Formulas: ratio, proportion and percent, grade 8

Ratio and proportion

A ratio is also a fraction
$\displaystyle a:b$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
Cross-multiplying a proportion
$\displaystyle a\cdot d$ $\displaystyle =$ $\displaystyle b\cdot c$ $\displaystyle \frac{a}{b}=\frac{c}{d}$
Quantities in the ratio $\displaystyle a:b$ – $\displaystyle k$ the size of one part
$\displaystyle ak:bk$ $\displaystyle =$ $\displaystyle a:b$
Directly proportional quantities – the ratio $\displaystyle k$ stays the same (the unit price, for one)
$\displaystyle \frac{y}{x}=k$

At the scale $\displaystyle 1:n$ – $\displaystyle a$ the length in the drawing, $\displaystyle b$ the real length, in the same unit
$\displaystyle b$ $\displaystyle =$ $\displaystyle a\cdot n$
An area at the scale $\displaystyle 1:n$ – it grows $\displaystyle n^2$ times, not $\displaystyle n$ times
$\displaystyle S_b$ $\displaystyle =$ $\displaystyle S_a\cdot n^2$

$\displaystyle p$ percent are $\displaystyle p$ hundredths
$\displaystyle p\%$ $\displaystyle =$ $\displaystyle \frac{p}{100}$
A rise of $\displaystyle p\%$ – $\displaystyle a$ the amount before the change, $\displaystyle b$ the amount after it
$\displaystyle b$ $\displaystyle =$ $\displaystyle (100\%+p\%)\cdot a$
A fall of $\displaystyle p\%$ – $\displaystyle a$ the amount before the change, $\displaystyle b$ the amount after it
$\displaystyle b$ $\displaystyle =$ $\displaystyle (100\%-p\%)\cdot a$
The percent change – a part of the amount before the change
$\displaystyle p\%=\frac{b-a}{a}$
Two changes one after the other – a rise of $\displaystyle p\%$ , then a rise of $\displaystyle q\%$ (for a fall: minus instead of plus)
$\displaystyle c$ $\displaystyle =$ $\displaystyle (100\%+p\%)\cdot(100\%+q\%)\cdot a$