Formulas: percent, ratio and scale, grade 6

A percent is a hundredth

$\displaystyle a$ percent are $\displaystyle a$ hundredths
$\displaystyle a\%$ $\displaystyle =$ $\displaystyle \frac{a}{100}$
The whole
$\displaystyle 100\%$ $\displaystyle =$ $\displaystyle 1$
A half
$\displaystyle 50\%$ $\displaystyle =$ $\displaystyle \frac{1}{2}$
A quarter
$\displaystyle 25\%$ $\displaystyle =$ $\displaystyle \frac{1}{4}$
Three quarters
$\displaystyle 75\%$ $\displaystyle =$ $\displaystyle \frac{3}{4}$
A tenth
$\displaystyle 10\%$ $\displaystyle =$ $\displaystyle \frac{1}{10}$
A fifth
$\displaystyle 20\%$ $\displaystyle =$ $\displaystyle \frac{1}{5}$
An eighth
$\displaystyle 12\frac{1}{2}\%$ $\displaystyle =$ $\displaystyle \frac{1}{8}$

$\displaystyle a\%$ of the whole quantity $\displaystyle b$
$\displaystyle a\%\times b$ $\displaystyle =$ $\displaystyle \frac{a}{100}\times b$
The percent and the quantity can change places: 18% of 50 is 50% of 18
$\displaystyle a\%\times b$ $\displaystyle =$ $\displaystyle b\%\times a$

What percent $\displaystyle a$ is of $\displaystyle b$ – reduce or expand to the denominator 100; the numerator $\displaystyle p$ is the number of percent
$\displaystyle \frac{a}{b}=\frac{p}{100}=p\%$

A ratio is a quotient – it can be written as a fraction
$\displaystyle a:b$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
Equal ratios – both numbers multiplied or divided by the same number $\displaystyle k$
$\displaystyle a:b$ $\displaystyle =$ $\displaystyle (a\times k):(b\times k)$ $\displaystyle k\neq0$
A quantity $\displaystyle c$ shared in the ratio $\displaystyle a:b$ – the first share ( $\displaystyle a$ parts out of $\displaystyle a+b$ )
$\displaystyle \frac{c}{a+b}\times a$
A quantity $\displaystyle c$ shared in the ratio $\displaystyle a:b$ – the second share ( $\displaystyle b$ parts out of $\displaystyle a+b$ )
$\displaystyle \frac{c}{a+b}\times b$

At the scale $\displaystyle 1:n$ – $\displaystyle a$ the length in the drawing, $\displaystyle b$ the real length, in the same unit
$\displaystyle b$ $\displaystyle =$ $\displaystyle a\times n$