XXIII Ratio and Proportion (Algebra)

Out of the two ways of comparing the magnitudes of two numbers, subtraction and division, only the latter is relevant for describing proportions.

Ratio Notations

$a \div b$, $\frac a b$, $a : b$

Rates remain the same regardless of their units, e.g.:
3 hours to 2 hours $= \frac 3 2$
9 metres to 6 metres $= \frac 3 2$
108° to 72° $= \frac 3 2$

so

$\frac a b = \frac{am}{bm}$

$\frac a b = \frac {a \div m}{b \div m}$

Numbers are proportionate when fractions are equal:

$\frac a b = \frac c d$

Continued Proportion

Given a sequence of

$\frac a b = \frac b c = \frac c d = \text{\textellipsis}$

Since the numbers are in a continued proportion they are all multiples of the same number $n$.

Example

$n = 3$
$\frac 2 6 = \frac {6}{18} = \frac {18}{54} = \text{/textellipsis}$

Mean Proportional

Given $a$, $b$ and $c$ such that

$\frac a b = \frac b c$

$\Rarr b$ is the mean proportional between $a$ and $c$. Therefore

$b^2 = ac$
$b = \sqrt[2]{ac}$

Other Theorems on Proportions

• Given $\frac a b = \frac c d$

since multiplying by the LCD yields

$\frac a b bd = \frac c d bd$

and the $b$ and $d$ cancel, it follows that

$ad=bc$

• If two fractions are equal, so are their reciprocals

$\frac a b = \frac c d \rarr \frac b a = \frac d c$

• Another pair of proportional fractions can be made by rearranging the fractions

$\frac 2 4 = \frac {6}{12} = \frac 1 2$

$\frac 2 6 = \frac {4}{12} = \frac 1 3$

Manipulation With a Common Value

$\frac a b = \frac c d = \frac e f = k$

$\frac 2 4 = \frac 3 6 = \frac 4 8 = 0.5$

so

$a = bk$ ($2 = 4 \times 0.5$)
$c = dk$ ($3 = 6 \times 0.5$)

Therefore

$a + c + e = bk + dk + fk$
$= k(b+d+f)$

$\frac {a+c+e}{b+d+f} = k$