Ratios vs Fractions

Several middle school math teachers have told me that there’s an important distinction between fractions and ratios that students don’t get. When I ask them what it is, the teachers can’t tell me; “it’s complicated”, they say. I’ve been troubled by that response. For me, ratios and fractions both involve division, and they certainly look very similar.

Edward Manfre at The Learning Chest offers this excellent example of the difference, in a personal e-mail: “If you go 3 for 6 in the first game of a doubleheader and 1 for 4 in the second game, your overall ratio of hits/at-bats is 4/10. But the sum of the ‘fractions’ 3/6 and 1/4 is obviously not 4/10.”

For the purposes of discussion, I’m going to use cocktail mixes instead; it’s the same concept, but I think it will make the examples clearer.

A common recipe for mimosas calls for one part juice to one part alcohol; a common recipe for rum and Coke calls for two parts cola to one part rum. So if you drink one of each, what portion of your drinks are alcohol (ignoring proof)?

The answer to this question relies on knowing how big each drink each one is. If each “part” is the same size, then that’s two parts of alcohol in five parts total, or 2/5 alcohol. If the two drinks are the same number of ounces (say, six ounces), then you have one drink that’s three ounces of alcohol and one drink that’s two ounces of alcohol; the combination would be 5/12 alcohol. We could of course have some other combination of drink sizes. One point being, we can’t add the ratios without knowing what total quantity is represented.

Another point, though, is that there’s no way of combining these two drinks in such a way that the final concoction will be \(\frac{1}{2} + \frac{1}{3} = \frac{5}{6}\) alcohol, because neither drink alone is more than half alcohol.

To take an even simpler example, let’s say we have two pizzas, each of which is half pepperoni. What portion of the entire order is pepperoni? Half, of course. But if we added the fractions involved, we’d get a whole. That’s because, when we add the two pizzas together, we do have a whole pepperoni pizza, but we have two pizzas in toto.

It’s not entirely that ratios and fractions work differently: This is arguably a mathematical mirage brought on by ignoring the units. If we have two halves of a pizza and add them together, we have one whole pizza, but we have a half of “two pizzas”. Fractions are implied to be portions of a consistent unit; when we combine ratios, though, we also combine the base objects that they’re ratios of.

Let’s take the case again where we have a six ounce mimosa and a six ounce rum and Coke: We have five ounces of alcohol, and if we define “drink” as “a six ounce beverage”, then we do indeed have 5/6 of a drink’s worth of alcohol between our two cocktails. But that 5/6 of a drink represent 5/12 of two drinks that we have. It’s not that we can’t add the ratios; it’s that we have to also take into account the fact that the base unit has changed.

This makes the mathematics more complicated. Looking back at the original example, from baseball, to determine total batting average through the addition of fractions, we’d have to first decide on some base unit for the two statistics. Let’s say a “full outing” is four at-bats.

If you know your per-game average, your overall average for two games would be the solution to \(x\) for this set of equations: \[avg_1 \cdot out_1 = wgt_1 \\ avg_2 \cdot out_2 = wgt_2 \\ x \cdot (out_1 + out_2) = (wgt_1 + wgt_2)\] where \(avg\) is the average of a particular game, \(out\) is the number of outings in that game, and \(wgt\) is a “weight” that is the product of those two.

If we did this in the example at hand, we would get: \[ \frac{3}{6} \cdot \frac{3}{2} = \frac{9}{12} = \frac{3}{4} \\ \frac{1}{4} \cdot \frac{2}{2} = \frac{2}{8} \ \frac{1}{4} \\ x \cdot \frac{5}{2} = \frac{3}{4} + \frac{1}{4} = 1 \\ \rightarrow x = \frac{2}{5}\]

We can collapse this into a single line, and generalize it across any number of games: \[ x = \frac{\Sigma_{k=1}^n avg_k \cdot out_k}{\Sigma_{k=1}^n out_k} \]

I say “we can”, but in practice for this example, we don’t. To figure out batting average, we just count up the number of times someone went to bat and then count up the number of times they got a hit. My point is that adding ratios could be done comparably to adding fractions, just more complicated because we have to adjust for the combining base quantities. (I am also not suggesting pressing the average seventh grader to use formulas like this one.)

So long as the denominators of ratios are multiples of the same unit, adding ratios is a much simpler process than all of this, though. If I went 3-for-4 in game one, 2-for-3 in game two, 0-for-4 in game three, and 1-for-5 in game four, then I went 6-for-16 overall. If my “parts” are the same size in cocktails, then a mimosa (one part alcohol, one part OJ) and a rum and Coke (one part alcohol, two parts cola) have two parts alcohol to three parts non-alcohol, together.

Overall, then, the difference between ratios and fractions is that fractions represent portions of a consistently sized “whole” (the denominator is constant, while the numerator aggregates), while adding ratios require being mindful that the unit aggregates, not just the numerator.

It strikes me that this is only problematic in addition. Multiplication is not a problem. Champagne is about 12% alcohol, while standard rum is about 40% alcohol. So a mimosa is about 6% alcohol while a rum and Coke is about 13% alcohol. This is because multiplication is not dependent on units (or rather, because multiplication combines units as it combines quantities) while addition requires comparable units.