A. Ratios And Fractions
Similarities and Differences
Fractions may be identified with two term ratios, and vice-versa as well,
sometimes. Two fraction are equal or equivalent when and only when the
corresponding two term ratios are equal or equivalent. But fractions can
be added, subtracted, multiplied and divided while the same operations
are not defined for two- and multiple term ratios. While we may call a
fraction, a ratio or a rational number, ratios are different. Triple term ratios exist, but triple term fractions do no
exist. Three and more -term ratios cannot be identified with
fractions.
The following lesson cover the properties of two term and
multiple term ratios.
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Fractions
As (two term) Ratios and Fractions Versus Ratios: Fractions are
often called ratios, and vice-versa. But the vice-versa only holds for
two term ratios. This lesson identifies fractions with two-term ratios
and contrasts the properties of fractions and two-term ratios. (Ratios
cannot be added, subtracted or compared, but like fractions, the terms
in ratios can be raised or lowered).
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Implied or
Derived Ratios - New Fractions and Ratios from Old: If two
fractions are equivalent then their reciprocals are also equivalent.
Likewise if a pair of two-term ratios are equivalent, interchanging the
first and second terms of each ratio in the pair leads to a pair of
equivalent ratios. Beyond that, more equivalent ratios can also be
generated from a pair of ratios. Food for thought: How may
equivalent fractions or ratios may be formed from the relations ad =
bc?
-
Multiple Ratios:
Multiple Term Ratios - Three Term Ratios to be precise. We read the
triple ratio a : b :c as a to b to c. We further write
a : b: c :: A: B:
C
to say two triple ratios a : b: c and
A: B: C are equal or equivalent when and only
when
there are other ways to say when two triple ratios are
equal or equivalent.
Note: Triple ratios or triple proportionalities
occur between the sides of similar triangles. More generally,
multiple ratios or proportionalities occur between the sides of
similar triangles.
The discussion of ratios or multiple ratios is best
understood besides a discussion of proportionality.
Inner Versus Outer Terms - small point: In the
discussion of equality of ratios a : b = A: B written in
that order, the inner terms are small b and big A while the outer terms
are small a and big B. In contrast, if we rewrite the equality as
A: B = a : b, we find the inner and outer terms are
interchanged. However, the equality requires the product of the inner
and outer terms be equal, that is aB = Ab. That equality is not
affected by rewriting a : b = A: B as A: B = a
: b, and the resulting swap of inner and outer terms
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
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Arithmetic
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Algebra
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Calculus Starter Lessons
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They cover basic topics in ways likely to complement your
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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