# Rate, multipart

A comparison between **more than two** *quantities* (multitudes or magnitudes) of a **different** type.

The quantities being compared are called the **“terms”** of the rate.

Multipart rates are also known as multi-*term* or multi-*step* rates.

Multipart rates are usually written using the **forward slash**, /, to separate the terms: “𝑎 units / 𝑏 units / 𝑐 units…”

As with rates, the forward slash (a.k.a. *solidus* ), /, is read aloud as **“per”**, which just means “for each”.

## Usage of Multipart Rates

Multipart rates are **everywhere in science**—from physics to biology, and all fields in between.

Whenever you see some scientific calculation with **more than two units in it**, then you are dealing with a multipart rate.

**Acceleration** under gravity is a simple example:
9.8 metres *per* second *per* second

(Yes, two terms are seconds, relating to *time*, but they need to be accounted for *separately*.)

It is typically measured in metres (m) *per* second (s) *per* second (s).

**Acceleration under gravity**, in “basic” rate format, is written:
9.8 m / 1 s / 1 s

This “basic” format for rates is ideal for *recording* your observations.

You group **terms together** by how they happen in the real world, as a **number and its unit**, *together*.

However, this is not rates are written in scientific calculations.

Why? Well, it is easier if we convert these rates of terms into another format, to allow **calculations** to be made.

What is this other format?

**“Rational number and grouped units”** format.

So, starting with: 9.8 m / 1 s / 1 s

First, move the **numbers from each term to the left**, and calculate their result (where “per” means division):
9.8 ÷ 1 ÷ 1m / s / s

Then, you keep *all* **units to the right**, as a group:
= 9.8 m / s / s
keeping track of the arithmetical operations you did on the numbers, so you can convert *back* to physical reality.

(In this case, all we did was divide everything, so the units are written as metres/second/second.)

### Multipart rates with repeated terms

**Speed** is the *distance* you travel in a given unit of *time*.

It is a **rate of change** of distance travelled *per* time, measured in units like miles per hour (mph), or metres per second (m/s).

**Acceleration** is the *rate of change* of your speed (either increasing or decreasing) over *time*.

A “rate of change” *of* a “rate of change”.

As mentioned, acceleration can be thought of as a multipart rate, one with *repeated* terms (the seconds):
9.8 m / 1 s / 1 s
(for acceleration under *gravity*)

The **International System of Units (SI)** uses the “rational number and grouped units” format, with the multipart rate written as:
= 9.8 ÷ 1 ÷ 1 m/s/s
= 9.8 m/s/s
= 9.8 m/s**²**

But, ultimately, the SI format is derived from the original **“rate of terms”** notation, i.e.:
9.8 m / 1 s / 1 s
and I would advise keeping this original formulation in mind when thinking in rates—to avoid confusion by the jumble of units on the right-hand side of the SI format.

### Multipart rates with unique terms

While acceleration has **repeated terms** (metres per *second* per *second* ), there are multipart rates with **unique terms**, i.e. *no* repeated terms.

One close to my own heart is **“VO _{2} max”**, a name derived from three abbreviations:

**“V”**for*volume*,**“O**for_{2}”*oxygen*, and**“max”**for*maximum*.

VO_{2} max measures **cardiovascular fitness** and represents the maximum rate of oxygen consumption achieved during physical effort.

It is used to test **fitness in athletes**, but also in a healthcare setting, to assess patient’s cardiovascular function etc.

For athletes, measuring their VO_{2} max is a **fun** experience.

They strap a mask on your mouth and nose, one hooked to a **machine measuring your breathing**.

Then, they stick you on a treadmill, and keep upping the speed until you **cannot keep up**—the point of physical collapse, basically.

(They don’t go so hard on hospital patients, for *obvious* reasons.)

VO_{2} max is a **multipart rate**, with three completely *different* terms:

**milliliters (mL)**of oxygen consumed per**kilograms (kg)**of body weight per**minutes (min)**of time.

For a test that took 20 minutes to exhaust an 80 kg athlete, we can record the multipart rate: 90,000 mL / 80 kg / 20 min

And, as above, we can convert this to **SI format**:
= 90,000 ÷ 80 ÷ 20 mL/kg/min
= 1,125 ÷ 20 mL/kg/min
= **56.25** mL/kg/min
to give a result in terms of **unit values**.