Formulas

Elementary Algebra

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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 \({\large\frac{a}{c}\normalsize} = {\large\frac{b}{d}\normalsize}\)
  8. Denominator addition property
    If \({\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\), then \({\large\frac{a + b}{b}\normalsize} = {\large\frac{c + d}{d}\normalsize}\)
  9. Denominator subtraction property
    If \({\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\), then \({\large\frac{a – b}{b}\normalsize} = {\large\frac{c – d}{d}\normalsize}\)
  10. General numerator-denominator addition property
    If \({\large\frac{a}{b}\normalsize} = {\large\frac{c}{d}\normalsize}\), then \({\large\frac{{ma + nb}}{{pa + qb}}\normalsize} = {\large\frac{{mc + nd}}{{pc + qd}}\normalsize}\;\;\) \(\left( {{p^2} + {q^2} \ne 0} \right)\)
  11. 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 \({\lambda _1}{b_1} + {\lambda _2}{b_2} + \ldots \) \(+\, {\lambda _n}{b_n} \ne 0\)