Speed & Mental Maths
No-calculator fluency: the four operations, fraction/decimal/percent conversion, percentage shortcuts.
This page is a sequence, not a document: try a question cold, learn the methods, drill until they’re automatic, then prove it under a clock. Skipping straight to reading is the least effective thing you can do here — that’s not opinion, it’s the testing effect.
Try one first
Before any teaching — a real question tells you honestly where you stand.
Calculate: 3 + 4 × 5
The methods
Every speed & mental maths pattern the tests use — the method in a few lines, then a worked example.
Order of operations
Apply order of operations: multiplication and division before addition and subtraction.
Worked example
Calculate: 6 + 9 × 4
Answer: 42
Multiplication before addition: 9 × 4 = 36, then 6 + 36 = 42. The 60 foil works left to right ((6+9)×4) — ignoring order of operations.
Mental percentages
Build the percentage from 10% (÷10) and 5% (half of 10%): e.g. 15% = 10% + 5%.
Worked example
What is 15% of 80?
Answer: 12
15% = 10% + 5% = 8 + 4 = 12. The 8 foil stops at 10%; 16 uses 20%; 1.2 slips the decimal.
Comparing magnitudes
Convert everything to decimals (or percentages) before comparing.
Worked example
Which of these is the largest? 0.6, 5/8, 62%, 0.59, 3/5
Answer: 5/8
Convert all to decimals: 0.6, 0.625, 0.62, 0.59, 0.6. The largest is 5/8 = 0.625. The 62% foil looks biggest as a percentage; 0.59 has the most digits but is smallest.
Estimation
Round to one or two leading digits, compute, and match the order of magnitude.
Worked example
Estimate 397 × 21. Which is closest?
Answer: 8,400
Round 397 to 400: 400 × 21 = 8,400 (actual 8,337). The 84,000 and 800 foils slip the magnitude by a factor of ten.
Fraction–decimal–percent
Divide numerator by denominator for the decimal; multiply by 100 for the percentage.
Worked example
What is 3/8 as a percentage?
Answer: 37.5%
3 ÷ 8 = 0.375 = 37.5%. The 0.375% foil forgets to multiply by 100; 62.5% is the complement (5/8); the others invert the fraction.
Furthest from the middle
Find the middle (median) value, then the option with the largest distance from it — not the largest value.
Worked example
Which number is furthest from the middle value? 2, 18, 20, 21, 30
Answer: 2
The middle (median) value is 20. Distances: 2→18, 18→2, 21→1, 30→10. The furthest is 2 (gap 18). The 30 foil takes the maximum, not the furthest.
Missing operator
Try each operator between the numbers and keep the one that makes the equation true.
Worked example
Which operator makes this true? 18 ? 6 = 12
Answer: −
18 − 6 = 12. (Deliberately 4 options — there are only four basic operators, a natural ceiling like the age problem.)
Rounding
Look at the digit below the rounding place: 5 or more rounds up, less rounds down.
Worked example
Round 6,847 to the nearest hundred.
Answer: 6,800
The tens/units are 47, below 50, so round down to 6,800. The 6,900 foil rounds the wrong way; the others round to the wrong place value.
Drill it
Instant feedback, named mistakes when you miss. Free questions to start; the full bank with premium.
Prove it under a clock
A skill isn't test-ready until it survives time pressure, mixed with everything else.
Your mastery map on the dashboard tracks this area from every answer you give — including the ones on this page.