Percentage

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Percentage meaning & conversions

x% of a number

\frac{x}{100} \times N

Turn the percent into a fraction first; cancel where you can.

Whole from a part

\text{whole} = \frac{\text{part} \times 100}{x}

x is the percent that the part stands for.

A as a percent of B

\frac{A}{B} \times 100\%

B is the quantity that follows the word 'of'.

Remaining part

\text{rest} = 100\% - \text{given}\%

Only when both percentages are of the same whole.

Percentage increase / decrease & successive change

Per cent change

\frac{\text{new} - \text{old}}{\text{old}} \times 100

Old value below the line.

Multiplier

\text{new} = \text{old} \times \frac{100 \pm x}{100}

Plus for a rise, minus for a fall.

Successive changes

a + b + \frac{ab}{100}

a and b carry their own signs.

Same x% up and down

\text{net loss} = \frac{x^2}{100}\%

x% rise followed by x% cut.

Fraction with both parts changed

\text{new} = \text{old} \times \frac{\text{top chip}}{\text{bottom chip}}

'x% more than' ↔ 'x% less than' & chains

x% more, reversed

\frac{100x}{100+x}\% \text{ less}

x% more than becomes this much less than.

x% less, reversed

\frac{100x}{100-x}\% \text{ more}

x% less than becomes this much more than.

Per cent of a per cent

\frac{a}{100} \times \frac{b}{100} = \frac{ab}{10000}

Each 'of' multiplies.

Population growth, depreciation & elections

Growth for n years

P\left(1 + \frac{r}{100}\right)^n

r% added every year, compounding.

Depreciation for n years

P\left(1 - \frac{r}{100}\right)^n

r% of value lost every year.

Value n years ago

\frac{\text{now}}{\left(1 \pm \frac{r}{100}\right)^n}

Divide by the chip power to go back.

Election margin

\text{margin votes} = (\text{winner}\% - \text{rival}\%) \times \text{valid votes}

Percents on valid votes only.

Marks, income–expenditure–savings & price–consumption

Maximum marks (two candidates)

M = \frac{(a+b) \times 100}{y - x}

a = shortfall below pass, b = marks above pass, x and y are the two percents.

Pass mark from failing

\text{pass} = \frac{x}{100}M + a

a is the shortfall.

Consumption cut after a price rise

\frac{100x}{100 + x}\%

Budget unchanged, price up x%.

Extra quantity after a price cut

\frac{100c}{100 - c}\%

Budget unchanged, price down c%.