Ratio and proportion

1. Overview

Ratio and proportion are fundamental tools for comparing quantities and scaling values. Ratios express the relative sizes of two or more quantities, while a proportion states that two ratios are equal. Mastering these concepts is crucial for solving real-world problems involving scaling, sharing, and comparing different amounts. This section covers simplifying ratios, dividing quantities in given ratios, and applying proportional reasoning in various contexts, as required by the IGCSE Cambridge Mathematics syllabus.

Key Definitions

• Ratio: A way of comparing two or more quantities of the same kind, shown using the colon ( : ) symbol.
• Proportion: A statement that two ratios or fractions are equal.
• Simplest Form: A ratio where the numbers are integers and have no common factors other than 1.
• Unitary Method: A technique where you first find the value of a single unit and then multiply to find the required value.

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Core Content

Simplifying Ratios

To simplify a ratio, divide all parts by their Highest Common Factor (HCF). Important: Before simplifying, ensure all quantities are in the same units.

Worked example 1 — Simplifying with Units

Question: Simplify the ratio $400 \text{ cm} : 2 \text{ m}$.

1. Convert to the same units: $2 \text{ m} = 200 \text{ cm}$
   • Reason: To compare, both quantities must be in the same unit.
2. Write the ratio: $400 \text{ cm} : 200 \text{ cm}$
   • Reason: Express the comparison using the same units.
3. Divide by the HCF (200): $400 \div 200 : 200 \div 200$
   • Reason: Simplify by dividing by the highest common factor.
4. Final answer: $2 : 1$
   • Reason: The simplified ratio.

Worked example 2 — Simplifying a Three-Part Ratio

Question: Simplify the ratio $12 : 18 : 30$

1. Identify the HCF: The highest common factor of 12, 18, and 30 is 6.
   • Reason: Find the largest number that divides all parts of the ratio.
2. Divide each part by the HCF: $12 \div 6 : 18 \div 6 : 30 \div 6$
   • Reason: Divide each term by the HCF to reduce the ratio.
3. Simplify: $2 : 3 : 5$
   • Reason: Perform the division.
4. Final answer: $2 : 3 : 5$

Dividing a Quantity in a Given Ratio

To divide a total amount into a specific ratio (e.g., $a : b$):

1. Add the parts to find the total number of shares ($a + b$).
2. Divide the total quantity by the total number of shares to find the value of one share.
3. Multiply the value of one share by each part of the ratio.

Worked example 3 — Dividing Money

Question: Divide $$120$ in the ratio $3 : 5$.

1. Total parts: $3 + 5 = 8 \text{ parts}$
   • Reason: Determine the total number of shares.
2. Value of one part: $$120 \div 8 = $15$
   • Reason: Calculate the value of a single share.
3. Multiply by ratio parts:
   • $3 \times $15 = $45$
     • Reason: Calculate the value of the first share.
   • $5 \times $15 = $75$
     • Reason: Calculate the value of the second share.
4. Check: $$45 + $75 = $120$
   • Reason: Verify that the sum of the parts equals the original amount.
5. Final answer: $$45 : $75$

Worked example 4 — Sharing sweets

Question: Amy and Ben share 45 sweets in the ratio 2:3. How many sweets does Ben receive?

1. Total parts: $2 + 3 = 5 \text{ parts}$
   • Reason: Determine the total number of shares.
2. Value of one part: $45 \div 5 = 9 \text{ sweets}$
   • Reason: Calculate the value of a single share.
3. Ben's share: $3 \times 9 = 27 \text{ sweets}$ * Reason: Calculate the value of Ben's share.
4. Final answer: $27 \text{ sweets}$

Proportional Reasoning in Context (The Unitary Method)

If you know the cost of a certain number of items, find the cost of one item first.

Worked example 5 — Unitary Method

Question: If 6 pens cost $$4.50$, calculate the cost of 10 pens.

1. Find the cost of 1 pen: $$4.50 \div 6 = $0.75$
   • Reason: Determine the cost of one pen.
2. Find the cost of 10 pens: $10 \times $0.75 = $7.50$
   • Reason: Calculate the cost of ten pens.
3. Final answer: $$7.50$

Worked example 6 — Scaling a Recipe

Question: A recipe for 4 people requires 200g of flour. How much flour is needed for 6 people?

1. Find the flour needed per person: $200 \text{g} \div 4 = 50 \text{g}$
   • Reason: Determine the amount of flour required for one person.
2. Find the flour needed for 6 people: $6 \times 50 \text{g} = 300 \text{g}$
   • Reason: Calculate the total amount of flour needed for six people.
3. Final answer: $300 \text{g}$

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Extended Content (Extended curriculum only)

Advanced Simplification

In the Extended curriculum, you may be asked to simplify ratios involving fractions or decimals. To simplify, multiply all parts by a common multiple to eliminate fractions/decimals first. You may also need to simplify ratios involving algebraic terms.

Worked example 7 — Fractional Ratios

Question: Simplify the ratio $\frac{1}{2} : \frac{3}{4} : 2$.

1. Find a common denominator: The lowest common multiple of 2 and 4 is 4.
   • Reason: Identify a common denominator for the fractions.
2. Convert all parts: $\frac{2}{4} : \frac{3}{4} : \frac{8}{4}$
   • Reason: Express each term with the common denominator.
3. Multiply by 4 to remove denominators: $4 \times \frac{2}{4} : 4 \times \frac{3}{4} : 4 \times \frac{8}{4}$
   • Reason: Eliminate the fractions by multiplying by the common denominator.
4. Simplify: $2 : 3 : 8$
   • Reason: Perform the multiplication.
5. Final answer: $2 : 3 : 8$

Worked example 8 — Algebraic Ratios

Question: Express the ratio $3x^2 : 6x$ in its simplest form.

1. Identify common factors: Both sides can be divided by $3$ and by $x$.
   • Reason: Find the common factors in both terms.
2. Divide by $3x$: $\frac{3x^2}{3x} : \frac{6x}{3x}$
   • Reason: Divide both terms by the common factor.
3. Simplify: $x : 2$
   • Reason: Simplify the expression.
4. Final answer: $x : 2$

Worked example 9 — Ratios with Decimals

Question: Simplify the ratio $0.2 : 1.5$

1. Multiply by 10 to remove decimals: $0.2 \times 10 : 1.5 \times 10$
   • Reason: Eliminate the decimals by multiplying by a power of 10.
2. Simplify: $2 : 15$
   • Reason: Perform the multiplication.
3. Final answer: $2 : 15$

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Key Equations

Total Shares = $part_1 + part_2 + ... + part_n$

Value of One Share = $\frac{\text{Total Quantity}}{\text{Total Shares}}$

Value of Specific Part = Value of One Share $\times$ Ratio Number

Note: No formulas for ratio and proportion are provided on the IGCSE formula sheet; they must be memorised.

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Common Mistakes to Avoid

• ❌ Wrong: Simplifying $500 \text{ ml} : 2 \text{ Litres}$ to $250 : 1$.
• ✅ Right: Convert units first. $500 \text{ ml} : 2000 \text{ ml} = 1 : 4$. Always convert to consistent units before calculating.
• ❌ Wrong: Using the ratio $2 : 3$ to find the smaller share by multiplying the total by $\frac{3}{5}$.
• ✅ Right: Read the question carefully. If the ratio is $A:B = 2:3$, $A$ is the smaller part. Using the ratio backward leads to an incorrect solution.
• ❌ Wrong: Skipping the "total parts" step and dividing the quantity by the ratio numbers directly.
• ✅ Right: Always sum the parts of the ratio first to find the denominator.
• ❌ Wrong: Forgetting to simplify the ratio to its simplest form at the end of the calculation.
• ✅ Right: Always check if the resulting ratio can be further simplified by dividing by the HCF.
• ❌ Wrong: Assuming that if $a:b = 2:5$, then $a = 2$ and $b = 5$.
• ✅ Right: Understand that $a$ and $b$ are multiples of 2 and 5, respectively. You need more information (like the total) to find their exact values.

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Exam Tips

• Command Words:
  • "Simplify": Divide down to the lowest possible whole numbers.
  • "Divide in the ratio": Show the total shares calculation clearly.
  • "Express as a ratio of $1 : n$": Divide both sides of the ratio by the first number so the left side becomes 1 (even if $n$ becomes a decimal).
• Real-world Contexts: Be prepared for questions involving map scales (e.g., $1 : 25,000$), currency exchange, or fuel consumption (flow rates).
• Calculator Tip: For non-calculator papers, ratios are usually designed to have "nice" whole number totals for the shares. If your "one share" value is a messy decimal, re-check your addition of the parts.
• Show Working: Even if you can do the math in your head, write down the "Total parts" and "Value of one part." If you make a small subtraction error later, you can still gain marks for the correct method.
• Units: Always include units in your final answer, especially in real-world problems.
• Double-Check: After dividing a quantity in a given ratio, add the individual parts to ensure they sum up to the original quantity. This helps catch arithmetic errors.