Revision: numerical reasoning

What the area tests

The standard test has 6 numerical reasoning questions out of 50, 12% of the test. Bocconi says advanced mathematics is not required and the calculations are simple; the challenge is often to read a chart or table correctly. No calculator is allowed. 75 minutes for 50 questions is 90 seconds a question on average; our pacing suggestion, not a Bocconi figure, is about 60 seconds for a numerical reasoning item, and estimation is what makes that possible. Estimation settles many items, but not all: when two options are close, compute exactly.

A four-step extraction protocol

Step	Check	Why
1. Units	Thousands, millions, or percentage shares?	A figure in €M read as € is off by a factor of a million
2. Axes	Two vertical axes? Which series uses which?	Reading a series against the wrong scale gives a wrong value
3. Footnotes	Exclusions, currency conversions, base years	A footnote can change what a column means
4. Filter	Which rows and columns does the question name?	Most cells are irrelevant to any one question

Shares of shares and pie charts

A share of a share multiplies: if equities are 45% of a fund and technology is 20% of equities, technology is $0.45 \times 0.20 = 0.09$, 9% of the fund. In a pie chart the whole is $360^\circ$, so $1\% = 3.6^\circ$: a 15% slice is $54^\circ$, a 25% slice $90^\circ$, and a $108^\circ$ slice is $108 \div 3.6 = 30\%$.

Index numbers

An index sets a base year to 100. Two series both indexed to 100 in 2020 show growth, not size: if Country A is at 140 in 2025 and Country B at 125, A grew 40% and B 25% since 2020, but which economy is larger cannot be told. Growth between two later years is measured from the earlier value: from 120 to 150 is $\frac{150 - 120}{120} = \frac{30}{120} = 25\%$, not 30%. The 30 is index points.

Estimation techniques

• Round to two significant figures. $48\,912 \div 7\,034 \approx 49\,000 \div 7\,000 = 7$ (exactly, about 6.95).
• Round both terms the same way. A fraction keeps its value when top and bottom are scaled by the same factor, so if you round one up, round the other up by about the same percentage.
• Build percentages from 10%, 5% and 1%. 10% of 1,105 is 110.5 and 1% is about 11, so 143 is about $10\% + 3\% = 13\%$ of 1,105 (exactly 12.94%).

Reciprocal benchmarks

Fraction	Percent	Fraction	Percent
$\frac{1}{6}$	16.67%	$\frac{1}{11}$	9.09% ($\frac{2}{11} = 18.18\%$, $\frac{3}{11} = 27.27\%$)
$\frac{1}{7}$	14.29% ($\frac{2}{7} \approx 28.57\%$)	$\frac{1}{12}$	8.33%
$\frac{1}{8}$	12.5% ($\frac{3}{8} = 37.5\%$)	$\frac{1}{15}$	6.67%
$\frac{1}{9}$	11.11% ($\frac{2}{9} = 22.22\%$)	$\frac{1}{16}$	6.25%

Percentages are rounded to two decimals.

Percent change and percentage points

	Relative change	Percentage points
Formula	$\frac{\text{new} - \text{old}}{\text{old}} \times 100\%$	new % minus old %
Tax rate from 20% to 25%	$\frac{5}{20} = +25\%$	+5 points, or +500 basis points

The base of a relative change is always the starting value: the one after "from" or "than". One percentage point is 100 basis points, so one basis point is 0.01 of a percentage point.

Recovery is asymmetric

Change	Multiplier	Reverse multiplier	Change needed to get back
−10%	$0.90$	$\frac{10}{9} \approx 1.111$	+11.11%
−20%	$0.80$	$\frac{5}{4} = 1.25$	+25%
−25%	$0.75$	$\frac{4}{3} \approx 1.333$	+33.33%
+25%	$1.25$	$\frac{4}{5} = 0.80$	−20%

Products, quotients and weights

• Revenue = price × volume. Price up 15% and volume down 10% gives $1.15 \times 0.90 = 1.035$: revenue up 3.5%, not 5%.
• Real change = nominal multiplier ÷ price multiplier. A salary up 10% when prices rise 10% gives $\frac{1.10}{1.10} = 1$: no change in purchasing power.
• A weighted change is the weighted average of the parts' changes. A portfolio of 60% bonds returning +5% and 40% shares returning −10% returns $0.6 \times 5\% + 0.4 \times (-10\%) = 3\% - 4\% = -1\%$.
• A reverse base divides by the combined multiplier, never subtracts the percents.