TMUA without a calculator: mental maths tips

Why exact answers matter

Without a calculator, answers are often best kept exact: fractions, surds and multiples of pi. A long decimal in your working is a prompt to check your route. Practise leaving answers as 3/4, 2 sqrt(3) or 5 pi, and learn to compare exact values rather than convert them.

Quick arithmetic habits

• Percentages: find 10% and 5% and combine. For 15% of 240, 10% is 24 and 5% is 12, so the answer is 36.
• Squares: use (a - b)^2. For 19^2, write (20 - 1)^2 = 400 - 40 + 1 = 361.
• Near-round numbers: 99 x 101 = (100 - 1)(100 + 1) = 9999.
• Learn squares to 20, cubes to 10, and powers of 2 to 2^10 = 1024.

Fractions

Find a common denominator and keep it simple: 3/4 + 5/6 = 9/12 + 10/12 = 19/12. Cancel common factors before multiplying, not afterwards, as it keeps numbers small. If you can cancel a fraction, check for a shared factor in every term.

Surds

• Simplify using square factors: sqrt(48) = sqrt(16 x 3) = 4 sqrt(3).
• Difference of two squares: (3 + sqrt(2))(3 - sqrt(2)) = 9 - 2 = 7.
• Rationalise: 1 / (sqrt(5) - 1) = (sqrt(5) + 1) / ((sqrt(5) - 1)(sqrt(5) + 1)) = (sqrt(5) + 1) / 4.

Estimating

When options are far apart, a rough size is enough. Since 49 < 50 < 64, sqrt(50) lies between 7 and 8. Bounds like these can rule out options without finishing the calculation.

Algebra without slips

Write each step on its own line. Keep brackets until you are sure of the signs, and check at the end by substituting a simple value. Before expanding a large expression, look for a common factor you can take out, as that often keeps the numbers small.

How to practise

Spend ten minutes at the start of a study session on arithmetic and surd drills with no calculator. Then do questions from the <a href="/tmua/topics/">topic pages</a> in the same way. If you reach for the calculator by habit at home, put it in another room while you practise.