10-Week Mathematics Catch-Up Programme (CAPS)

Week 1 – Number Skills & Algebra Foundations (Grade 8 Review)

Day 1: Diagnostic Assessment & Number Sense
Day 2: Factors, Multiples, Integers & BODMAS ✔
Day 3: Fractions, Decimals and Percentages (below)
Day 4: Algebra Basics
Day 5: Weekly Test

Week 1 – Day 3

Fractions, Decimals and Percentages

Duration: 1 hour

Learning Outcomes

By the end of today's lesson you should be able to:

Simplify fractions.
Convert between fractions, decimals and percentages.
Add, subtract, multiply and divide fractions.
Solve percentage increase and decrease problems.
Apply these skills to everyday situations.

Part A – Study Guide

1. What is a Fraction?

A fraction represents part of a whole.

Examples:

\frac12,\quad \frac34,\quad \frac58

The top number is called the numerator.

The bottom number is called the denominator.

2. Simplifying Fractions

Divide both the numerator and denominator by their highest common factor.

Example

Simplify

\frac{12}{18}

The HCF of 12 and 18 is 6.

\frac{12\div6}{18\div6} = \frac23

Answer:

\boxed{\frac23}

3. Equivalent Fractions

These fractions have the same value.

\frac12=\frac24=\frac36=\frac48

Multiply or divide both numbers by the same amount.

4. Converting Fractions to Decimals

Divide the numerator by the denominator.

Example

\frac34

3 ÷ 4 = 0.75

Answer

0.75

5. Converting Decimals to Percentages

Multiply by 100.

Example

0.65

=65%

6. Converting Percentages to Fractions

Write over 100 and simplify.

Example

40%

=\frac{40}{100} =\frac25

7. Adding Fractions

Fractions must have the same denominator.

Example

\frac14+\frac24 =\frac34

If denominators are different:

Example

\frac13+\frac16

Common denominator = 6

\frac26+\frac16 =\frac36 =\frac12

8. Subtracting Fractions

Example

\frac56-\frac26 =\frac36 =\frac12

9. Multiplying Fractions

Multiply across.

Example

\frac23\times\frac45 = \frac{8}{15}

10. Dividing Fractions

Turn the second fraction upside down.

Multiply.

Example

\frac34\div\frac25=\frac34\times\frac52=\frac{15}{8}= 1\frac78

11. Percentage Increase

Increase R250 by 20%.

Step 1

20% of 250

=0.20 ×250

=50

Step 2

250+50

=R300

12. Percentage Decrease

Reduce R600 by 15%.

15%

=0.15 ×600

=90

600−90

=R510

Common Mistakes

❌ Adding the denominators:

\frac12+\frac13=\frac25

Incorrect.

Always find a common denominator.

❌ Cancelling when adding fractions.

You only cancel when multiplying fractions.

❌ Forgetting to simplify your final answer.

Always check whether the fraction can be reduced.

Part B – Worked Examples

Example 1

Simplify

\frac{15}{25}

HCF = 5

\frac35

Example 2

Convert

\frac58

to a decimal.

5 ÷8

=0.625

Example 3

Convert

0.84

to a percentage.

84%

Example 4

Calculate

\frac25+\frac15

=\frac35

Example 5

Calculate

\frac34-\frac18

Convert

\frac34=\frac68

\frac68-\frac18=\frac58

Example 6

Calculate

\frac47\times\frac{14}{15}

Cancel before multiplying:

4 and 14 → divide both by 2

2 and 7 → divide both by 7

Result:

\frac{8}{15}

Example 7

A shirt costs R480. It is discounted by 25%.

How much do you pay?

25% of 480

=120

480−120

=R360

Part C – Practice Questions

Section A – Simplifying Fractions

Simplify $\frac{8}{12}$
Simplify $\frac{18}{24}$
Simplify $\frac{21}{28}$
Simplify $\frac{45}{60}$
Simplify $\frac{16}{40}$

Section B – Convert

$\frac12$ to a decimal
$\frac34$ to a decimal
$\frac15$ to a decimal
0.75 to a percentage
0.42 to a percentage
60% to a fraction
25% to a fraction
80% to a decimal
15% to a decimal
Write 0.125 as a fraction in its simplest form.

Section C – Fraction Calculations

$\frac14+\frac24$
$\frac35+\frac15$
$\frac78-\frac38$
$\frac56-\frac16$
$\frac12+\frac14$
$\frac23+\frac16$
$\frac45-\frac25$
$\frac34\times\frac25$
$\frac57\times\frac{14}{15}$
$\frac89\div\frac23$

Section D – Percentages

Find 15% of R240.
Increase R320 by 10%.
Decrease R500 by 30%.
A pair of shoes costs R850. They are discounted by 20%. What is the sale price?
A learner scored 36 out of 45 in a test. What percentage did they achieve?

Challenge Questions

Arrange these from smallest to largest:

\frac23,\quad0.70,\quad68%

A recipe needs $\frac34$ cup of milk. You only have a $\frac14$ -cup measuring cup. How many times must you fill it?
A jacket costs R1 200. It is reduced by 15% and then VAT of 15% is added to the reduced price. What is the final price?
Which is larger:

\frac79 \text{ or } 75%?

Explain your answer.

Simplify:

\frac23+\frac14-\frac16

Answers

$\frac23$
$\frac34$
$\frac34$
$\frac34$
$\frac25$
0.5
0.75
0.2
75%
42%
$\frac35$
$\frac14$
0.8
0.15
$\frac18$
$\frac34$
$\frac45$
$\frac12$
$\frac23$
$\frac34$
$\frac56$
$\frac25$
$\frac{3}{10}$
$\frac23$
$\frac43$ or $1\frac13$
R36
R352
R350
R680
80%
68%, 0.70, $\frac23$
3 times
R1 173.00
$\frac79$ (≈77.8%) is larger than 75%
$\frac34$

Parent's Note

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

Convert confidently between fractions, decimals, and percentages.
Perform all four operations with fractions.
Solve percentage increase and decrease problems without a calculator where appropriate.
Explain why each step is taken, not just perform the calculations.

For the remaining lessons, I'll keep this same format so each one is self-contained and ready to use in a one-hour study session. As we progress, I'll also begin introducing questions that resemble those found in Grade 10 and Grade 11 CAPS examinations, so the transition to higher-level mathematics feels gradual rather than overwhelming.