# Proportions

Real numbers: $$a$$, $$b$$, $$c$$, $$d$$, $$m$$, $$n$$, $$p$$, $$q$$, $${\lambda _i}$$
1. A proportion is the equality of two ratios:
$${\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}$$
The terms $$a$$ and $$d$$ are called the extremes and $$b$$ and $$c$$ are called the means.
2. Means-Extremes (Cross-Products) property
The product of the means $$b$$ and $$c$$ is equal to the product of the extremes $$a$$ and $$d:$$$${\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\;$$ $$\Rightarrow \;ad = bc$$
3. Equivalent converse relation
If $$ad = bc$$, then $${\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}$$
4. Rule of Three
The Rule of Three is used to find the $$4$$th term in a proportion when the other $$3$$ terms are known:
If $${\large\frac{a}{b}\normalsize} = {\large\frac{c}{x}\normalsize},$$ then $$x = {\large\frac{bc}{a}\normalsize}.$$
5. Reciprocal property
If $${\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}$$, then $${\large\frac{b}{a}\normalsize} = {\large\frac{d}{c}\normalsize}$$
6. Extremes Switching property
If $${\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}$$, then $${\large\frac{d}{b}\normalsize} = {\large\frac{c}{a}\normalsize}$$
7. Means Switching property
If $${\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}$$, then $${\large\frac{a}{c}\normalsize} = {\large\frac{b}{d}\normalsize}$$
If $${\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}$$, then $${\large\frac{a + b}{b}\normalsize} = {\large\frac{c + d}{d}\normalsize}$$
If $${\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}$$, then $${\large\frac{a – b}{b}\normalsize} = {\large\frac{c – d}{d}\normalsize}$$
If $${\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}$$, then $${\large\frac{{ma + nb}}{{pa + qb}}\normalsize} = {\large\frac{{mc + nd}}{{pc + qd}}\normalsize}\;\;$$ $$\left( {{p^2} + {q^2} \ne 0} \right)$$
$${\large\frac{{{a_1}}}{{{b_1}}}\normalsize} = {\large\frac{{{a_2}}}{{{b_2}}}\normalsize} = \cdots = {\large\frac{{{a_n}}}{{{b_n}}}\normalsize} =$$ $${\large\frac{{{\lambda _1}{a_1} + {\lambda _2}{a_2} + \ldots + {\lambda _n}{a_n}}}{{{\lambda _1}{b_1} + {\lambda _2}{b_2} + \ldots + {\lambda _n}{b_n}}}\normalsize}$$,
where $${\lambda _1}{b_1} + {\lambda _2}{b_2} + \ldots$$ $$+\, {\lambda _n}{b_n} \ne 0$$