Why This Matters for Grade 10#
Fractions, decimals and percentages appear everywhere in Grade 10:
- Algebra: Simplifying $\frac{2x^2}{4x}$ and working with algebraic fractions
- Finance: $A = P(1 + i)^n$ where rates move between decimal and percentage form
- Probability: $P(A)=\frac{n(A)}{n(S)}$ and converting to percentages
- Trigonometry: Values like $\sin 30° = \frac{1}{2}=0.5=50\%$
If you’re shaky on fractions, fix it now — every topic in Grade 10 (and beyond) depends on these skills.
1. BODMAS / Order of Operations#
Brackets → Orders (powers) → Division & Multiplication → Addition & Subtraction
Multiplication and division have equal priority — work left to right. Same for addition and subtraction.
Worked Example 1#
$3 + 2 \times 4 = 3 + 8 = 11$ (NOT $20$ — multiplication happens before addition)
Worked Example 2#
$2(3 + 4)^2 = 2(7)^2 = 2(49) = 98$
Step by step: brackets first → then the power → then multiplication.
Worked Example 3#
$24 \div 6 \times 2 = 4 \times 2 = 8$ (left to right, NOT $24 \div 12 = 2$)
The Biggest BODMAS Trap#
$$(2 + 3)^2 \neq 2^2 + 3^2$$$$(2 + 3)^2 = 5^2 = 25 \qquad \text{but} \qquad 2^2 + 3^2 = 4 + 9 = 13$$You cannot distribute a power over addition. This mistake reappears in exponents, surds, and algebraic expansion — fix it here.
2. Fraction Operations — Quick Reference#
For the full fraction toolkit (multiplying, dividing, LCD, complex fractions, cancelling rules), see:
👉 Appendix: Fractions & Algebraic Fractions Toolkit
Here’s a quick summary of the key operations:
Multiplying Fractions#
Multiply top × top, bottom × bottom. Simplify before or after.
$$\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$$Dividing Fractions#
Flip the second fraction and multiply (Keep-Change-Flip):
$$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$$Adding/Subtracting Fractions#
You MUST have a common denominator (LCD):
$$\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$$Worked Example 4 — Mixed Operations#
$$\frac{2}{3} + \frac{1}{2} \times \frac{3}{4}$$BODMAS: multiplication first, then addition.
$$= \frac{2}{3} + \frac{3}{8} = \frac{16}{24} + \frac{9}{24} = \frac{25}{24}$$The Cancelling Rule#
You can only cancel factors, never terms:
$$\frac{2x}{4x^2} = \frac{1}{2x} \quad ✓ \qquad \text{(cancelling the factor } 2x\text{)}$$$$\frac{x + 2}{x + 4} \neq \frac{2}{4} \quad ✗ \qquad \text{(you cannot cancel terms!)}$$3. Decimals ↔ Fractions ↔ Percentages#
The Conversion Triangle#
| From → To | Method | Example |
|---|---|---|
| Fraction → Decimal | Divide top by bottom | $\frac{3}{8} = 3 \div 8 = 0.375$ |
| Decimal → Percentage | Multiply by 100 | $0.375 \times 100 = 37.5\%$ |
| Percentage → Decimal | Divide by 100 | $37.5\% \div 100 = 0.375$ |
| Percentage → Fraction | Write over 100, simplify | $37.5\% = \frac{375}{1000} = \frac{3}{8}$ |
| Decimal → Fraction | Write as place value, simplify | $0.375 = \frac{375}{1000} = \frac{3}{8}$ |
Must-Know Conversions#
| Fraction | Decimal | Percentage |
|---|---|---|
| $\frac{1}{2}$ | $0.5$ | $50\%$ |
| $\frac{1}{4}$ | $0.25$ | $25\%$ |
| $\frac{3}{4}$ | $0.75$ | $75\%$ |
| $\frac{1}{5}$ | $0.2$ | $20\%$ |
| $\frac{1}{8}$ | $0.125$ | $12.5\%$ |
| $\frac{1}{3}$ | $0.\overline{3}$ | $33.\overline{3}\%$ |
| $\frac{2}{3}$ | $0.\overline{6}$ | $66.\overline{6}\%$ |
| $\frac{1}{10}$ | $0.1$ | $10\%$ |
4. Recurring Decimals → Fractions#
A recurring decimal is a decimal where one or more digits repeat forever.
Method#
- Let $x$ = the recurring decimal
- Multiply by the appropriate power of 10 to shift the repeating block
- Subtract to eliminate the repeating part
- Solve for $x$
Worked Example 5#
Convert $0.\overline{3}$ to a fraction.
Let $x = 0.333...$
$10x = 3.333...$
$10x - x = 3.333... - 0.333...$
$9x = 3$
$x = \frac{3}{9} = \frac{1}{3}$
Worked Example 6#
Convert $0.\overline{27}$ to a fraction.
Let $x = 0.272727...$
$100x = 27.272727...$
$100x - x = 27$
$99x = 27$
$x = \frac{27}{99} = \frac{3}{11}$
Worked Example 7#
Convert $0.1\overline{6}$ to a fraction.
Let $x = 0.1666...$
$10x = 1.666...$
$100x = 16.666...$
$100x - 10x = 16.666... - 1.666...$
$90x = 15$
$x = \frac{15}{90} = \frac{1}{6}$
5. Percentage of an Amount#
Worked Example 8#
$15\%$ of $R800 = \frac{15}{100} \times 800 = R120$
Worked Example 9 — Finding the Percentage#
A test is marked out of 80. A learner scores 52. What percentage is this?
$\frac{52}{80} \times 100 = 65\%$
Worked Example 10 — Finding the Original Amount#
After a 20% discount, a shirt costs R160. What was the original price?
$\text{Original} \times (1 - 0.20) = 160$
$\text{Original} \times 0.80 = 160$
$\text{Original} = \frac{160}{0.80} = R200$
6. Percentage Increase & Decrease#
The Multiplier Method#
- Increase by $p\%$: New = Original $\times (1 + \frac{p}{100})$
- Decrease by $p\%$: New = Original $\times (1 - \frac{p}{100})$
Worked Example 11#
A population of 5000 increases by 12%. Find the new population.
$5000 \times 1.12 = 5600$
Worked Example 12#
A car worth R180 000 depreciates by 15% per year. Find its value after 1 year.
$180\,000 \times 0.85 = R153\,000$
The Link to Finance#
This is exactly how interest and depreciation work:
- Simple interest: $A = P(1 + in)$ — add the same percentage each year
- Compound interest: $A = P(1 + i)^n$ — multiply by the same factor each year
This connection makes percentage calculations one of the most important skills in Grade 10.
🚨 Common Mistakes#
| Mistake | Why it’s wrong | Fix |
|---|---|---|
| Forgetting BODMAS | $2 + 3 \times 4 = 14$, not $20$ | Multiplication before addition |
| Mixing decimal and percentage | $0.08 = 8\%$, not $0.8\%$ | $\times 100$ to go decimal → percentage |
| “Percentage of” vs “percentage change” | $20\%$ of $50 = 10$, but a $20\%$ increase on $50 = 60$ | “Of” = multiply; “increase” = add to original |
| Cancelling terms instead of factors | $\frac{x+2}{x+4} \neq \frac{2}{4}$ | Only cancel common factors |
| $(a+b)^2 = a^2 + b^2$ | This is WRONG: $(3+4)^2 = 49 \neq 25$ | Powers don’t distribute over addition |
| Finding original after percentage change | $R160$ after $20\%$ off is NOT $160 \times 1.20 = 192$ | Divide by the multiplier: $\frac{160}{0.80} = 200$ |
💡 Pro Tip: The Interest Rate Conversion#
In Finance questions, you always need the rate as a decimal:
$8\% = 0.08$ (divide by 100)
$12.5\% = 0.125$
$0.5\% \text{ per month} = 0.005$
Get this conversion automatic — it’s needed in every Finance calculation.
🔗 Related Grade 10 topics:
- Finance — percentage increase/decrease IS interest and depreciation
- Probability — probability values are fractions that convert to percentages
- Algebra — algebraic fractions use all these fraction skills