I have long been of the opinion that ratios and fractions are effectively the same thing, but I’ve recently changed that belief.
A ratio is a relationship between two values that states the relative size. These can generally be expressed in fractional form, but they need not be. How much sense the units make depends on the values being compared.
For instance, assume I have a high class that consists of thirty total students, twenty of whom are juniors and the rest of whom are seniors. Then 20/30, or 2/3, of the students in my class are juniors. The ratio of juniors to the total class is 2 : 3. However, I could also discuss the ratio of juniors to seniors: 2 : 1, that is, for every senior in my class, there are two juniors. But there’s no easy unit here. While I could express the ratio as 2/1, it doesn’t make sense to say “For every senior in my class, two of them are juniors.”
The latter case still does have a unit, though: 2 juniors per senior. Any confusion lies in the notion of fractions, not ratios.
The archetypal fraction is (1) a rational number (2) between zero and one (3) written in a specific form. I say “archetypal” because there’s a lot of wiggle room here. A few years ago, I asked my Twitter followers about two values, and these were the results:
This was hardly a scientific poll, but with a thousand voters, most of whom were academically involved with mathematics in some fashion, there’s enough data here to reach the conclusion that there’s no consensus on what a “fraction” really is.
Some people pointed out that we don’t know whether \(\frac{\pi}e\) is rational or not, but it probably isn’t (wouldn’t it be truly astounding if it were?). It is, nonetheless, greater than one, so it fails at least one, probably two, of the three archetypal criteria.
Meanwhile, 0.7 passes two of those criteria, but fails on form.
If there is not a single required attribute to be a fraction, the conclusion is that all numbers are fractions. Whether or not this is problematic is a matter of personal linguistic taste.
I feel that what’s happened, historically, is the conflation of the concepts of “fraction” and “fractional form” that may have followed down a similar path to the muddying of “number”. At some point in time, “number” implied a counting number, that is, a positive integer; that’s still the case in some mathematical fields. With such a meaning, “fraction” and “ratio of parts to a whole” would be the same concept, meeting all three criteria above.
However, as “number” got muddled, so too did “fraction”. And so it would seem that “fraction” could mean “any real number” or even any number at all.
Bring in the archetypes, bring in the sandwiches.
It’s been a popular social media debate to argue about whether all manner of things are sandwiches. There is, to be sure, an archetype of “sandwich”: two independent slides of bread with some sort of protein-focused filling, intended to be eaten by hand (without utensils). Vegetables and sauces are an option without deviating from that archetype. So a BLT is a sandwich, and matches all the criteria.
Hot dogs meet most of the criteria, but the bread is a single bun. A taco meets some of the criteria (it has an external starch which is only bread-adjacent, and is not independent, but it has a filling). So too does a “lettuce wrap” (which houses a protein and can be eaten with the hand, but which lacks the external starch). Ice-cream sandwiches lack bread. And so on. A common conclusion is that “sandwich-like” examples can be designed that violate each of the details, so everything is a sandwich (including non-food items).
This would be like concluding that, since there are dogs in the world that violate any coherent definition of “dog”, everything is a dog.
What it really means is: There are better examples of fractions, and sandwiches, and dogs, and there are less-good examples of each. Natural language not only can be, but must be, flexible in allowing in those less-good examples. It’s okay.
This conversation has not yet brought in the notion of units. I feel like the idea is that fractions are numeric values; even if any numeric value can be a fraction, it still has to be a numeric value.
Ratios, in contrast, often at least imply a unit. If the ratio is parts to a whole, then we’re often comfortable ignoring the units, but they’re still there. For instance, a fairly standard way of introducing fractions is with pizzas: Of a six-slice pizza, I eat two slices. I have eaten 1/3 of a whole pizza. If I do that every day, I am in the habit of eating 1/3 of every pizza I encounter.
The ratio of “pizza I eat” to “pizza I encounter” is 1 : 3 (which is often written as 1/3). The portion of the pizzas I eat is 1/3. Because I’m comparing parts to a whole, I can generally leave off the unit.
This topic came up recently in terms of the trigonometric ratios. Each angle measurement corresponds to three values attached to its terminal side (in standard position): Its slope, its rise, and its run. The latter two of these has a length measurement, and we would generally measure them with the same units.
The sine of an angle, for instance, is a ratio involving its rise. If we started at the origin and moved one unit along the terminal side of an angle with a measurement of \(\theta\), we would be at a height of \(\sin\theta\) units. If we moved 4 feet along a \(30^{\circ}\) incline, we would be at a height of 2 feet (\(4\sin 30^{\circ} = 2\)).
The relevance of this to ramps in construction is fairly clear: If we want a particular height and a particular incline, we can determine how long the ramp has to be. If we can measure the incline and the distance travelled, we can figure out how high we are.
Sine and cosine are both part-to-whole ratios, and hence their absolute values must be between 0 and 1. Tangent is a part-to-part ratio, and can be any real number.
To recap: Every given angle corresponds to the slope of its terminal side, which in turn represents a ratio between the relative height and width of any point on the ray; such point is some distance from the origin. Sine is height : distance, cosine is width : distance, and tangent is height : width.
Given the point (5, 12), which is 13 units from the origin on a ray with a rotation of \(\theta\) radians. Then \(\sin\theta = 12/13\), \(\cos\theta = 5/13\), and \(\tan\theta = 12/5\):
These are ratios from the standpoint that they reflect the relative size of values, that is, a rate of change between an independently measured value and a second value.
Another example comes from calculus: \(\frac{\text{d}y}{\text{d}x}\), the derivative. High school calculus teachers are consistently reminded not to call this a fraction, but what is it?
It’s a ratio of the instantaneous rates of changes. The reason it’s not a fraction in the sense discussed above is because it’s not a real number; neither \(\text{d}x\) nor \(\text{d}y\) are real numbers because they represent infinitesimally small values. They are numbers, but they are not real numbers, so their ratio is not the ratio of real numbers.
A further confusion is over what \(\text{d}\) represents. If we see it as a function that returns the instantaneous rate of change of its argument with regards to another value, we can (and sometimes do) write that argument in parentheses: \(\frac{\text{d}(y)}{\text{d}(x)}\), which in turn helps explain why we write the second derivative the way we do.
The second derivative is the rate of change of the rate of change of the dependent value, with regards to the rate of change of the independent value evaluated twice. We are differentiating the dependent value twice, but we’re only differentiating the independent value once. Writing this with parentheses: \[\frac{\text{d}(\text{d}(y))}{\text{d}(x)\cdot\text{d}(x)}=\frac{\text{d}^2y}{\text{d}x^2}\]
This reasoning is based on the notion that the derivative is a ratio (not a fraction) of infinitesimal values.
So, in practice, ratios and fractions may look similar, and the mathematics involved overlap a lot (especially if we consider anything in fractional form to be a fraction). The difference is largely conceptual.