Chapter: Ratio and Proportion
Welcome to the guide on Ratio and Proportion! Whether you are mixing paint, adapting a recipe for more friends, sharing prize money, or working out currency exchange rates on holiday, ratio and proportion are skills you use in everyday life. Don't worry if maths hasn't always been your favourite subject; we will break everything down step-by-step so you can tackle these exam questions with confidence.
1. Understanding and Simplifying Ratios
A ratio compares two or more quantities of the same kind. It shows the relative sizes of the quantities.
We write ratios using a colon \(:\) which is read aloud as "to". For example, if a fruit bowl has \(3\) apples and \(5\) bananas, the ratio of apples to bananas is \(3 : 5\).
Key Rule for Ratios
Order matters! The ratio of apples to bananas is \(3 : 5\), but the ratio of bananas to apples is \(5 : 3\).
Simplifying Ratios
Just like simplifying fractions, you can simplify a ratio by dividing all parts by their Highest Common Factor (HCF).
Example 1: Simplify the ratio \(12 : 18\).
Step 1: Find a common factor of \(12\) and \(18\). Both numbers can be divided by \(6\).
Step 2: Divide both sides: \(12 \div 6 = 2\) and \(18 \div 6 = 3\).
Answer: \(2 : 3\)
Ratios with Different Units
Before simplifying, always make sure the units are the same!
Example 2: Simplify the ratio \(50\text{ cm} : 2\text{ m}\).
Step 1: Convert to the same units. Since \(1\text{ m} = 100\text{ cm}\), \(2\text{ m} = 200\text{ cm}\).
Step 2: Write the ratio with numbers only: \(50 : 200\).
Step 3: Divide both sides by \(50\): \(50 \div 50 = 1\) and \(200 \div 50 = 4\).
Answer: \(1 : 4\)
Writing Ratios in the Form \(1 : n\) or \(n : 1\)
Sometimes an exam question asks you to write a ratio in the form \(1 : n\) or \(n : 1\). This means one of the numbers must be turned into \(1\), even if it creates a decimal or fraction for the other number.
Example 3: Write \(4 : 10\) in the form \(1 : n\).
Step 1: To make the first number \(1\), divide both numbers by \(4\).
Step 2: \(4 \div 4 = 1\) and \(10 \div 4 = 2.5\).
Answer: \(1 : 2.5\)
Key Takeaway: Always simplify ratios by dividing by the highest common factor. Ensure units match before simplifying, and remember that the order of the numbers is crucial.
2. Sharing Amounts in a Given Ratio
A classic exam question asks you to share a total amount between people or groups in a given ratio. We use a simple 3-step method: Add, Divide, Multiply (remember the mnemonic: A-D-M).
The 3-Step Method (A-D-M)
1. Add: Add up the total number of parts in the ratio.
2. Divide: Divide the total amount by the total number of parts to find the value of \(1\) part.
3. Multiply: Multiply the value of \(1\) part by each number in the ratio.
Example: Share \(\text{\textsterling}120\) between Adam and Ben in the ratio \(3 : 5\).
Step 1 (Add parts): \(3 + 5 = 8\text{ total parts}\)
Step 2 (Find \(1\) part): \(\text{\textsterling}120 \div 8 = \text{\textsterling}15\)
Step 3 (Multiply):
Adam gets \(3 \times \text{\textsterling}15 = \text{\textsterling}45\)
Ben gets \(5 \times \text{\textsterling}15 = \text{\textsterling}75\)
Quick Check: \(\text{\textsterling}45 + \text{\textsterling}75 = \text{\textsterling}120\). It adds up to the original total!
Working Backwards (When One Share or Difference is Given)
Don't fall into the trap of automatically adding the parts together! Read the question carefully to see if you are given the total, one person's share, or the difference between shares.
Example (One Share Given): Flour and sugar are mixed in the ratio \(5 : 2\). If \(350\text{ g}\) of flour is used, how much sugar is needed?
Step 1: Flour represents \(5\) parts. So, \(5\text{ parts} = 350\text{ g}\).
Step 2: Find \(1\) part: \(350\text{ g} \div 5 = 70\text{ g}\).
Step 3: Sugar is \(2\) parts: \(2 \times 70\text{ g} = 140\text{ g}\).
Answer: \(140\text{ g}\) of sugar.
Example (Difference Given): Chloe and Dan share money in the ratio \(7 : 4\). Chloe receives \(\text{\textsterling}18\) more than Dan. How much money did they share in total?
Step 1: Find the difference in parts: \(7 - 4 = 3\text{ parts}\).
Step 2: Find \(1\) part: \(\text{\textsterling}18 \div 3 = \text{\textsterling}6\).
Step 3: Find the total parts: \(7 + 4 = 11\text{ parts}\).
Step 4: Total money: \(11 \times \text{\textsterling}6 = \text{\textsterling}66\).
Answer: \(\text{\textsterling}66\)
Key Takeaway: Before calculating, ask yourself: "Am I given the total, one single part, or the difference between parts?"
3. Direct Proportion
Two quantities are in direct proportion if, when one increases, the other increases at the same rate. For example, if you double the number of items you buy, you double the total cost.
Method 1: The Unitary Method
The unitary method means finding the value of one single unit first, then multiplying by the required quantity.
Example (Recipe Adaptation): A pancake recipe for \(4\) people requires \(200\text{ g}\) of flour and \(300\text{ ml}\) of milk. How much flour and milk are needed for \(6\) people?
Step 1: Find the amount needed for \(1\) person (divide by \(4\)):
Flour for \(1\) person: \(200\text{ g} \div 4 = 50\text{ g}\)
Milk for \(1\) person: \(300\text{ ml} \div 4 = 75\text{ ml}\)
Step 2: Multiply by \(6\) to find the amount for \(6\) people:
Flour: \(50\text{ g} \times 6 = 300\text{ g}\)
Milk: \(75\text{ ml} \times 6 = 450\text{ ml}\)
Answer: \(300\text{ g}\) of flour and \(450\text{ ml}\) of milk.
Method 2: Best Buy / Value for Money
To find which product is better value, compare the cost per unit (e.g., price per \(100\text{ g}\) or price per item), or the quantity per unit of money (e.g., grams per \(\text{\textsterling}1\)).
Example:
Pack A: \(6\) cans for \(\text{\textsterling}4.50\)
Pack B: \(8\) cans for \(\text{\textsterling}5.60\)
Which pack is better value?
Cost per can for Pack A: \(\text{\textsterling}4.50 \div 6 = \text{\textsterling}0.75\) (\(75\text{p}\) per can)
Cost per can for Pack B: \(\text{\textsterling}5.60 \div 8 = \text{\textsterling}0.70\) (\(70\text{p}\) per can)
Conclusion: Pack B is better value because each can is cheaper.
Key Takeaway: In direct proportion, find the value of \(1\) unit first (the unitary method) to make scaling up or comparing easy.
4. Inverse Proportion (Indirect Proportion)
Two quantities are in inverse proportion if, when one increases, the other decreases at the same rate. A common example is workers completing a job: if you double the number of workers, the time taken to finish is halved!
The Key Rule for Inverse Proportion
The product of the two variables remains constant:
\(\text{Quantity } A \times \text{Quantity } B = \text{Constant}\)
Example: It takes \(3\) builders \(8\) days to build a wall. How many days would it take \(4\) builders working at the same pace?
Step 1: Calculate the total workload in "builder-days":
\(3\text{ builders} \times 8\text{ days} = 24\text{ builder-days}\)
Step 2: Divide the total workload by the new number of builders:
\(\text{Time taken} = 24 \div 4 = 6\text{ days}\)
Answer: It would take \(6\) days.
Common Mistake: Do not multiply \(8\) by \(\frac{4}{3}\). More builders must mean less time, not more!
Key Takeaway: For inverse proportion problems, multiply the given pair together to find the constant total, then divide by the new value.
5. Formal Algebraic Proportion
In higher-tier questions, proportion is written using the proportionality symbol \(\propto\).
Direct Proportion: \(y \propto x\)
When \(y\) is directly proportional to \(x\), we write:
\(y \propto x \implies y = kx\)
where \(k\) is the constant of proportionality.
Example: \(y\) is directly proportional to \(x\). When \(x = 5\), \(y = 35\). Find \(y\) when \(x = 8\).
Step 1: Set up the equation: \(y = kx\)
Step 2: Substitute the known values to find \(k\):
\(35 = k \times 5 \implies k = \frac{35}{5} = 7\)
Step 3: Write the full formula: \(y = 7x\)
Step 4: Use the formula with the new value (\(x = 8\)):
\(y = 7 \times 8 = 56\)
Answer: \(y = 56\)
Inverse Proportion: \(y \propto \frac{1}{x}\)
When \(y\) is inversely proportional to \(x\), we write:
\(y \propto \frac{1}{x} \implies y = \frac{k}{x}\)
Example: \(y\) is inversely proportional to \(x\). When \(x = 3\), \(y = 12\). Find the value of \(y\) when \(x = 9\).
Step 1: Set up the equation: \(y = \frac{k}{x}\)
Step 2: Substitute known values to find \(k\):
\(12 = \frac{k}{3} \implies k = 12 \times 3 = 36\)
Step 3: Write the formula: \(y = \frac{36}{x}\)
Step 4: Substitute \(x = 9\):
\(y = \frac{36}{9} = 4\)
Answer: \(y = 4\)
Proportion Involving Squares, Cubes, and Roots
The same steps apply when variables involve powers or roots:
- If \(y\) is proportional to the square of \(x\): \(y = kx^2\)
- If \(y\) is inversely proportional to the square of \(x\): \(y = \frac{k}{x^2}\)
- If \(y\) is proportional to the square root of \(x\): \(y = k\sqrt{x}\)
Key Takeaway: Always follow the 4-step method: (1) Write the equation with \(k\), (2) Substitute known numbers to find \(k\), (3) Rewrite the complete formula, (4) Substitute the target number to find the answer.
Common Pitfalls to Avoid
- Forgetting to match units: Mixing centimetres with metres or pence with pounds will lead to incorrect answers.
- Dividing by the wrong number: Only divide by the total number of parts if you are given the total amount.
- Direct vs. Inverse confusion: Always ask yourself: "If this value goes up, should the other go up or down?" If more workers are working, time must decrease!