Percentages

1.1 The core percentage formula

A percentage expresses how much A is relative to B, scaled to 100:

Percentage = (A ÷ B) × 100

A and B can be any two quantities. In practice it is almost always more convenient to work with the rate, the decimal equivalent of a percentage:

rate = Percentage ÷ 100

Converting percentages to rates: 35% → 0.35, 7.5% → 0.075, 110% → 1.10. The three working forms you will use constantly:

Percentage = (Part ÷ Total) × 100

1.2 Part-to-whole: three problem types

Every percentage question gives you two of the three quantities and asks for the third. Identify what is given and select the matching rearrangement.

Given rate & total   →   Part  = rate × Total
Given part & rate    →   Total = Part ÷ rate
Given part & total   →   rate  = Part ÷ Total

Example — find the Part: a region has 3 400 000 inhabitants and a poverty rate of 22%. At-risk persons = 0.22 × 3 400 000 = 748 000. Example — find the Total: 84 600 people represent 6% of a city's workforce. Workforce = 84 600 ÷ 0.06 = 1 410 000. Example — find the rate: 156 000 out of 780 000 workers are self-employed. Rate = 156 000 ÷ 780 000 = 0.20 = 20%.

1.3 Complementary percentages

All categories in a complete distribution must sum to 100%. When one category is missing, subtract the sum of all known shares from 100%.

Missing share = 100% − sum of all known shares

Example: energy mix shows Nuclear 31%, Gas 24%, Renewables 28%, Coal 9%. The remaining 'Other' = 100 − 31 − 24 − 28 − 9 = 8%.

1.4 Percentages above 100%

A percentage can exceed 100% when A is larger than B. This is common when comparing a value to a benchmark it exceeds.

Percentage = (A ÷ B) × 100

Example: a factory produces 780 units against a target of 600. Output as % of target = (780 ÷ 600) × 100 = 130%. This means output was 30% above target.

1.5 Percentage increase applied to a rate

When a rate itself changes by a percentage, apply that percentage to the rate — do NOT add percentage points.

New rate = Old rate × (1 + change/100)

Example: an interest rate of 4% is increased by 15%. Increase in pp = 4% × 0.15 = 0.6 pp. New rate = 4.6% (NOT 4% + 15% = 19%).

1.6 Progressive tax brackets

In a multi-bracket system, each rate applies only to the income slice within that bracket. Total tax is the sum of each bracket's contribution.

Tax per bracket = bracket income slice × bracket rate
Total tax       = sum of all bracket taxes

Example: brackets 0% up to €15 000, then 22% on €15 001–€45 000, then 38% above €45 000. For income of €60 000: Bracket 1 = €0. Bracket 2 = €30 000 × 0.22 = €6 600. Bracket 3 = €15 000 × 0.38 = €5 700. Total tax = €12 300.