Working without a calculator

At the test centre you are given a notebook and pen for rough working. The notebook is not erasable; unlike a whiteboard, you cannot wipe it clean and reuse space. This matters more than it sounds: candidates who rely on crossing out and rewriting can run out of room under pressure, and the visual clutter of crossed-out working makes it easy to copy the wrong number into the next step.

Practise with the same setup at home. Take a standard A4 notebook, hold it in landscape orientation, and use a thin marker rather than a biro. Landscape orientation gives you wider lines that mirror the natural width of mathematical working. A thin marker forces you to write clearly and large; cramped pencil working is hard to read back under time pressure. Practise keeping your working structured: one calculation per row, clear column alignment for arithmetic.

When exact arithmetic looks messy, round each value to 1 significant figure and compute an estimate. ESAT answer options are usually spread far enough apart that the correct answer is immediately distinguishable from a 1 s.f. estimate.

Before estimating, check how spread out the options are. If the options are 1900, 2000, 2100, and 2200, a 1 s.f. estimate won't resolve them; you need the exact value. Always look at the option spread before deciding how much precision to aim for.

Division by a decimal less than 1 trips up many candidates. Remember: dividing by 0.5 is multiplying by 2; dividing by 0.25 is multiplying by 4; dividing by 0.1 is multiplying by 10. Always convert to whole-number division first.

Multiply by 5: multiply by 10, then halve. $36\times 5=360÷2=180$.

Multiply by 25: multiply by 100, then divide by 4. $36\times 25=3600÷4=900$.

Multiply by 9: multiply by 10, subtract the original. $36\times 9=360-36=324$.

Multiply by 11 (2-digit number): the middle digit is the sum of the two outer digits (with carry if $\ge 10$). $36\times 11=3,(3+6),6=396$.

Squaring numbers near a round number: $(n\pm k{)}^2=n^2\pm 2nk+k^2$.

$49^2=(50-1{)}^2=2500-100+1=2401$.

$98^2=(100-2{)}^2=10000-400+4=9604$.

Symmetric multiplication via difference of two squares: if two numbers are equidistant from a midpoint, $(m-k)(m+k)=m^2-k^2$.

$48\times 52=(50-2)(50+2)=2500-4=2496$.

$17\times 23=(20-3)(20+3)=400-9=391$.

Percentage shortcuts:

• 1% = divide by 100; 10% = divide by 10
• Build any percentage: 15% = 10% + half of 10%; 7.5% = half of 15%
• Compound changes: a 20% increase then a 10% decrease is ×1.2 × 0.9 = ×1.08, i.e., an 8% net increase
• "Percentage of" works both ways: 7% of 300 = 300% of 7 = 21 (useful when one number is easier to percentage)