Mathematics (9260) | Ratio and proportion

Hello and Welcome to Ratio and Proportion!

Get ready to dive into the world of comparisons and scaling! Ratio and proportion are fundamental skills in Mathematics (9260) and they are used everywhere—from following a recipe and mixing paint to reading maps and scaling models.

Don't worry if this seems tricky at first! We will break down every concept step-by-step. By the end of this chapter, you’ll be confident in sharing money, comparing speeds, and solving real-world scaling problems.

Why is Ratio and Proportion important?

• It helps you scale things up or down accurately (like making a tiny model car from a full-sized car).
• It is essential for understanding finance and budgeting (sharing costs fairly).
• It forms the basis for speed, density, and flow rates in Science and Physics.

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Part 1: Understanding Ratio (Comparing Quantities)

The Basics of Ratio

A ratio is a way of comparing two or more quantities of the same kind. We use a colon (:) to separate the quantities.

Key Features of Ratios

• Comparison: Ratios compare parts to other parts (e.g., 2 parts flour to 3 parts sugar).
• Order Matters: The order in which the numbers are written is essential. A ratio of 2:3 is completely different from 3:2.
• Units: All quantities in a ratio must be measured using the same units before you start calculating! (You can't compare 1 meter to 50 centimeters until you convert the meter to 100 centimeters).

Example: If a class has 10 boys and 15 girls, the ratio of boys to girls is \(10 : 15\).

Ratio vs. Fractions – A Quick Look

Students often confuse ratios and fractions. Remember:

If the ratio of Blue paint to Yellow paint is \(2 : 3\):

• Ratio: Compares Blue (2 parts) to Yellow (3 parts).
• Fraction: Compares a part to the whole mixture (total parts = 5). So, the fraction of Blue paint is \(\frac{2}{5}\).

Simplifying Ratios

Just like fractions, ratios must always be simplified to their lowest terms using whole numbers. This makes them easier to compare.

Step-by-Step: How to Simplify a Ratio

Goal: Divide all parts of the ratio by the same number until no common factors remain (find the Highest Common Factor, HCF).

Example 1: Simplify the ratio \(18 : 24\).

1. Find the HCF of 18 and 24. (The largest number that divides both is 6).
2. Divide both sides by 6:
   \(18 \div 6 = 3\)
   \(24 \div 6 = 4\)
3. The simplified ratio is \(3 : 4\).

Dealing with Decimals and Fractions in Ratios

If your ratio contains decimals or fractions, you must eliminate them first to get whole numbers.

Example 2: Simplify \(1.5 : 2\).

Multiply both sides by 10 (or 2, to eliminate the 0.5) to make 1.5 a whole number.

• \(1.5 \times 2 = 3\)
• \(2 \times 2 = 4\)
• The simplified ratio is \(3 : 4\).

Common Mistake to Avoid: Different Units!

You cannot simplify a ratio until all units match.

If the ratio is 50 cm to 2 m:

Step 1: Convert 2 m to 200 cm. The ratio is \(50 : 200\).
Step 2: Simplify by dividing both by 50.
The simplified ratio is \(1 : 4\).

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Part 2: Sharing a Quantity in a Given Ratio

This is one of the most common exam questions. You will be given a total amount (e.g., money or weight) and a ratio, and asked to share the amount fairly according to that ratio.

Step-by-Step: The Total Parts Method

Let’s share £60 in the ratio \(2 : 3\).

Step 1: Calculate the Total Number of Parts

Add the numbers in the ratio together.
Total parts = \(2 + 3 = 5\) parts.

Step 2: Find the Value of One Part (The Unit Value)

Divide the total amount by the total number of parts.
Value of 1 part = \(\frac{\text{Total Amount}}{\text{Total Parts}}\)
Value of 1 part = \(\frac{£60}{5} = £12\).

Step 3: Calculate the Value of Each Share

Multiply the value of one part by the number in the ratio for each person/item.

• Share 1 (2 parts): \(2 \times £12 = £24\)
• Share 2 (3 parts): \(3 \times £12 = £36\)

Step 4: Check Your Answer

Add the shares back together to make sure they equal the original total: \(£24 + £36 = £60\). (It works!)

Memory Aid: T. U. M. (Total parts, Unit value, Multiply).

Working Backwards: Finding the Original Amount

Sometimes you are given the size of one share and asked to find the total amount.

Example: Amy and Ben share money in the ratio \(4 : 7\). If Amy receives £20, how much did they share in total?

1. Identify the known share: Amy corresponds to the '4' in the ratio.
2. Find the Unit Value: If 4 parts = £20, then 1 part = \(\frac{£20}{4} = £5\).
3. Find the Total Amount: The total parts are \(4 + 7 = 11\).
4. Total Amount = \(11 \times £5 = £55\).

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Part 3: Ratio and Unit Form (n:1 and 1:n)

In real life, ratios are often simplified so that one side is equal to 1. This is called the Unit Ratio. It makes comparing values very easy.

Type A: Simplifying to \(1 : n\)

We often use this when comparing prices (e.g., how much does 1 kg cost?).

Example: The cost of 3 cakes is £4.50. Write the ratio of the number of cakes to the cost in the form \(1 : n\).

1. Start with the ratio: Cakes : Cost = \(3 : 4.50\).
2. To change the left side (3) to 1, divide both sides by 3.
3. \(3 \div 3 = 1\)
4. \(4.50 \div 3 = 1.50\)
5. The ratio is \(1 : 1.50\). This tells us 1 cake costs £1.50.

Type B: Simplifying to \(n : 1\)

We use this to find out how many of the first item corresponds to one of the second item.

Example: A map scale is 15 cm : 3 km. Write the scale in the form \(n : 1\) (where \(n\) is the number of cm per km).

1. Start with the ratio: \(15 : 3\).
2. To change the right side (3) to 1, divide both sides by 3.
3. \(15 \div 3 = 5\)
4. \(3 \div 3 = 1\)
5. The ratio is \(5 : 1\). This means 5 cm on the map represents 1 km in real life.

Key Takeaway: To get the '1', divide all parts by the number on the side you want to become 1.

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Part 4: Proportion (Scaling and Relationships)

Proportion deals with how two quantities relate to each other as they change. We look at two main types: Direct and Inverse.

Analogy: If you double the ingredients (ratio), you double the cake (proportion).

A. Direct Proportion

Two quantities are in direct proportion if they increase or decrease at the same rate. When one quantity doubles, the other quantity also doubles.

Relationship: Quantity A \(\propto\) Quantity B (A is proportional to B).

Real-World Example (Direct Proportion)

Cost and Quantity: The more ice creams you buy, the more money you spend.

Step-by-Step: Solving Direct Proportion Problems (The Unitary Method)

Example: 5 pencils cost £3. How much do 8 pencils cost?

1. Find the Cost of 1 Unit (The Unitary Method): Divide the cost by the number of items.
   Cost of 1 pencil = \(\frac{£3}{5} = £0.60\).
2. Scale Up to Find the Answer: Multiply the unit cost by the required quantity (8 pencils).
   Cost of 8 pencils = \(8 \times £0.60 = £4.80\).

Using a Conversion Ratio/Multiplier (Advanced Method)

You can also set up a multiplier (k) between the two quantities.

If we know 5 pencils (\(P\)) cost £3 (\(C\)), we can find the multiplier \(k\):
\(C = kP\)
\(3 = k \times 5 \Rightarrow k = \frac{3}{5} = 0.6\) (This is the price per pencil!)

Now apply the multiplier to 8 pencils:
\(C = 0.6 \times 8 = £4.80\).

B. Inverse Proportion (Indirect Proportion)

Two quantities are in inverse proportion if, as one quantity increases, the other decreases at a proportional rate.

Relationship: Quantity A \(\propto \frac{1}{\text{Quantity B}}\).

Real-World Example (Inverse Proportion)

Workers and Time: The more workers you hire, the less time it takes to complete the job.

Step-by-Step: Solving Inverse Proportion Problems

The key here is that the Total Work remains constant.
Total Work = Quantity A \(\times\) Quantity B.

Example: 4 builders can finish a wall in 9 hours. How long would it take 6 builders to finish the same wall?

1. Calculate the Total Work (Total Builder-Hours):
   \(4 \text{ builders} \times 9 \text{ hours} = 36 \text{ builder-hours}\). (This is the constant amount of work needed).
2. Use the Total Work to find the new time:
   Time = \(\frac{\text{Total Work}}{\text{New Quantity}}\)
   Time for 6 builders = \(\frac{36 \text{ hours}}{6 \text{ builders}} = 6 \text{ hours}\).

Notice that increasing the number of builders (from 4 to 6) decreased the time (from 9 hours to 6 hours).

How to Spot the Difference!

If you multiply A, you must multiply B. (Direct)

If you multiply A, you must divide B. (Inverse)

Concept	Direct Proportion	Inverse Proportion
Relationship	A and B move in the same direction.	A and B move in opposite directions.
Method to Solve	Find the value of 1 (divide), then multiply.	Find the total constant (multiply), then divide.
Math Idea	Ratio \(\frac{A}{B}\) is constant.	Product \(A \times B\) is constant.

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Common Mistakes and Final Tips

1. Ignoring Units

Always ensure ratios (especially involving time, distance, or money) are in the same units before simplifying or calculating. Convert everything to the smaller unit first (e.g., hours to minutes, metres to centimetres).

2. Forgetting the Total Parts

When sharing an amount in a ratio (e.g., 2:5), students sometimes divide the total amount by 2 or 5 instead of dividing by the total parts (\(2+5=7\)).

3. Mixing Up Direct and Inverse

Ask yourself: "Does more of A mean more of B?" If yes, it's Direct. If no (more of A means less of B), it's Inverse.

Encouragement

You have mastered the core concepts of number comparison and scaling! Practice these step-by-step methods, especially the unitary method, and you will find ratio and proportion questions becoming straightforward. Keep up the fantastic work!