find the actual area of a park that is \( 3 \text{ cm}^2 \) on the map.
1. Convert distance scale to km: \( 1 \text{ cm} : 20,000 \text{ cm} \rightarrow 1 \text{ cm} : 0.2 \text{ km} \).
2. Square it for area: \( 1^2 \text{ cm}^2 : (0.2)^2 \text{ km}^2 \rightarrow 1 \text{ cm}^2 : 0.04 \text{ km}^2 \).
3. Multiply by 3: \( 3 \text{ cm}^2 \rightarrow 3 \times 0.04 = 0.12 \text{ km}^2 \).
Final Answer: \( 0.12 \text{ km}^2 \)

Key Takeaway: Always square the linear scale before working with areas!

3. Direct and Inverse Proportion

Proportion describes how two quantities change in relation to each other.

Direct Proportion

When two quantities are in direct proportion, as one increases, the other increases at the same rate. Their ratio remains constant.

The Formula: \( y = kx \)
(where \( k \) is a constant called the constant of proportionality)

Real-world analogy: The more apples you buy (x), the more money you pay (y). Double the apples, double the cost!

Inverse Proportion

When two quantities are in inverse proportion, as one increases, the other decreases. Their product remains constant.

The Formula: \( y = \frac{k}{x} \) or \( xy = k \)

Real-world analogy: The more workers you have (x), the less time it takes to build a wall (y). If you double the workers, you halve the time!

Steps to Solve Proportion Problems

1. Identify the type: Is it direct (\( y = kx \)) or inverse (\( y = \frac{k}{x} \))?
2. Find k: Plug in the pair of values given in the question to solve for \( k \).
3. Form the equation: Rewrite the formula with the value of \( k \) you found.
4. Solve: Use the equation to find the unknown value requested in the question.

Example (Direct): If \( y \) is directly proportional to \( x \), and \( y = 10 \) when \( x = 2 \), find \( y \) when \( x = 7 \).
Step 1: \( y = kx \)
Step 2: \( 10 = k(2) \rightarrow k = 5 \)
Step 3: Equation is \( y = 5x \)
Step 4: When \( x = 7, y = 5(7) = 35 \).
Final Answer: 35

Did you know?
In a graph, direct proportion is always a straight line passing through the origin (0,0). Inverse proportion creates a curve (called a hyperbola) that never touches the x or y axes!

Key Takeaway: Always find the constant \( k \) first. It is the "key" that unlocks the rest of the problem.

Final Quick Tips for Success

- Units: Always check if units are consistent before you start calculating.
- Labels: When working with ratios, label your columns (e.g., "Map : Actual") to avoid mixing them up.
- Read Carefully: Does the question say "directly proportional to \( x \)" or "directly proportional to the square of \( x \)"? If it says square, your formula becomes \( y = kx^2 \).
- Don't Panic: If a problem looks complex, break it down. Find \( k \) first, and the rest will follow!