Percentage
🔒 Log in to trackhigh importance~2 Q in Tier 121 formulas⚡ 15 shortcuts5 subtopics
Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.
Percentage meaning & conversions
Per cent of a number
x\% \text{ of } N = \frac{xN}{100}
Fraction to per cent
\frac{p}{q} \times 100\%
Per cent to fraction
x\% = \frac{x}{100}
Commutativity
x\% \text{ of } y = y\% \text{ of } x
Recovering the base
N = \frac{\text{part} \times 100}{\text{percentage}}
Percentage increase / decrease & successive change
Percentage change
\%\ \text{change} = \frac{\text{final} - \text{initial}}{\text{initial}} \times 100
Successive change
a + b + \frac{ab}{100}
use signs: + for increase, − for decrease
Same % up and down
+x\% \text{ then } -x\% \Rightarrow \text{net} = -\frac{x^2}{100}\%
Multiplier form
\text{final} = \text{initial} \times \frac{100 + a}{100} \times \frac{100 - b}{100}
Original value back
\text{initial} = \text{final} \times \frac{100}{100 + a}
'x% more than' ↔ 'x% less than' & chains
More → less
A = B\left(1 + \frac{x}{100}\right) \Rightarrow B = A\left(\frac{100}{100+x}\right),\ \text{less by } \frac{100x}{100+x}\%
Less → more
A = B\left(1 - \frac{x}{100}\right) \Rightarrow \text{B is } \frac{100x}{100-x}\% \text{ more than A}
Chain of percentages
A = \frac{a}{100}B,\ B = \frac{b}{100}C \Rightarrow A = \frac{ab}{100}\% \text{ of } C
Population growth, depreciation & elections
Growth / decay
P_n = P\left(1 \pm \frac{r}{100}\right)^n
Two different rates
P_2 = P\left(1 + \frac{r_1}{100}\right)\left(1 + \frac{r_2}{100}\right)
Depreciated value back
P = \text{present} \times \left(\frac{100}{100-r}\right)^n
Election margin
\text{margin} = (\text{winner share} - \text{loser share}) \times \text{valid votes}
Marks, income–expenditure–savings & price–consumption
Pass marks equation
\frac{pM}{100} + f = \frac{qM}{100} - e
Savings identity
\text{savings} = \text{income} - \text{expenditure}
Price up r%, expenditure fixed
\text{reduce consumption by } \frac{100r}{100+r}\%
Price cut & extra quantity
\text{reduced price} = \frac{r\% \text{ of outlay}}{\text{extra quantity}},\quad \text{old price} = \frac{\text{reduced price}}{1 - r/100}