Study Notes: Ratio and Proportion (Numbers and the Number System)
Hello future mathematician! This chapter, Ratio and Proportion, is incredibly important because it's used everywhere—from cooking and mixing paint to calculating exchange rates and reading maps. It is the language of comparison!
Don't worry if this seems tricky at first. We will break down every concept into clear, simple steps. Ratio and proportion are just special ways of looking at multiplication and division. Let's get started!
1. Understanding Ratios: The Basics of Comparison
What is a Ratio?
A ratio compares the sizes of two or more quantities. It tells you how much of one thing there is compared to another.
Key Concept: A ratio is written using a colon (:).
- If you mix cordial using 1 part syrup and 4 parts water, the ratio of syrup to water is 1 : 4.
- If there are 5 girls and 7 boys in a room, the ratio of girls to boys is 5 : 7.
Crucial Point: Order Matters!
The ratio 5 : 7 (Girls : Boys) is completely different from 7 : 5 (Boys : Girls). Always read the question carefully to ensure your numbers are in the correct order.
Ratios vs. Fractions
A ratio compares parts to parts (e.g., Syrup to Water). A fraction compares a part to the whole (e.g., Syrup to the Total Drink).
Example: If the ratio is 1 : 4 (Syrup : Water), the total number of parts is \(1 + 4 = 5\).
The fraction of syrup in the drink is \(1/5\).
The fraction of water in the drink is \(4/5\).
Quick Takeaway: Ratios use a colon (:) to compare parts. The order is fixed!
2. Simplifying Ratios
Simplifying a ratio is just like simplifying a fraction: you divide all parts of the ratio by the same number until they cannot be divided any further. We look for the Highest Common Factor (HCF).
Step-by-Step: Simplifying Ratios
Example 1: Simplify the ratio 12 : 18
- Find the HCF of 12 and 18. The largest number that divides both is 6.
- Divide both sides by the HCF (6):
\(12 \div 6 = 2\)
\(18 \div 6 = 3\) - The simplified ratio is 2 : 3.
Dealing with Different Units (A Common Mistake!)
You must convert all parts of the ratio to the same unit before simplifying. You cannot compare apples and oranges!
Example 2: Simplify the ratio 50 cm : 2 m
- Choose the smallest unit (cm). Convert 2 m into cm.
(Remember: 1 m = 100 cm)
\(2 \text{ m} = 2 \times 100 = 200 \text{ cm}\) - The ratio is now 50 : 200.
- Simplify by dividing by the HCF (50):
\(50 \div 50 = 1\)
\(200 \div 50 = 4\) - The simplified ratio is 1 : 4.
Handling Ratios with Three or More Parts
The principle remains the same: find a number that divides all parts equally.
Example: Simplify 15 : 25 : 30. The HCF is 5.
\(15 \div 5 : 25 \div 5 : 30 \div 5 = \mathbf{3 : 5 : 6}\)
Quick Review Box: Simplifying
- Rule 1: Convert to the same units first.
- Rule 2: Divide all parts by the HCF.
3. Dividing a Quantity in a Given Ratio
This is where ratios become very practical. You might need to divide money, ingredients, or prizes fairly based on contributions or effort. We use the Total Parts Method.
Step-by-Step: Sharing a Quantity
Example: Divide £40 in the ratio 3 : 5.
This means the first person gets 3 parts and the second person gets 5 parts.
- Find the Total Number of Parts:
\(3 + 5 = 8\) total parts. - Find the Value of One Part (The Unitary Value):
Divide the total quantity by the total parts.
\(£40 \div 8 = £5\)
This means 1 part is worth £5. - Calculate the Individual Shares:
Person 1 (3 parts): \(3 \times £5 = £15\)
Person 2 (5 parts): \(5 \times £5 = £25\) - Check Your Answer:
\(£15 + £25 = £40\). (It adds up correctly!)
Tip for Struggling Students: Always draw a box or circle around the value of one part (like the £5 above). This is the key number you multiply everything by!
Applying Ratio Differences
Sometimes the question asks for the difference between the shares.
Example: Alan and Ben share some money in the ratio 7 : 4. If Alan receives £18 more than Ben, how much did they share in total?
- Find the Difference in Parts:
Alan has 7 parts, Ben has 4 parts. Difference in parts: \(7 - 4 = 3\) parts. - Find the Value of One Part:
We know 3 parts correspond to the £18 difference.
Value of 1 part: \(£18 \div 3 = £6\). - Calculate the Total Amount:
Total parts: \(7 + 4 = 11\) parts.
Total money: \(11 \times £6 = £66\).
Key Takeaway: When dividing a quantity, first find the total number of parts, and then find the value of one part.
4. Direct Proportion
When two quantities are in direct proportion, they increase or decrease together at the same rate.
Analogy: If you double the amount of paint you buy, you double the cost. If you halve the time you spend running, you halve the distance covered (assuming a constant speed).
We often use the Unitary Method (finding the value of 1) or the Ratio Method to solve these problems.
Method 1: The Unitary Method (Find the Cost of 1)
Example: 5 pens cost £3. How much do 8 pens cost?
- Find the cost of 1 (The Unitary Value):
\(£3 \div 5 \text{ pens} = £0.60\) per pen. - Use the unitary value to find the required amount:
Cost of 8 pens: \(8 \times £0.60 = £4.80\).
Method 2: Using the Ratio Equality (Setting up Fractions)
If \(A\) is directly proportional to \(B\), then the ratio \(A/B\) is constant.
\( (\text{Old Pens} / \text{Old Cost}) = (\text{New Pens} / \text{New Cost}) \)
Example (Same problem: 5 pens cost £3. How much do 8 pens cost (x)?):
Set up the equality:
\( \frac{5}{3} = \frac{8}{x} \)
Solve for \(x\):
\( 5x = 8 \times 3 \)
\( 5x = 24 \)
\( x = \frac{24}{5} = 4.80 \)
The cost is £4.80.
Did You Know? Direct proportion is fundamentally linked to linear graphs passing through the origin (0, 0). The constant ratio you find is called the constant of proportionality.
Key Takeaway (Direct Proportion): If one goes UP, the other goes UP. Use the Unitary Method (find 1) for the simplest calculation.
5. Inverse Proportion
Inverse proportion (sometimes called indirect proportion) is the opposite of direct proportion. As one quantity increases, the other quantity decreases.
Analogy: If you are building a wall, the more workers you hire, the less time it takes to finish the job.
The key to inverse proportion is that the product of the two quantities is constant. (Quantity 1 \(\times\) Quantity 2 = Constant Total Work).
Step-by-Step: Solving Inverse Proportion
Example: It takes 4 builders 12 days to build a wall. How long would it take 6 builders to build the same wall? (Assume they all work at the same speed.)
- Calculate the Total Work (The Constant):
Total Work (Builder-Days) = Builders \(\times\) Time (Days)
\(4 \text{ builders} \times 12 \text{ days} = 48 \text{ builder-days}\).
This means 48 days of work are needed in total. - Use the Total Work to find the new time:
If you now have 6 builders, divide the total work by the new number of builders.
Time = Total Work \(\div\) New Builders
\(48 \div 6 \text{ builders} = 8 \text{ days}\).
Check: Does the answer make sense? Yes! More builders (6) means less time (8 days), which confirms it is inverse proportion.
Common Mistake to Avoid in Proportion
Students often treat inverse proportion problems as direct proportion.
The Trick:
Direct Proportion: Divide first, then multiply (The Unitary Method).
Inverse Proportion: Multiply first (to find the constant Total Work), then divide.
Quick Review Box: Proportion
- Direct: \(A \propto B\) (Ratio \(A/B\) is constant).
- Inverse: \(A \propto 1/B\) (Product \(A \times B\) is constant).
6. Ratio and Scale
Scale is a specific use of ratio, often used on maps, blueprints, or models.
Understanding Scale Notation
A scale written as 1 : 50,000 means:
1 unit on the map represents 50,000 of the same units in real life.
Example: A map has a scale of 1 : 10,000. If the distance between two towns on the map is 5 cm, what is the real distance in kilometres?
- Use the Ratio:
Map distance is 5 cm. Real distance is \(5 \times 10,000 = 50,000 \text{ cm}\). - Convert Units (cm to km):
(Remember: 100 cm = 1 m, and 1000 m = 1 km)
First, convert to meters: \(50,000 \div 100 = 500 \text{ m}\).
Next, convert to kilometres: \(500 \div 1000 = 0.5 \text{ km}\).
The real distance is 0.5 km.
You have covered all the essential elements of Ratio and Proportion for your IGCSE exam! Remember to practice converting units, as this is often where marks are lost. Keep practicing, and you will master this topic!