60-second revision
- Order: Vinculum (bar), Brackets, Of, Division, Multiplication, Addition, Subtraction (VBODMAS).
- "Of" is solved before division: 12 ÷ 3 of 2 = 12 ÷ 6 = 2.
- Division and multiplication are worked left to right when "of" is not involved: 48 ÷ 6 × 4 = 32.
- a² - b² = (a + b)(a - b) turns big squares into one easy product.
- Near-100 trick: 96 × 104 = 100² - 4² = 9,984.
- Multiply by 25: multiply by 100 and divide by 4. Multiply by 5: multiply by 10 and halve.
- In approximation, round each number to the nearest friendly value, then pick the closest option.
The order of operations (VBODMAS)
| Letter | Stands for | Example |
|---|---|---|
| V | Vinculum (bar over numbers) | a bar over 7 - 3 means (7 - 3) |
| B | Brackets: ( ), then { }, then [ ] | inner bracket first |
| O | Of (multiplication written as "of") | 1/4 of 20 = 5 |
| D | Division | left to right with M |
| M | Multiplication | left to right with D |
| A | Addition | left to right with S |
| S | Subtraction | left to right with A |
Division and multiplication have equal rank, so work them from left to right. The same applies to addition and subtraction. Worked example: 36 - 6 × 2 + 18 ÷ 3 = 36 - 12 + 6 = 30.
Algebraic identities that save time
| Identity | Use it for |
|---|---|
| (a + b)² = a² + 2ab + b² | 103² = 10,000 + 600 + 9 = 10,609 |
| (a - b)² = a² - 2ab + b² | 97² = 10,000 - 600 + 9 = 9,409 |
| a² - b² = (a + b)(a - b) | 47 × 53 = 50² - 3² = 2,491 |
| (a + b)³ and (a - b)³ | cube of numbers near a round base |
| a³ + b³ = (a + b)(a² - ab + b²) | fractions like (a³ + b³)/(a² - ab + b²) = a + b |
Multiplication shortcuts
- By 5: multiply by 10, then halve. 486 × 5 = 4,860 ÷ 2 = 2,430.
- By 25: multiply by 100, then divide by 4. 64 × 25 = 6,400 ÷ 4 = 1,600.
- By 11 (two-digit number): write the digits apart and put their sum between them. 53 × 11 = 5 (5 + 3) 3 = 583; carry if the sum is 10 or more.
- Near a base of 1,000: 998 × 997. Deficits are 2 and 3. Left part 998 - 3 = 995, right part 2 × 3 = 006, so 995,006.
- Two numbers equally spaced around a round number: 96 × 104 = 100² - 4² = 9,984.
Percentages and fractions inside simplification
Swap common percentages for fractions before calculating: 12.5% = 1/8, 37.5% = 3/8, 62.5% = 5/8, 16.67% = 1/6, 33.33% = 1/3, 66.67% = 2/3, 83.33% = 5/6. Example: 37.5% of 816 = 3/8 × 816 = 306. Also remember x% of y = y% of x: 18% of 50 = 50% of 18 = 9.
Chains of fractions multiply directly: 2/3 of 3/5 of 450 = 2/5 of 450 = 180. Cancel before you multiply.
Finding the missing number ("?")
Simplify the known side fully, then undo operations on the "?" side in reverse order. Example: ? ÷ 8 + 14 = 30% of 120. Right side = 36, so ? ÷ 8 = 22 and ? = 176.
Approximation: estimate, do not calculate
- Round each number to the nearest easy value: 49.86% to 50%, 803 to 800, 24.97 to 25, 6.02 to 6.
- Round square roots to the nearest perfect square: sqrt 1,295 is close to sqrt 1,296 = 36.
- Keep decimals that matter: 7.5 × 7.5 should stay 56.25, not 64.
- Look at the gap between options. If options are far apart, rough rounding is enough; if they are close, round more carefully.
- Round one number up and the other down in a product to cancel errors (e.g. 49 × 61 ~ 50 × 60).
Worked example: 49.86% of 803 + 15.02 × 5.97 is about 50% of 800 + 15 × 6 = 400 + 90 = 490.
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