International Mathematics (0607) | Ratio and proportion

Welcome to the World of Ratio and Proportion!

Hi there! This chapter is all about comparing quantities and seeing how things scale up or down—skills you use every day, whether you're following a recipe or reading a map.
Ratio and proportion are fundamental mathematical tools that help you solve real-world problems involving sharing, scaling, and comparative measure. Let’s dive in!

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1. Understanding Ratios (C1.10/E1.10)

A ratio is a way to compare the size of two or more quantities of the same kind. It shows how much of one thing there is compared to another.

Key Terms and Notation

If you mix paint, using 1 part red paint for every 3 parts white paint, the ratio of red to white is 1 to 3, written as: \(1 : 3\).

• Parts: The numbers in the ratio (1 and 3).
• Order Matters: The ratio \(1:3\) (Red:White) is very different from \(3:1\) (White:Red). Always check the order requested in the question!

1.1 Simplifying Ratios

Ratios should always be given in their simplest form, just like fractions. You simplify a ratio by dividing all parts by the Highest Common Factor (HCF).

Example: Simplifying \(20 : 30 : 40\)
The HCF of 20, 30, and 40 is 10.
Divide all parts by 10:
\(20 \div 10 : 30 \div 10 : 40 \div 10\)
Simplified ratio: \(2 : 3 : 4\).

Important Rule: Units Must Match!

You cannot simplify a ratio until all quantities are measured using the same units.

Example: Simplify \(50 \text{ cm} : 2 \text{ m}\)
1. Convert metres to centimetres: \(2 \text{ m} = 200 \text{ cm}\).
2. Write the ratio using the same unit: \(50 : 200\).
3. Simplify (divide by 50): \(1 : 4\).

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2. Dividing a Quantity in a Given Ratio (C1.10/E1.10)

This is often called the "sharing" problem. When you need to split a total amount according to a ratio, follow these simple steps:

Step-by-Step: The Total Shares Method

Problem: Share $450 in the ratio \(2 : 3 : 4\).

Step 1: Find the Total Number of Shares (Parts)
Add up all the numbers in the ratio: \(2 + 3 + 4 = 9\) total shares.

Step 2: Find the Value of One Share
Divide the total quantity by the total number of shares:
\( \text{Value per share} = \frac{\$450}{9} = \$50 \).

Step 3: Calculate Each Person's Amount
Multiply the value of one share by the number of shares for each part of the ratio:

• Part 1: \(2 \times \$50 = \$100\)
• Part 2: \(3 \times \$50 = \$150\)
• Part 3: \(4 \times \$50 = \$200\)

Check: \(100 + 150 + 200 = 450\). The total matches! (Always check your work!)

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3. Proportional Reasoning in Context (C1.10/E1.10)

Proportional reasoning means applying ratios to practical situations to maintain balance or scale things correctly.

3.1 Recipes and Ingredients

Ratios are essential for scaling recipes up or down. If a recipe uses a ratio of flour to sugar of \(5:2\), this ratio must stay constant regardless of whether you make a large batch or a small one.

Scaling Example: A recipe needs 150g of sugar and 375g of flour. If you only have 50g of sugar, how much flour do you need?

1. Find the ratio (Flour : Sugar): \(375 : 150\).
2. Simplify the ratio: Divide by 75, giving \(5 : 2\).
3. Set up the proportion: \(\frac{\text{Flour}}{\text{Sugar}} = \frac{5}{2}\).
4. Use the new amount of sugar (50g): \(\frac{\text{Flour}}{50} = \frac{5}{2}\).
5. Solve for Flour: \(\text{Flour} = 50 \times \frac{5}{2} = 125 \text{ g}\).

3.2 Map Scales

Map scales are often given as ratios, like \(1 : 50\ 000\). This means that 1 unit on the map represents 50 000 of the same units in reality.

• If the distance on the map is 3 cm, the real distance is: \(3 \times 50\ 000 = 150\ 000 \text{ cm}\).
• Since \(100 \text{ cm} = 1 \text{ m}\) and \(1\ 000 \text{ m} = 1 \text{ km}\): \(150\ 000 \text{ cm} = 1500 \text{ m} = 1.5 \text{ km}\).

3.3 Determining Best Value

To find the best value, you must compare the cost per unit (e.g., cost per kilogram, cost per litre). This uses a rate.

Example: Which is better value?
A: 500g of coffee for $8.00
B: 750g of coffee for $11.50

• Calculate the price per gram (or per 100g, or per kg). Let's use 1g.
• A: \(\frac{\$8.00}{500 \text{ g}} = \$0.016\) per gram.
• B: \(\frac{\$11.50}{750 \text{ g}} \approx \$0.0153\) per gram.

Since B has a lower price per gram, B is the better value.

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4. Rates (C1.11/E1.11)

A rate is a special type of ratio where the two quantities being compared have different units. For example, kilometres per hour (km/h) or dollars per litre (\$/L).

4.1 Speed and Time

Speed is the most common rate calculation you will encounter. You must know the relationship:

$$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$ $$ \text{Distance} = \text{Speed} \times \text{Time} $$ $$ \text{Time} = \frac{\text{Distance}}{\text{Speed}} $$

Common Mistake: Time Conversion!

When calculating speed, time must be in a single unit (usually hours or seconds), not hours and minutes.

Example: A car travels 45 km in 1 hour 30 minutes. What is the average speed?

1. Convert time to hours: 30 minutes = 0.5 hours. Total time = 1.5 hours.
2. Calculate speed: \(\text{Speed} = \frac{45 \text{ km}}{1.5 \text{ h}} = 30 \text{ km/h}\).

4.2 Other Common Rates

• Hourly Rates of Pay: Earning (\$) per hour (h).
• Fuel Consumption: Distance travelled (km) per amount of fuel used (L), often expressed as km/L.
• Flow Rates: Volume (e.g., litres or cm³) per time (e.g., second or minute), often expressed as L/min or g/cm³ (which is density, a type of rate).
• Exchange Rates: The ratio used to convert one currency to another (e.g., 1 USD : 0.85 EUR).

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5. Extended Content: Algebraic Proportion / Variation (E2.8)

If you are studying the Extended syllabus, you need to understand how quantities relate algebraically using proportionality.

The symbol for proportionality is \(\propto\) (read as "is proportional to").

The Key Principle: The Constant \(k\)

Whenever you see \(\propto\), you replace it with an equals sign and a constant, \(k\). This constant \(k\) is called the constant of proportionality.

Don't worry if this seems tricky at first—the method is always the same: Find k first!

5.1 Direct Proportion

Two quantities, \(y\) and \(x\), are in direct proportion if they increase or decrease together at the same rate. When one doubles, the other doubles.

Linear Direct Proportion

Statement: \(y\) is directly proportional to \(x\).
Notation: \(y \propto x\)
Equation: \(y = kx\)

Analogy: The more hours you work (\(x\)), the more money you earn (\(y\)).

Other Direct Proportions

Direct proportion can involve powers or roots of \(x\):

• \(y\) is proportional to the square of \(x\): \(y \propto x^2 \implies y = kx^2\)
• \(y\) is proportional to the cube root of \(x\): \(y \propto \sqrt[3]{x} \implies y = k\sqrt[3]{x}\)

5.2 Inverse Proportion

Two quantities, \(y\) and \(x\), are in inverse proportion if as one increases, the other decreases (and vice versa).

Linear Inverse Proportion

Statement: \(y\) is inversely proportional to \(x\).
Notation: \(y \propto \frac{1}{x}\)
Equation: \(y = \frac{k}{x}\)

Analogy: The more workers on a job (\(x\)), the less time it takes to finish (\(y\)).

Other Inverse Proportions

Inverse proportion can also involve powers or roots:

• \(y\) is inversely proportional to the square of \(x\): \(y \propto \frac{1}{x^2} \implies y = \frac{k}{x^2}\)

5.3 Step-by-Step: Solving Variation Problems

Problem: \(P\) is directly proportional to \(T\). When \(T=5\), \(P=20\). Find \(P\) when \(T=12\).

1. Write the variation statement and equation:
   \(P \propto T \implies P = kT\)
2. Substitute the known values to find \(k\):
   \(20 = k(5)\)
   \(k = \frac{20}{5} = 4\)
3. Write the complete formula:
   \(P = 4T\)
4. Use the formula to solve for the unknown:
   When \(T=12\): \(P = 4(12) = 48\).

Did you know? Understanding direct and inverse proportion is crucial for studying Physics, especially when dealing with forces, pressure, and electrical circuits!