In which year, the expenditures on the social services have increased at the…

Concept — The rate of increase between two consecutive periods is always measured against the earlier value: rate of increase = (later value − earlier value) ÷ earlier value × 100. It is not the same as the absolute rise, because a smaller rise on a small base can still be a larger rate than a bigger rise on a large base. The base year's value must therefore sit in the denominator every time.

Application — The "Social Services" row of the table gives that head’s share of total government expenditure: 11.06 in 2007-08, 12.94 in 2008-09, 13.06 in 2009-10 and 14.02 in 2010-11. Since 2007-08 is the earliest column, the table supplies no earlier value for it and no rate of increase can be formed for that year. For the remaining three years the year-on-year rate works out as follows.

1. 2008-09: (12.94 − 11.06) ÷ 11.06 × 100 = 1.88 ÷ 11.06 × 100 ≈ 17.00%

2. 2009-10: (13.06 − 12.94) ÷ 12.94 × 100 = 0.12 ÷ 12.94 × 100 ≈ 0.93%

3. 2010-11: (14.02 − 13.06) ÷ 13.06 × 100 = 0.96 ÷ 13.06 × 100 ≈ 7.35%

Cross-check — Both quantities that drive the ratio favour 2008-09: it carries the largest rise (1.88 against 0.12 and 0.96) and it also sits on the smallest base (11.06), so a bigger numerator is divided by a smaller denominator. Reading "increase" as the plain rise in percentage points would point to the same year, so the two readings agree here and the ranking is stable.

Hence the expenditure on social services rose at the highest rate in 2008-09, at roughly 17 per cent over the preceding year.