Ratios and Rates

Let’s Start with a Question! 🤔

Have you ever wondered which shop offers better value - 500g of cheese for £6 or 750g for £10? Or how to adjust a recipe that serves 4 people to feed 6? The answers lie in understanding ratios and rates - two of the most practical mathematical tools you’ll use throughout your life!

What Are Ratios?

A ratio is a comparison of two or more quantities that shows their relative sizes. Ratios tell us “how much of one thing compared to another.”

For example, if a class has 12 boys and 15 girls:

• The ratio of boys to girls is 12:15 (read as “12 to 15”)
• We can simplify this to 4:5 by dividing both numbers by 3
• This tells us that for every 4 boys, there are 5 girls

Ways to Write Ratios

The same ratio can be written in three different forms:

1. Using a colon: 3:4 (most common)
2. Using the word “to”: 3 to 4
3. As a fraction: 3/4

All three forms mean exactly the same thing!

Why Are Ratios Important?

Ratios appear everywhere in daily life:

• Cooking (recipe proportions: 2 cups flour to 1 cup sugar)
• Maps (scale ratios: 1 cm represents 10 km)
• Art (mixing paint colours: 3 parts blue to 1 part yellow)
• Sports (win-loss ratios, player statistics)
• Finance (comparing prices, profit margins)

What Are Rates?

A rate is a special type of ratio that compares two quantities with different units. The most common rates have “per” in them - meaning “for each” or “for every one.”

Common examples:

• Speed: 60 kilometres per hour (km/h)
• Price: £3 per kilogram
• Heart rate: 72 beats per minute
• Wage: £12 per hour
• Fuel efficiency: 50 miles per gallon

Unit Rates

A unit rate is a rate where the second quantity is 1. Unit rates make it easy to compare different options!

Example: If 3 apples cost £1.50, the unit rate (price per apple) is £1.50 ÷ 3 = £0.50 per apple.

Teacher’s Insight 👨‍🏫

Here’s what I’ve learned from teaching thousands of students: The secret to mastering ratios and rates is understanding that they’re all about relationships between quantities. When my students stop thinking “3:4 is just two numbers” and start thinking “for every 3 of these, I have 4 of those,” everything becomes clearer.

My top tip: Always pay attention to order in ratios! “Boys to girls is 3:2” is very different from “Girls to boys is 2:3.” The order matters because we’re describing a specific relationship. Also, practice simplifying ratios just like you simplify fractions - it makes them much easier to understand and compare!

Key Vocabulary

• Ratio: A comparison of two or more quantities (e.g., 3:4)
• Rate: A ratio comparing quantities with different units (e.g., £5 per hour)
• Unit rate: A rate where the second quantity is 1 (e.g., £0.50 per apple)
• Proportion: An equation stating that two ratios are equal (e.g., 2:3 = 4:6)
• Equivalent ratios: Ratios that represent the same relationship (2:3, 4:6, 6:9 are equivalent)
• Simplest form: A ratio reduced to its smallest whole numbers (12:18 simplifies to 2:3)
• Scale: A ratio showing the relationship between a drawing/model and the real thing

Understanding Ratios in Depth

Simplifying Ratios

Just like fractions, ratios can be simplified by dividing both terms by their greatest common factor (GCF).

Example: Simplify 18:24

• GCF of 18 and 24 is 6
• 18 ÷ 6 = 3, 24 ÷ 6 = 4
• Simplified ratio: 3:4

Equivalent Ratios

Ratios are equivalent when they represent the same relationship. You create equivalent ratios by multiplying or dividing both terms by the same number.

Example: 2:3 is equivalent to 4:6, 6:9, 8:12, etc. (All found by multiplying both terms by 2, 3, 4, etc.)

Three-Part Ratios

Ratios can compare more than two quantities!

Example: A paint mixture uses red:yellow

in the ratio 5:3:2

• For every 5 parts red, there are 3 parts yellow and 2 parts blue
• Total parts = 5 + 3 + 2 = 10 parts

Worked Examples

Example 1: Simplifying a Ratio

Problem: A fruit bowl contains 6 apples and 9 oranges. What is the ratio of apples to oranges in simplest form?

Solution: 2:3

Detailed Explanation:

• Initial ratio of apples to oranges: 6:9
• Find the GCF of 6 and 9: GCF = 3
• Divide both terms by 3: 6 ÷ 3 = 2, 9 ÷ 3 = 3
• Simplest form: 2:3
• This means for every 2 apples, there are 3 oranges

Think about it: Simplifying makes ratios easier to understand and compare!

Example 2: Finding Equivalent Ratios

Problem: If the ratio of cats to dogs is 3:5 and there are 15 cats, how many dogs are there?

Solution: 25 dogs

Detailed Explanation:

• Original ratio: cats
  = 3:5
• We need: 15:?
• Find the multiplier: 15 ÷ 3 = 5
• Multiply the second term by the same factor: 5 × 5 = 25
• There are 25 dogs

Think about it: Whatever you do to one part of a ratio, you must do to all parts to maintain the relationship!

Example 3: Calculating a Unit Rate (Speed)

Problem: A car travels 180 kilometres in 2 hours. What is the unit rate in kilometres per hour?

Solution: 90 km/h

Detailed Explanation:

• Distance = 180 km, Time = 2 hours
• Unit rate = Distance ÷ Time
• 180 ÷ 2 = 90 km/h
• This means the car travels 90 kilometres every hour

Think about it: Unit rates give us a standard way to measure and compare speeds!

Example 4: Comparing Unit Rates (Shopping)

Problem: Shop A sells 500g of cheese for £6. Shop B sells 750g of cheese for £10. Which shop offers better value per gram?

Solution: Shop A offers better value

Detailed Explanation:

• Shop A: £6 for 500g → £6 ÷ 500 = £0.012 per gram (or £1.20 per 100g)
• Shop B: £10 for 750g → £10 ÷ 750 = £0.0133 per gram (or £1.33 per 100g)
• Shop A has the lower unit price, so it’s better value

Think about it: Converting to unit rates makes it easy to compare different package sizes!

Example 5: Sharing in a Given Ratio

Problem: £60 is to be shared between Ali and Ben in the ratio 2:3. How much does each person get?

Solution: Ali gets £24, Ben gets £36

Detailed Explanation:

• Ratio is 2:3, so total parts = 2 + 3 = 5
• Ali gets 2 parts out of 5: (2/5) × £60 = £24
• Ben gets 3 parts out of 5: (3/5) × £60 = £36
• Check: £24 + £36 = £60 ✓

Think about it: The ratio 2:3 means Ali gets 2/5 of the total and Ben gets 3/5!

Example 6: Scaling a Recipe

Problem: A recipe for 4 people uses 2 eggs and 300g of flour. How much is needed for 6 people?

Solution: 3 eggs and 450g of flour

Detailed Explanation:

• Current ratio: 4 people uses 2 eggs and 300g flour
• Scale factor: 6 ÷ 4 = 1.5 (multiply everything by 1.5)
• Eggs: 2 × 1.5 = 3 eggs
• Flour: 300 × 1.5 = 450g
• For 6 people, use 3 eggs and 450g flour

Think about it: Recipes are all about maintaining the same ratios of ingredients!

Example 7: Map Scale

Problem: On a map with scale 1:50,000, two cities are 8 cm apart. What is the actual distance in kilometres?

Solution: 4 km

Detailed Explanation:

• Scale 1:50,000 means 1 cm on map = 50,000 cm in reality
• Map distance: 8 cm
• Actual distance: 8 × 50,000 = 400,000 cm
• Convert to metres: 400,000 ÷ 100 = 4,000 m
• Convert to kilometres: 4,000 ÷ 1,000 = 4 km

Think about it: Map scales use ratios to represent large distances on small pieces of paper!

Common Misconceptions & How to Avoid Them

Misconception 1: “The order in a ratio doesn’t matter”

The Truth: Order is crucial! The ratio “boys to girls is 3:2” means something completely different from “girls to boys is 3:2.”

How to think about it correctly: Always pay attention to what’s being compared to what. Write it out in words first if needed: “boys

” or “girls.”

Misconception 2: “Ratios and differences are the same thing”

The Truth: If boys

is 3:2, this does NOT mean there’s 1 more boy than girls! It means for every 3 boys, there are 2 girls. The actual difference depends on the total numbers.

How to think about it correctly: Ratios compare relative sizes using multiplication/division, not addition/subtraction.

Misconception 3: “You can add ratios like fractions”

The Truth: 2:3 + 4:5 ≠ 6:8. Ratios represent relationships, not values you can simply add together.

How to think about it correctly: Ratios describe proportions. To combine situations, you need to think about the total quantities involved.

Misconception 4: “Unit rates are always per hour or per item”

The Truth: Unit rates can be “per” anything! Per second, per square metre, per person, per £1 spent, etc.

How to think about it correctly: A unit rate is any rate where the second quantity equals 1. It’s a tool for standardizing comparisons.

Common Errors to Watch Out For

Memory Aids & Tricks

The Colon Means “To”

When you see 3:4, read it as “3 to 4” - this helps you remember it’s a comparison.

The “For Every” Trick

Replace the colon with “for every” to understand relationships:

• 2:5 becomes “for every 2 of these, there are 5 of those”

Simplifying Ratios Like Fractions

Treat ratios just like fractions when simplifying: find the GCF and divide!

• 12:18 → divide by 6 → 2:3 (just like 12/18 = 2/3)

Unit Rate Formula

Unit Rate = Total Amount ÷ Number of Units

• £15 for 3 kg → £15 ÷ 3 = £5 per kg

The “Times Table” Check

If you can’t see the GCF, check if the numbers appear in the same times table:

• 6:9 are both in the 3 times table → GCF is 3

Practice Problems

Easy Level (Basic Ratios)

1. Simplify the ratio 8:12 Answer: 2:3 (divide both by 4: GCF is 4)

2. Write the ratio of 5 cats to 3 dogs using a colon. Answer: 5:3

3. If the ratio is 2:5 and the first quantity is 6, what’s the second quantity? Answer: 15 (multiply by 3: 6 is 2×3, so second is 5×3 = 15)

4. A recipe uses flour and sugar in the ratio 4:1. If you use 8 cups of flour, how much sugar? Answer: 2 cups (8 is 4×2, so sugar is 1×2 = 2)

Medium Level (Rates and Conversions)

5. A car travels 150 km in 3 hours. What is the speed in km/h? Answer: 50 km/h (150 ÷ 3 = 50)

6. 6 pens cost £4.50. What is the cost per pen? Answer: £0.75 per pen (£4.50 ÷ 6 = £0.75)

7. Juice costs £2.40 for 1.5 litres. What’s the price per litre? Answer: £1.60 per litre (£2.40 ÷ 1.5 = £1.60)

8. Share £40 in the ratio 3:5. How much does each person get? Answer: £15 and £25 (3+5=8 parts; £40÷8=£5 per part; 3×£5=£15, 5×£5=£25)

Challenge Level (Multi-Step Problems)

9. On a map with scale 1:25,000, two points are 6 cm apart. What’s the real distance in metres? Answer: 1,500 m or 1.5 km (6 × 25,000 = 150,000 cm = 1,500 m)

10. Which is better value: 400g for £3.20 or 650g for £4.80? Answer: 650g for £4.80 (400g: £0.008/g or £0.80/100g; 650g: £0.0074/g or £0.74/100g - lower is better)

Real-World Applications

In the Kitchen 👨‍🍳

Scenario: A smoothie recipe for 2 people uses 1 banana, 200ml milk, and 100g berries. You need to make smoothies for 5 people.

Solution:

• Scale factor: 5 ÷ 2 = 2.5
• Bananas: 1 × 2.5 = 2.5 ≈ 3 bananas (round up)
• Milk: 200 × 2.5 = 500 ml
• Berries: 100 × 2.5 = 250g

Why this matters: Cooking and baking rely entirely on maintaining correct ratios of ingredients. Understanding ratios means you can adjust any recipe for any number of people!

At the Shops 🛒

Scenario: Comparing two deals: Brand A - 6 yoghurts for £2.70, Brand B - 9 yoghurts for £3.60. Which is better value?

Solution:

• Brand A: £2.70 ÷ 6 = £0.45 per yoghurt
• Brand B: £3.60 ÷ 9 = £0.40 per yoghurt
• Brand B is better value!

Why this matters: Calculating unit prices helps you get the best value for money and avoid being fooled by bulk pricing that isn’t actually cheaper!

Planning Travel 🚗

Scenario: You’re driving 240 km. Your car uses petrol at a rate of 12 km per litre, and petrol costs £1.50 per litre. How much will the journey cost?

Solution:

• Petrol needed: 240 ÷ 12 = 20 litres
• Cost: 20 × £1.50 = £30

Why this matters: Understanding rates helps you budget for journeys, compare different vehicles’ fuel efficiency, and plan trips!

In Construction 🏗️

Scenario: A builder mixes cement, sand, and gravel in the ratio 1:2:3. To make 600 kg of concrete mix, how much of each is needed?

Solution:

• Total parts: 1 + 2 + 3 = 6
• Each part: 600 ÷ 6 = 100 kg
• Cement: 1 × 100 = 100 kg
• Sand: 2 × 100 = 200 kg
• Gravel: 3 × 100 = 300 kg

Why this matters: Construction, engineering, and manufacturing all depend on precise ratios to ensure quality and safety!

Understanding Maps 🗺️

Scenario: A map has scale 1:100,000. A walking route measures 4.5 cm on the map. How far is the actual walk in kilometres?

Solution:

• Actual distance: 4.5 × 100,000 = 450,000 cm
• Convert to metres: 450,000 ÷ 100 = 4,500 m
• Convert to kilometres: 4,500 ÷ 1,000 = 4.5 km

Why this matters: Map reading is an essential life skill for navigation, hiking, and understanding geography!

Study Tips for Mastering Ratios and Rates

1. Practice Identifying the Order

Always write out “A to B” or “A

” clearly. The order matters!

2. Learn to Spot the GCF Quickly

Practice finding greatest common factors - it makes simplifying ratios much faster.

3. Master Unit Rates First

Once you can calculate unit rates confidently, comparing options becomes easy.

4. Use Real Shopping Examples

When shopping with family, calculate and compare unit prices. It makes the maths relevant!

5. Draw Bar Models

Visual representations help! Draw bars to show ratio relationships - it makes problems clearer.

6. Check Your Work With Different Methods

If you solve using one method, verify with another. Ratios give you multiple approaches!

7. Practice Recipe Scaling

Try doubling, halving, or adjusting recipes. It’s practical and reinforces ratio skills!

How to Check Your Answers

1. Reverse the operation: If you divided to find a unit rate, multiply back to check
2. Check proportions: Ensure your answer maintains the same ratio relationship
3. Use equivalent fractions: Convert ratios to fractions to verify
4. Add up parts: In sharing problems, do your parts add up to the total?
5. Common sense check: Does your answer make sense in context?

Example: If the ratio is 2:3 and you calculated 10 and 12, something’s wrong! 10:12 simplifies to 5:6, not 2:3. The correct answer would be 10 and 15.

Extension Ideas for Fast Learners

• Explore inverse proportions (as one increases, the other decreases)
• Study golden ratio and its appearance in nature and art
• Calculate exchange rates between different currencies
• Work with compound ratios (combining multiple ratios)
• Investigate scale models and architectural plans
• Learn about gear ratios in machines and bicycles
• Study population density (ratio of people to land area)
• Explore aspect ratios in photography and film

Parent & Teacher Notes

Building Understanding: Ratios and rates are abstract concepts. Use concrete materials (counters, blocks) to show relationships physically before moving to numbers alone.

Common Struggles: If a student struggles, check if they:

• Understand basic fractions and multiplication/division
• Can identify and find the GCF of two numbers
• Understand the concept of “for every” relationships
• Can distinguish between additive and multiplicative thinking

Differentiation Tips:

• Struggling learners: Use visual models, start with simple whole number ratios (1:2, 1:3)
• On-track learners: Progress to three-part ratios and real-world rate problems
• Advanced learners: Challenge with complex proportions, scale drawings, and inverse relationships

Real-World Connection: Help students see ratios everywhere - in sports statistics, cooking, shopping, construction, and art. When students realize how practical these skills are, motivation increases dramatically!

Assessment Tips: Look for understanding beyond just calculation:

• Can students explain what a ratio represents?
• Do they maintain order and relationship?
• Can they choose appropriate methods for different problems?

Remember: Ratios and rates are fundamental to proportional reasoning - a critical thinking skill used in science, economics, engineering, and countless other fields. Mastering this topic opens doors to advanced mathematics and practical problem-solving! 🌟