Calculating quickly by hand

Why hand calculation is a skill of its own

Calculators are not permitted in the Online Bocconi Test, and Bocconi describes the numerical reasoning calculations as simple. Simple does not mean instant: a multiplication done digit by digit takes time you could spend reading the data. This lesson makes the common calculations short and shows how to check them quickly.

Look at the options before you calculate

The options tell you how precise you need to be. If they are 6%, 12%, 25% and 40%, a rough estimate decides the question; if two of them are 8.75% and 9%, you need the exact value. Decide the precision first, then calculate only as far as you need.

Percentages by pieces

Build any percentage from easy pieces: 10% (move the decimal point one place), 5% (half of 10%), 1% (move it two places).

• 18% of 350 = 10% (35) + 5% (17.5) + 3% ($3 \times 3.5 = 10.5$) = 63.
• 2.5% of 36,000 = $2.5 \times 360 = 900$.

A useful swap: $x$% of $y$ equals $y$% of $x$. So 16% of 25 is 25% of 16, which is 4.

Fractions worth knowing as percentages

Fraction	Percent	Fraction	Percent
1/2	50%	1/8	12.5%
1/3	33.3%	1/10	10%
1/4	25%	1/12	8.3%
1/5	20%	1/20	5%
1/6	16.7%	1/25	4%

With these, 87.5% of 640 is 640 minus an eighth of 640: $640 - 80 = 560$. And dividing by 0.125 is the same as multiplying by 8.

Multiplying and dividing by splitting

Split one factor into easy parts: $24 \times 15 = 24 \times 10 + 24 \times 5 = 240 + 120 = 360$. Or use a nearby round number: $49 \times 18 = 50 \times 18 - 18 = 900 - 18 = 882$.

For division, simplify first: $\frac{40\,500}{450} = \frac{4\,050}{45} = 90$. Cancel common factors before dividing. To divide by 5, double and divide by 10.

Comparing fractions without dividing

To tell which of $\frac{a}{b}$ and $\frac{c}{d}$ is larger (all numbers positive), compare $a \times d$ with $c \times b$. The larger product goes with the larger fraction. This is often faster than turning both into decimals, and it is exact.

Percentage change in one line

Percentage change = change ÷ starting value. Work out the change first, then look for an easy relationship with the base: an increase of 45 on 900 is $\frac{45}{900} = \frac{1}{20} = 5\%$.

Averages from a guess

To average a list of similar numbers, choose a round guess, add up the differences from it, and correct. For 96, 103, 99 and 106 with a guess of 100, the differences are −4, +3, −1 and +6, which sum to +4; spread over 4 numbers that is +1, so the mean is 101.

Check the size of the answer

Before choosing, ask whether the answer has the right order of magnitude. A 2.5% charge on €36,000 cannot be €9,000 (that would be 25%). A percentage below 100% of a number must be smaller than the number; 5% growth on a million is 50,000, not 500,000. Most hand-calculation errors are misplaced decimal points, and this check catches them.

Round late, and round both numbers the same way

If you must round, round once, near the end, and when rounding two numbers in a division, round both the same way (both up or both down) so the errors partly cancel. Rounding the numerator up and the denominator down pushes the result off twice.