# IGCSE Higher: Ratio and Proportion

## Prior Knowledge for IGCSE Mathematics: Ratios and Proportions

1. Understanding Basic Fractions
• Concept: Knowing how to interpret and manipulate fractions.
• Example: Simplify $$\frac{12}{16}$$. $\frac{12}{16} = \frac{3}{4}$
2. Simplifying Ratios
• Concept: Understanding how to simplify ratios to their simplest form.
• Example: Simplify the ratio $$8:12$$. $8:12 \Rightarrow \frac{8}{4}:\frac{12}{4} = 2:3$
3. Converting Between Fractions, Decimals, and Percentages
• Concept: Understanding how to convert between fractions, decimals, and percentages.
• Example: Convert $$0.75$$ to a fraction and a percentage. $0.75 = \frac{75}{100} = \frac{3}{4}, \quad 0.75 \times 100 = 75\%$
4. Understanding Proportions
• Concept: Knowing that proportions are equations that state two ratios are equal.
• Example: Solve for $$x$$ in the proportion $$\frac{3}{4} = \frac{x}{8}$$. $\frac{3}{4} = \frac{x}{8} \Rightarrow 3 \times 8 = 4x \Rightarrow 24 = 4x \Rightarrow x = 6$
5. Basic Arithmetic Operations
• Concept: Performing addition, subtraction, multiplication, and division.
• Example: Calculate $begin:math:text$ 25 \\times 4 – 10 $end:math:text$. $25 \times 4 – 10 = 100 – 10 = 90$

## Success Criteria for IGCSE Mathematics

1. Divide a Quantity in a Given Ratio
• Objective: Students should be able to divide a given quantity in a specified ratio.
• Example: Share £416 in the ratio 5:3. $\text{Total parts} = 5 + 3 = 8 \\ \text{Value of one part} = \frac{416}{8} = 52 \\ \text{Share for 5 parts} = 5 \times 52 = 260 \\ \text{Share for 3 parts} = 3 \times 52 = 156$
2. Use the Process of Proportionality to Evaluate Unknown Quantities
• Objective: Students should be able to use proportions to find unknown quantities.
• Example: If 4 apples cost £1.20, find the cost of 7 apples. $\text{Cost of one apple} = \frac{1.20}{4} = 0.30 \\ \text{Cost of 7 apples} = 7 \times 0.30 = 2.10$
3. Calculate an Unknown Quantity from Quantities that Vary in Direct Proportion
• Objective: Students should be able to find an unknown value when quantities vary directly.
• Example: $$s$$ varies directly as $$t$$. If $$s = 12$$ when $$t = 3$$, find $$s$$ when $$t = 5$$. $\frac{s_1}{t_1} = \frac{s_2}{t_2} \Rightarrow \frac{12}{3} = \frac{s}{5} \Rightarrow 4 = \frac{s}{5} \Rightarrow s = 20$
4. Solve Word Problems about Ratio and Proportion
• Objective: Students should be able to solve real-world problems involving ratios and proportions.
• Example: A map has a scale of 1:50,000. If the distance between two towns on the map is 4 cm, find the actual distance. $\text{Actual distance} = 4 \times 50,000 = 200,000 \text{ cm} = 2 \text{ km}$

## Key Concepts for IGCSE Mathematics

1. Understanding Ratios
• Concept: Knowing that a ratio is a way to compare quantities of the same kind by showing the relative sizes of two or more values.
• Example: The ratio 3:2 means for every 3 units of the first quantity, there are 2 units of the second quantity.
2. Direct Proportion
• Concept: Understanding that two quantities are directly proportional if they increase or decrease at the same rate. If $$y$$ is directly proportional to $$x$$, then $$y = kx$$ for some constant $$k$$.
• Example: If $$y = 3x$$, then $$y$$ is directly proportional to $$x$$. When $$x = 2$$, $$y = 6$$. $y = 3x \Rightarrow y = 3 \times 2 = 6$
3. Unitary Method
• Concept: Using the unitary method to find the value of a single unit and then using it to find the value of multiple units.
• Example: If 5 apples cost £1.50, the cost of one apple is $$\frac{1.50}{5} = £0.30$$. Therefore, the cost of 7 apples is $$7 \times 0.30 = £2.10$$. $\frac{1.50}{5} = 0.30 \\ 7 \times 0.30 = 2.10$
4. Solving Proportional Problems Using Cross-Multiplication
• Concept: Using cross-multiplication to solve problems involving proportions.
• Example: To solve $$\frac{a}{b} = \frac{c}{d}$$, cross-multiply to get $$ad = bc$$. $\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc$

## Common Misconceptions for IGCSE Mathematics

1. Misinterpreting Ratios
• Misconception: Students often misunderstand the order of ratios and may invert them, thinking that a ratio of 3:2 is the same as 2:3.
• Example: Given the ratio 3:2 to divide £100, a student might incorrectly calculate £60 and £40 instead of the correct £60 and £40.
2. Incorrectly Simplifying Ratios
• Misconception: Students might incorrectly simplify ratios, treating them as fractions and not simplifying both parts to the same degree.
• Example: Simplifying the ratio 8:12 to 4:12 instead of the correct simplification 2:3.
3. Confusing Direct and Inverse Proportion
• Misconception: Students often confuse direct proportion with inverse proportion, applying the wrong method to solve problems.
• Example: Given that $$y$$ is directly proportional to $$x$$, if $$y = 6$$ when $$x = 2$$, finding $$y$$ for $$x = 4$$ incorrectly as $$\frac{6}{2} \cdot 4 = 12$$ instead of the correct method: $y = 3x \Rightarrow y = 3 \cdot 4 = 12$
4. Errors in Cross-Multiplication
• Misconception: Students might make mistakes in cross-multiplication, either by incorrectly multiplying or misaligning terms.
• Example: To solve $$\frac{a}{b} = \frac{c}{d}$$, incorrectly performing the multiplication as $$a \cdot d = b \cdot c$$ instead of $$a \cdot d = b \cdot c$$.

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