10-Week Mathematics Catch-Up Programme (CAPS)

Week 2 – Day 4

Today's lesson introduces ratio, proportion, and rates—topics that are essential not only for Grade 9 but also for Grade 10 and Grade 11 Mathematics. They appear in financial mathematics, trigonometry (similar triangles), probability, maps and scale, and many real-world problem-solving questions. Show every step. Understanding why a method works is more important than memorising it.

Ratio, Proportion and Rates

Duration: 1 Hour

Grade Level: Grade 9 Foundation

Learning Outcomes

By the end of this lesson you should be able to:

• Simplify ratios.
• Compare quantities using ratios.
• Divide amounts in a given ratio.
• Solve problems involving direct proportion.
• Solve simple inverse proportion problems.
• Calculate unit rates and speed.

Part A – Study Guide

1. What is a Ratio?

A ratio compares two or more quantities.

Example:

There are 12 girls and 8 boys in a class.

The ratio of girls to boys is:

$12 : 8$

Simplify by dividing both numbers by 4:

$3 : 2$

Always write ratios in their simplest form.

2. Order Matters

The ratio of girls to boys is not the same as the ratio of boys to girls.

Example:

Girls : Boys = 3 : 2

Boys : Girls = 2 : 3

Always read the question carefully.

3. Dividing in a Given Ratio

Example:

Divide R240 in the ratio 5 : 3.

Step 1: Find the total number of parts.

$5 + 3 = 8$

Step 2: Find the value of one part.

$240 \div 8 = 30$

Step 3: Multiply.

First amount:

$5 \times 30 = 150$

Second amount:

$3 \times 30 = 90$

Answer:

R150 and R90

4. Direct Proportion

When one quantity increases, the other increases at the same rate.

Example:

If 4 books cost R120, what do 7 books cost?

Step 1: Find the cost of one book.

$120 \div 4 = 30$

Step 2: Multiply by 7.

$30 \times 7 = 210$

Answer:

R210

5. Inverse Proportion

When one quantity increases, the other decreases.

Example:

If 4 workers complete a job in 12 days, how long would 8 workers take?

Multiply workers × days:

$4 \times 12 = 48$

Divide by the new number of workers:

$48 \div 8 = 6$

Answer:

6 days

6. Unit Rates

A unit rate compares a quantity to one unit.

Examples:

• R60 for 3 kg = R20 per kg
• 180 km in 3 hours = 60 km/h

7. Speed

The formula is:

$\text{Speed}=\frac{\text{Distance}}{\text{Time}}$

Example:

A car travels 240 km in 3 hours.

$240 \div 3 = 80$

Answer:

80 km/h

Common Mistakes

Mistake 1

Only simplifying one part of a ratio.

Example:

12 : 8

❌ 6 : 8

Correct:

$3 : 2$

Mistake 2

Forgetting to add the parts before dividing.

Example:

Ratio 2 : 5

Total parts = 7, not 5.

Mistake 3

Confusing direct and inverse proportion.

More books → More cost (direct)

More workers → Less time (inverse)

Part B – Worked Examples

Example 1

Simplify:

18 : 30

Divide both by 6.

Answer:

3 : 5

Example 2

Divide R540 in the ratio 2 : 7.

Total parts:

$2+7=9$

One part:

$540\div9=60$

Amounts:

$2\times60=120$

$7\times60=420$

Answer:

R120 and R420

Example 3

If 6 T-shirts cost R450, how much do 10 T-shirts cost?

One T-shirt:

$450\div6=75$

Ten T-shirts:

$75\times10=750$

Answer:

R750

Example 4

Three workers paint a house in 18 days.

How many days will six workers need?

$3\times18=54$

$54\div6=9$

Answer:

9 days

Example 5

A cyclist travels 96 km in 4 hours.

Speed:

$96\div4=24$

Answer:

24 km/h

Example 6

A packet of 8 muffins costs R72.

What is the cost per muffin?

$72\div8=9$

Answer:

R9 per muffin

Part C – Practice Questions

Section A – Simplify the Ratios

1. 16 : 24

2. 15 : 45

3. 36 : 54

4. 28 : 42

5. 18 : 27

6. 40 : 60

7. 32 : 48

8. 25 : 35

9. 45 : 60

10. 56 : 72

Section B – Divide in the Given Ratio

11. Divide R180 in the ratio 2 : 4.

12. Divide R350 in the ratio 5 : 2.

13. Divide 72 sweets in the ratio 5 : 3.

14. Divide R540 in the ratio 4 : 5.

15. Divide 84 learners into groups in the ratio 4 : 3.

Section C – Direct Proportion

16. If 5 pens cost R80, how much do 8 pens cost?

17. If 12 litres of petrol cost R300, what do 20 litres cost?

18. If 7 notebooks cost R126, what do 15 notebooks cost?

19. If 9 kg of apples cost R225, what do 4 kg cost?

20. A train travels 360 km in 6 hours at a constant speed. How far will it travel in 9 hours?

Section D – Inverse Proportion

21. Four painters finish a job in 15 days. How long will 5 painters take?

22. Six taps fill a swimming pool in 8 hours. How long will 12 taps take?

23. Eight workers complete a task in 9 days. How long will 6 workers take?

24. Three machines produce an order in 20 hours. How long will 5 machines take?

25. Ten people can paint a hall in 6 hours. How long will 12 people take?

Section E – Rates

26. A car travels 420 km in 6 hours. Find its average speed.

27. A runner covers 15 km in 2.5 hours. Find the average speed.

28. A bag of 5 kg rice costs R140. What is the cost per kilogram?

29. Twelve cooldrinks cost R180. Find the cost of one cooldrink.

30. A taxi travels 180 km using 15 litres of fuel. How many kilometres does it travel per litre?

Challenge Questions

31. A father shares R1 800 between his three children in the ratio 2 : 3 : 5. How much does each child receive?

32. A recipe uses flour, sugar and butter in the ratio 4 : 2 : 1. If you use 800 g of flour, how much sugar and butter are needed?

33. If 15 bricks weigh 45 kg, what will 24 bricks weigh (assuming all bricks weigh the same)?

34. Eight workers can build a wall in 15 days. After 5 days, two workers leave the job. Assuming everyone works at the same rate, how many more days are needed to finish the wall?

35. A school has 420 learners. The ratio of boys to girls is 4 : 3. How many boys and how many girls are at the school?

Answers

Section A

1. 2 : 3

2. 1 : 3

3. 2 : 3

4. 2 : 3

5. 2 : 3

6. 2 : 3

7. 2 : 3

8. 5 : 7

9. 3 : 4

10. 7 : 9

Section B

11. R60 and R120

12. R250 and R100

13. 45 sweets and 27 sweets

14. R240 and R300

15. 48 learners and 36 learners

Section C

16. R128

17. R500

18. R270

19. R100

20. 540 km

Section D

21. 12 days

22. 4 hours

23. 12 days

24. 12 hours

25. 5 hours

Section E

26. 70 km/h

27. 6 km/h

28. R28/kg

29. R15 each

30. 12 km/L

Challenge Answers

31. Total parts = 10

• First child: R360
• Second child: R540
• Third child: R900

32. Flour : Sugar : Butter = 4 : 2 : 1

If 4 parts = 800 g, then 1 part = 200 g.

• Sugar = 400 g
• Butter = 200 g

33. One brick weighs:

$45 \div 15 = 3 \text{ kg}$

24 bricks weigh:

$24 \times 3 = 72 \text{ kg}$

34. Total work = $(8 \times 15 = 120)$ worker-days.

Work completed in first 5 days:

$8 \times 5 = 40$

Work remaining:

$120 - 40 = 80$

Workers remaining = 6

Days needed:

$80 \div 6 = 13\tfrac{1}{3}$

Answer: 13⅓ more days

35. Total parts = 7

One part:

$420 \div 7 = 60$

Boys:

$4 \times 60 = 240$

Girls:

$3 \times 60 = 180$

Parent's Notes

By the end of today's lesson, your student should be able to:

• Simplify and interpret ratios correctly.
• Divide quantities in a given ratio.
• Recognise when a problem involves direct or inverse proportion.
• Calculate unit rates such as cost per item or kilometres per litre.
• Apply proportional reasoning to real-life situations.