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13. Two Term RatiosWe may speak of two term and three or more term ratios. There is a big difference. Two term ratios may be identified with fractions. That identification may justify (I am not a historian and try to refrain from making remarks on the history of ideas in mathematics) calling unsigned and then signed fractions, rational numbers.
Two term ratios may be called binary ratios. What is a two term ratio?We read and declare A:B as the ratio A to B. We say one ratio A:B is the same as another ratio C:D when and only when the cross products AD = BC. Equality of two-term ratiosWe write A:B :: C:D when and only when AD = CD and read A:B :: C:D as the ratio A:C and C:D are equal. We could use the equal sign = in place of the old fashioned four dot symbol ::.
Two Term Ratios and FractionsNow the equality condition for ratios AD = BC holds when and only when
which in turn holds when and when only
So two ratios A:B and C:D are equal or equivalent when and only when the corresponding fractions (or compound fractions)
are equal or equivalent. So equality of two term ratios A:B and C:D may be cast as a comparision of fractions
Due to this correspondence, fractions where the numerators and denominators are both whole numbers are also called ratios.
Identification of Fractions and Binary (two-term) Ratios In many places around the world, the fraction
is called a ratio, and no difference is emphasized between the concept of a ratio A:B and the concept of a fraction. Even I will call a fraction a ratio, or vice-versa. Reasoning involving equivalent ratios written as A:B can also be done with equivalent fractions written as
Proportionality of Numerators and Denominators
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| M N |
= | A B |
are equal when and only when the first term M of the ratio M:N
| M | = | [ | A B |
] | N | = | kN |
is proportional to the second term N in the ratio M:N
Earlier writers identify a ratio m: n (read m to n) of a pair of numbers with the fraction
|
m |
That makes sense when considering m parts of equal value out of n parts of equal value. With this identification two ratios a:b and c:d are equal when and only when the corresponding fractions are equivalent
| a b |
= | c d |
(1) |
or have equal values. Here a and d are called the extremes of the ratio;
Therefore a:b = c:d implies c:d = a:b. Therefore a:b = c:d implies b:a = c:d (extremes swapped with means) and d:c = b:a as reciprocals of both sides in (1) must be equal.
Algebraic forward and backward views of the latter equation implies the following when two ratios a:b and c:d are equal.
| ad | = | cb | (2) | clear denominators in (1) by multiplying by bd. So product of extremes a and d equals the product of means |
| a c |
= | b d |
(3) | introduce denominators in (3) by
dividing by cd. So a:c = b:d. Swapping the means preserves equality. |
| d b |
= | c a |
(3) | introduce denominators in (2) by
dividing by ba. So d:c = b:a Swapping the extremes preserves equality. |
More on Scaling Ratios or raising terms
From the equivalent fraction raising terms property that
| A B |
= | nA nB |
we observe A: B = nA : nB when ever the first and second terms in a ratio A:B are multiplied by the same whole number n.
Compound fractions have a similar property:
| A B |
= | qA qB |
whenever q is a fraction (or real number). So A: B = qA : qB when ever the first and second terms in a ratio A:B are multiplied by the same fraction or real number q.
We can add, subtract, multiply and divide fractions written as
| A B |
But these arithmetic operations are not (to the best of my knowledge) defined for the ratios written as A:B.
We may also identify a fraction written as
| A B |
with a percentage or real number
Imagine a collection of q = m + n objects divided into disjoint subsets of m and n objects, respectively. Here the identification of the ratio m:q with the fraction
m
q
correctly gives the part as a fraction of the whole.
Imagine a collection of q = m + n objects divided into disjoint subsets of m and n objects, respectively. Here the identification of the ratio m:n with the fraction
m
n
is problematic. The ratio may be identified, if we must, with the compound fraction
|
m |
|
|
|
m |
All this is to suggest that a distinction or nuance exists between the ratio written as m:n and the fraction m/n. The question is how. The ratio notation does not distinguish between the ratio of a part to a whole and the ratio of complimentary parts.
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