# The Math of Coffee

## Milk before or after coffee?

# A Shocking Claim

My middle school French teacher (ma professeure) once told my class that coffee drinkers in France pour their milk into their cups *before* they pour their coffee.

I still don’t know whether this generalization is actually true. Yet, at the time, years before becoming an avid coffee drinker, twelve-year-old me was alarmed by the deeply unsettling claim.

“How do they know how much milk to add?” I asked. “Don’t you have to pour in the milk afterwards to achieve the right level of milkiness? This is madness.”

The other day, I found myself mindlessly doing what the French purportedly do. Upon realizing this and remembering that day in middle school, I began doubting my objections and thinking, of all things, about the math of pouring milk into coffee (or coffee into milk).

# The Ideal Ratio

Our first concern—the most important for someone who puts creamer in their coffee—is this: How do you achieve the optimal ratio between milk and coffee? Does adding the milk in first increase the likelihood that the ratio will be wrong?

Instinctively, we believe (or at least I once believed) that there is a linear relationship between the quantity of milk added and the level of milkiness of the coffee. In other words, adding twice the amount of milk changes the composition of the coffee twice as much:

Upon closer inspection, this turns out not to be true. The rate of milkiness is actually defined hyperbolically:

You can understand why this is so in two ways.

First, milk has a dilutive effect on coffee. Let’s think of the impact of each successive drop of milk. The first drop changes the composition of a cup with 100% coffee. The second drop, however, expends part of its dilutive impact on that added drop of milk. Therefore, as the cup fills up with milk, the incremental impact of adding even more milk goes down.

The other framing uses percentages. As the quantity of milk increases, the overall liquid in the cup increases. The percentage of milk is defined by a fight against an ever-rising denominator in this equation (where V stands for volume):

As an extreme example: If we begin with 10 ounces of coffee and add 5 ounces of milk, the composition of the liquid is now 66.6% coffee. But adding another 5 ounces of milk doesn’t bring the coffee down by another 33.3%. Instead, it drops it by only 16.6% to 50% coffee. The same quantity of milk has halved its impact on the milkiness of the beverage.

So back to our original question of whether the milk should go in first of second… If you’re worried about getting the ratio wrong by pouring the milk in first, don’t worry. The consequence of an error won’t be as significant as you might expect. Accidentally doubling the milk won’t result in coffee that’s half as strong.

The takeaway: If you enjoy very little milk in your coffee, pour the coffee first. Each drop of milk has a larger impact on the overall composition of the coffee and, therefore, its flavor and color. You’re more likely to achieve an undesired outcome if you blindly add the milk and hope for the best. Conversely, if you don’t mind a creamier cup of joe, pouring in the milk first won’t greatly impact the result.

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# The Temperature Factor

As anybody who adds milk to their coffee (or coffee to their milk) can attest, the process causes a noticeable drop in the temperature of the cup. So, a second interesting question, beyond the mere ratio of the two liquids, is this: Which causes the coffee to cool down faster, milk first or coffee first?

To answer this, it’s important to remember the physics rationale for why your cup of coffee cools down at all. Any time it comes into contact with something containing less thermal energy, molecules from the coffee transfer heat to that something. For a black cup of coffee, the “something” is the surrounding air in the room. For a cup with milk, it is the air in the room *and* the milk. In either case, it’s critical to note that the rate at which the temperature declines is *greater* if the difference in temperatures is greater. So if coffee starts out hotter, it will lose thermal energy faster.

Okay, back to the question then. Would changing the order of when the milk is added impact the drop in temperature?

Let’s arbitrarily say our coffee comes out at 150°F, our milk is at 30°, and our room is at a pleasant 70°.

In the coffee then milk situation:

The coffee starts at 150°. The rate of heat loss is defined by the difference between the coffee, 150°, and the room, 70°. Before the milk is poured, some heat is therefore lost to the room.

Let’s imagine our coffee has declined to 140° by the time the first splash of milk hits it.

The milk now brings the coffee down by, say, 20° to 120°.

The coffee continues to cool due to contact with the air, at a rate defined by the difference between the coffee (120°) and the room (70°)

In the milk then coffee situation:

The coffee is never given a chance to lose heat to the room before the milk impacts its temperature. (If anything, the milk actually warms up a bit while sitting in the cup; but let’s ignore that fact.)

As soon as the coffee does fill the cup, the milk cools it down by, say, 25° (for, unlike in step 3 above, here the milk has a greater impact, due to a greater difference between itself and the hotter coffee). The coffee is now at 125° (a

*higher*temperature than the 120° in the previous approach).The coffee cools due to contact with the air, at a higher starting point than step 4 above, but also at a higher rate of heat loss.

The conclusion? If you pour in your milk first, your coffee will start off hotter but cool down faster.

# Other Mathematical Quandaries Raised by Coffee

One can go on and on analyzing the mathematical questions raised by coffee. For instance:

When is the optimal time to drink an afternoon coffee and get that caffeine boost?

Given the extreme range of prices for espresso machines, what does the graph of cost to quality actually look like?

Why can cafés charge more for iced coffee despite the fact that they’re giving you less coffee?

# Whoops

It turns out my wife pours her milk into the cup first, and I never knew until I shared a draft of this article with her. She explained that the reason is not mathematical at all. Adding the coffee second causes the milk to get stirred in without the need for a spoon.

I suppose I have a tendency to overthink things…

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