Real numbers: \(a\), \(b\), \(c\), \(d\), \(m\), \(n\), \(p\), \(q\), \({\lambda _i}\)

- 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. - Means-Extremes (Cross-Products) property

The product of the means \(b\) and \(c\) is equal to the product of the extremes \(a\) and \(d:\)\({\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\;\) \(\Rightarrow \;ad = bc\) - Equivalent converse relation

If \(ad = bc\), then \({\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\) - 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 \(x = {\large\frac{bc}{a}\normalsize}.\) - Reciprocal property

If \({\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\), then \({\large\frac{b}{a}\normalsize} = {\large\frac{d}{c}\normalsize}\) - Extremes Switching property

If \({\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\), then \({\large\frac{d}{b}\normalsize} = {\large\frac{c}{a}\normalsize}\) - Means Switching property

If \({\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\), then \({\large\frac{a}{c}\normalsize} = {\large\frac{b}{d}\normalsize}\) - Denominator addition property

If \({\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\), then \({\large\frac{a + b}{b}\normalsize} = {\large\frac{c + d}{d}\normalsize}\) - Denominator subtraction property

If \({\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\), then \({\large\frac{a – b}{b}\normalsize} = {\large\frac{c – d}{d}\normalsize}\) - General numerator-denominator addition property

If \({\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\), then \({\large\frac{{ma + nb}}{{pa + qb}}\normalsize} = {\large\frac{{mc + nd}}{{pc + qd}}\normalsize}\;\;\) \(\left( {{p^2} + {q^2} \ne 0} \right)\) - Equal fractions property

\({\large\frac{{{a_1}}}{{{b_1}}}\normalsize} = {\large\frac{{{a_2}}}{{{b_2}}}\normalsize} = \cdots = {\large\frac{{{a_n}}}{{{b_n}}}\normalsize} =\) \( {\large\frac{{{\lambda _1}{a_1} + {\lambda _2}{a_2} + \ldots + {\lambda _n}{a_n}}}{{{\lambda _1}{b_1} + {\lambda _2}{b_2} + \ldots + {\lambda _n}{b_n}}}\normalsize}\),

where \({\lambda _1}{b_1} + {\lambda _2}{b_2} + \ldots \) \(+\, {\lambda _n}{b_n} \ne 0\)