P6 Mathematics - Term II

Topic 5: Fractions

Lesson 3: Ratio and Proportion

Duration: 45 minutes

Learning Objectives

By the end of this lesson, you will be able to:

• Define ratio and proportion
• Write ratios in different forms
• Identify and solve problems involving direct proportion
• Apply ratio and proportion to real-life situations in Uganda

Let's Count Together!

Look around our classroom:

Boys: Count them!
Girls: Count them!

Question: How can we compare the number of boys to girls?

What is a Ratio?

Ratio: A way of comparing two quantities

Example from our class:
If there are 15 boys and 20 girls...

We can say:

• "The ratio of boys to girls is 15 to 20"
• "The ratio of boys to girls is 15 : 20"

Three Ways to Write Ratios

Using words: "15 to 20"

Using a colon: 15 : 20

As a fraction: 15/20

All three mean the same thing!

More Ratio Examples

Windows to doors in this classroom

Desks to chairs

Textbooks to exercise books

What ratios can you find around you?

Simplifying Ratios

Just like fractions, ratios can be simplified!

Example: 4 : 6

Both numbers are divisible by 2

4 ÷ 2 : 6 ÷ 2 = 2 : 3

Simplified form: 2 : 3

Practice Simplifying

Simplify these ratios:

a) 6 : 9 = ?
b) 8 : 12 = ?
c) 15 : 20 = ?

Find the common factor and divide!

Ratios with Three Quantities

Example: 12 : 18 : 24

Find the common factor: All divisible by 6

12 ÷ 6 : 18 ÷ 6 : 24 ÷ 6

Simplified: 2 : 3 : 4

Ratios as Fractions

The ratio 2 : 3 tells us about parts

Total parts: 2 + 3 = 5 parts

First quantity: 2/5 of total
Second quantity: 3/5 of total

The ratio becomes fractions!

Example: Ratio to Fraction

15 learners: 6 boys and 9 girls

Ratio: 6 : 9 = 2 : 3 (simplified)

Boys as fraction: 6/15 = 2/5 of class
Girls as fraction: 9/15 = 3/5 of class

Check: 2/5 + 3/5 = 5/5 = 1 (whole class!)

Your Turn: Ratio Practice

In your exercise book:

1. Write the ratio of 5 red tops to 7 blue tops using colon notation
2. Simplify the ratio 10 : 15
3. If ratio of mangoes to oranges is 3 : 5, and there are 24 fruits total, how many are mangoes?

What is Proportion?

Proportion: When two ratios are equal

Example:
If 2 books cost 4,000 shillings,
then 6 books cost 12,000 shillings

The ratios are equal: 2:4,000 = 6:12,000

Understanding Proportion

Key idea: If one quantity increases, the other increases in the same ratio

More items = More cost (in same ratio)
More workers = More work done (usually!)
More ingredients = More food made

Solving Proportions: The Unitary Method

Method: Find the value of ONE unit first

Example:
If 3 kg of sugar costs 9,000 shillings,
how much does 5 kg cost?

Step 1: Find cost of 1 kg
Step 2: Find cost of 5 kg

Unitary Method: Step by Step

Problem: 3 kg = 9,000 shillings, find cost of 5 kg

Step 1: Cost of 1 kg
9,000 ÷ 3 = 3,000 shillings

Step 2: Cost of 5 kg
3,000 × 5 = 15,000 shillings

Answer: 5 kg costs 15,000 shillings

Another Unitary Example

Problem: 4 oranges cost 2,000 shillings. How much do 10 oranges cost?

Step 1: Cost of 1 orange
2,000 ÷ 4 = 500 shillings

Step 2: Cost of 10 oranges
500 × 10 = 5,000 shillings

Real-Life Ugandan Example

Recipe Problem:
A recipe for 4 people needs 2 cups of flour.
How much flour for 10 people?

Step 1: Flour per 1 person = 2 ÷ 4 = 0.5 cups
Step 2: Flour for 10 people = 0.5 × 10 = 5 cups

Your Turn: Proportion Practice

Solve in your exercise book:

Problem: If 5 pens cost 10,000 shillings, what do 8 pens cost?

Use the unitary method:

• Find cost of 1 pen
• Find cost of 8 pens

Cross Multiplication Method

Alternative method for proportions:

If 4/2,000 = 10/x

Cross multiply: 4 × x = 10 × 2,000
4x = 20,000
x = 5,000

Same answer, different method!

Summary: Ratio

Ratio: Compares two or more quantities

Written: Words, colon (:), or fraction

Simplified: Like fractions, divide by common factor

As fractions: Shows parts of the whole

Summary: Proportion

Proportion: When two ratios are equal

Unitary Method:

1. Find value of ONE unit
2. Multiply to get what you need

Use for: Pricing, recipes, rates, scaling

Real-Life Applications

Mixing juice: Concentrate to water (1:4 ratio)
Market prices: Proportional costs
Recipes: Scaling ingredients
Maps: Distances in proportion
Sharing: Dividing fairly by ratio

Quick Assessment

Answer in your exercise book:

1. Write ratio of 8 boys to 12 girls in simplest form
2. If ratio is 6 mangoes to 9 oranges, what fraction are mangoes?
3. If 3 books cost 12,000 shillings, how much do 5 books cost?

Challenge Problem

Ratio of Akello's money to Okello's money is 3:7

If Akello has 6,000 shillings, how much does Okello have?

Hint: Use the ratio to find the value of one part first!

Homework: Parts A and B

Part A: Writing and Simplifying Ratios

1. Write ratio of 10 chickens to 15 goats, then simplify
2. Simplify 12 : 18
3. In class of 40 learners, 16 are boys. Write ratio of boys to girls in simplest form
4. Farmer has 20 cows and 30 goats. What fraction are cows?

Homework: Parts B and C

Part B: Proportion Problems
5. If 4 notebooks cost 8,000 shillings, cost of 7 notebooks?
6. Car travels 80 km in 2 hours. How far in 5 hours?
7. Recipe for 8 people needs 6 cups flour. How much for 12 people?

Part C: Word Problem
8. Nakawa Market problem (oranges and mangoes)

Looking Ahead

Next Lesson:
We will learn about percentages and conversions

Connection:
Percentages are ratios out of 100!
(Per cent = per hundred)

Everything connects!

Key Takeaways

Ratio: Compares quantities (a : b)

Proportion: Equal ratios (if a:b = c:d)

Unitary Method: Find one, then multiply

Simplify: Always reduce to lowest terms

Think: Does my answer make sense?