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).

B. Two Term Ratios

We 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.

In the discussion of maps, scale factor (the relation between actual distance and distance on the map) may be expressed as a ratio or fraction.

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 ratios

We 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 ::.

Convention: The ratio notation A:B appears when and only when the scaling properties of the first and second term are important.

Two Term Ratios and Fractions

Now 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.

Rational numbers may be thought of as fractions whose numerators and denominators are provided by integers instead of whole numbers.

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
Or the first and second term in a ratio

Direct Proportionality: A number or quantity Z is directly proportional to another quantity X in several circumstances when and only when the quotient Z ÷ X = Z/X has a constant value k,.or equivalently, there is a constant k such that Z = k X. That is in each instance where we find or measure the value of X, the value of Z will be kX.

Fractions and Ratios scale in the same way. Therefore A:B = M:N when and only when

are equal when and only when the first term M of the ratio M:N

is proportional to the second term N in the ratio M:N

More on the Identification:

Earlier writers identify a ratio m: n (read m to n) of a pair of numbers with the fraction

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

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.

More on Scaling Ratios or raising terms

From the equivalent fraction raising terms property that

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:

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.

Differences between fractions A/B and ratios A:B

We can add, subtract, multiply and divide fractions written as

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

with a percentage or real number

Ratios of a part to the whole -YES

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.

Ratios of complementary parts - Problematic, Food for thought

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

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.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science