🔢 Mathematics: Number, Decimals & Percentages
ACER Level 1 · No calculator · Estimation and number sense beat exact arithmetic
📋 What You’ll Learn in This Lesson
- Fraction-decimal-percentage conversion table (memorise these 12)
- Percentage shortcuts that eliminate long multiplication
- Reverse percentage method — working backwards from the final amount
- HCF and LCM fast methods
- 3 fully worked ACER-style examples
🚫 No calculator. ACER maths is designed so the arithmetic works out to clean numbers. If you reach a messy decimal mid-calculation, you have likely made a wrong turn. Estimate first — it catches ~80% of careless errors.
📊 Fraction-Decimal-Percentage — Memorise These 12
| Fraction |
Decimal |
Percentage |
| 1/2 |
0.5 |
50% |
| 1/3 |
0.333 |
33.3% |
| 2/3 |
0.667 |
66.7% |
| 1/4 |
0.25 |
25% |
| 3/4 |
0.75 |
75% |
| 1/5 |
0.2 |
20% |
| 1/8 |
0.125 |
12.5% |
| 3/8 |
0.375 |
37.5% |
| 1/10 |
0.1 |
10% |
| 1/20 |
0.05 |
5% |
| 1/25 |
0.04 |
4% |
| 1/100 |
0.01 |
1% |
⚡ Percentage Shortcuts — No Long Multiplication Needed
| Percentage |
Method |
Example |
| 10% |
Divide by 10 |
10% of 340 = 34 |
| 5% |
Half of 10% |
5% of 340 = 17 |
| 1% |
Divide by 100 |
1% of 340 = 3.4 |
| 15% |
10% + 5% |
15% of 340 = 34 + 17 = 51 |
| 25% |
Divide by 4 |
25% of 340 = 85 |
| 20% |
Divide by 5 (or double 10%) |
20% of 340 = 68 |
| 12.5% |
Half of 25% |
12.5% of 340 = 42.5 |
| Any % |
% / 100 x number |
32% of 250: 0.32 x 250 = 80 |
Reverse Percentage: Original = Final amount ÷ (1 + rate)
Use when the price AFTER an increase or decrease is given
📝 Worked Example — Reverse Percentage — ACER Classic
Question
A price increased by 15% to $69. What was the original price?
Step 1
The final price ($69) = 115% of the original. So original x 1.15 = 69.
Step 2
Original = 69 / 1.15
Step 3
69 / 1.15 = 60. Check: $60 x 1.15 = $60 + $9 = $69 ✓
✅ Answer: $60
📝 Worked Example — LCM for Adding Fractions
Question
Calculate 3/8 + 5/12
Step 1
Find LCM(8, 12). Multiples of 8: 8, 16, 24. Multiples of 12: 12, 24. LCM = 24.
Step 2
Convert: 3/8 = 9/24 (x3). Convert: 5/12 = 10/24 (x2).
Step 3
9/24 + 10/24 = 19/24. Cannot simplify (19 is prime).
✅ Answer: 19/24
📝 Worked Example — Standard Form
Question
Express 3.6 x 10^4 as an ordinary number.
Step 2
3.6 x 10,000 = 36,000
✅ Answer: 36,000
⭐ ACER Examiner’s Trick: When a percentage question gives you the final value (after increase/decrease), divide by (1 + rate). When it asks for the percentage change: % change = (new – old) / old x 100. These two operations are reversed and ACER tests both.
📌 Quick Reference Card — Number, Decimals & Percentages
| % shortcut: 10% |
Divide by 10. Then build: 5% = half, 15% = 10%+5%, 20% = 2×10% |
| Fraction addition |
LCD method: convert to same denominator, then add numerators |
| Reverse % |
Original = Final / (1 + rate). E.g. after 15% rise: / 1.15 |
| Standard form |
3.6 x 10^4 = move decimal 4 right = 36,000 |
| Estimate first |
If answer options spread wide (e.g. 40, 80, 160, 320), estimate to eliminate 3 |
| Prime numbers |
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 |
