# Ratio, rate and proportion

## Terminology

Earlier, fractions (and division) were explained as expressions of a part of a whole or part of a unit. For instance, half of a cake or one third meters. Fractions can also be used to express a certain relationship between two quantities, not necessarily of the same unit.

### Ratio

A *ratio* shows a certain relationship between two quantities in the same unit of measurement.
For instance,
the number of dogs to the number of cats in a cats and dogs shelter. Suppose there are six dogs and eight cats in the shelter, then the *ratio* of
dogs to cats in that shelter is six to eight, denoted as 6:8 or as a fraction 6/8.
The ratio of cats to dogs is eight to six, denoted as 8:6.
Just like with fractions, we can simplify 6:8 to 3:4 and
8:6 to 4:3. So, for every 3 dogs there are 4 cats.

The ratio of the number of cats to the *total* number of cats and dogs in the shelter (6 + 8 = 14) is eight to fourteen,
8:14 (4:7).
Thus 4/7 of the animals in the shelter are a cat,
which relates closely to our original sense of a fraction as a part of a whole.

The two quantities in a ratio are of the same units. This means that the ratio itself is dimensionless (dimension 1). In the example above, both quantities are counts. Counts themselves are dimensionless, so the ratio is also dimensionless. But also a ratio of dimensioned quantities must be dimensionless. For example, a rise (vertical distance) of 10 meters for every run (horizontal distance) of 100 meters is equivalent to a slope of 10:100 (dimensionless or dimension 1). The slope ratio is in units of length. The ratio is the same for whatever unit of length (meters, inches, millimeters, miles, etc.) you use, as long as both quantities are measured in the same unit of length.

Ratios can be expressed as a percentage (see next part). A slope of 10:100 is a slope of 10%.

### Rate

A *rate* shows a certain relationship between two quantities in different units of measurement.
Examples are a speed of 25 kilometers per half an hour, or a nutrition fact like 80 kilo-calories per 100 grams.

*Temporal rate* ("per unit of time"), such as speed, is a common type of rate.

One of both quantities may be a count (dimensionless).
For example, "200 kilo-calories per serving".
A number of counts per unit of time is often referred to as "*frequency*".
For example, heart rate (beats per minute) is a frequency.

### Proportion

A *proportion* is an equation (an equality) of two ratios or rates.

$$\frac{a}{b}=\frac{c}{d}$$

As mentioned before, a ratio or rate describes a relationship between the two quantities. An adult literacy rate (is a ratio!) of 80% means that out of every 100 adults, 80 are literate. This implies that out of 1000 adults in the same population, 800 are literate. In other words: 80 is to 100 as literate adults is to the total number of adults:

$$\frac{80}{100}=\frac{\mathrm{literate\; adults}}{\mathrm{total\; number\; of\; adults}}$$

Not all ratios or rates describe a (directly) proportional relation! Six boys to fourteen children a in a group does not mean that doubling the group size automatically results in 12 boys per 28 children. Or a speedometer that reads 50 kilometers per hour, does not necessarily mean that you will travel exactly 100 kilometers in 2 hours.

## Directly proportional relationships

A rate of 25 kilometers per half an hour does not indicate what the actual relationship between the quantities
distance and time is. A distance of 20 kilometers may be covered in 15 minutes and 5 kilometers in the next 15 minutes.
In other words, the rate does not have to be equal, or *constant*, the whole given interval.

A rate represents an *average*: the total change over a given interval. For instance a travelled distance of 100 km in a time interval of 2 hours.
If this interval is finite, that is, it can be measured or expressed, and the relationship is constant,
the rate is equal at any sub-interval. The relationship is *directly proportional* or simply *proportional*.
If this interval is finite and the relationship is *not* constant, the rate is only an average over the interval.

It is fair to assume that a rate of 400 kilo-calories per 100 grams pie implies a constant rate for any mass of this sort of pie, and thus that for example eating 200 grams of this pie provides 800 kilo-calories energy, or 106 g pie provides 424 kcal. The amount of energy a piece of pie provides and the mass of this piece are proportional. For this particular pie the provided energy is to mass of pie as 400 kcal is to 100 g.

$$\frac{400\mathrm{kcal}}{100\mathrm{g}}=\frac{424\mathrm{kcal}}{106\mathrm{g}}=\frac{800\mathrm{kcal}}{200\mathrm{g}}=4\mathrm{kcal/g}$$

In the example above, the rate is always constant: 4 (kcal/g).
This constant is also called the *coefficient of proportionality* or *proportionality constant*.

Let's write this as a proportion (equation of rates) in which
`E` stands for the provided energy, `m`
stands for the mass and 4 is the proportionality constant:

$$\begin{array}{rl}\frac{E}{m}=& \frac{400}{100}\\ \frac{E}{m}\times m=& \frac{400}{100}\times m\\ E=& 4\times m\end{array}$$

So, a proportional relationship is a special case of a *linear function*.
Proportional quantities can be displayed as a straight line through the origin in a graph.
The steepness or *slope* of this line equals the proportionality constant.

`E` = c × `m`

When you look at the speedometer and see that the car drives 50 kilometers per hour, then that does not necessarily mean that
the car will actually drive a distance of 25 kilometers the coming half an hour or drove a distance of 50 kilometers the past hour.
It means that the car drives a "constant" speed of 50 km/h at that instantaneous moment, at that very tiny little time interval.
The rate or slope at that instantaneous moment is called the
*instantaneous rate of change*.
It is the *average* speed in that infinitely tiny time interval.

The instantaneous speed usually changes all the time. But if the car is on cruise control (a system that maintains a constant speed) the rate is 50 km/h at any moment and over any time interval. The speed is constant now, and the relationship is directly proportional.

Rates that are not directly proportional are outside scope of this course. Arithmetic word problems in education often involve proportional relationships. If you are sure that the quantities are actually proportional, then proportions (equations of ratios) can be used to calculate unknown values. More about this in the next section.

## Solving proportions

Although solving equations is part of algebra rather than arithmetic, solving proportions is often needed to solve the word problems that are generally part of primary education. Consider the next problem to be solved:

A car travelled 90 kilometers in 3 hours at a *constant* speed.
How many kilometers will this car travel in 5 hours at this same constant speed?

Common problems like this involve an equation of ratios (a proportion) to be solved:
`a` is to `b` as
`d` is to `e`.
The proportion for the problem above is:

$$\frac{90}{3}=\frac{d}{5}$$
in which `d` represents the asked travelled distance in 5 hours.

This linear equation can be solved by multiplying both sides of the equal sign by 5:

$$\begin{array}{rl}\frac{d}{5}=& \frac{90}{3}\\ \frac{d}{5}\times 5=& \frac{90}{3}\times 5\\ d=& 30\times 5=150\mathrm{km}\end{array}$$

The ratios must be the same, so 5 must be multiplied by 30 (`d` = 5 × 30), since 3 is also multiplied by 30 to get 90.
So, the proportionality constant is 30:

$$\frac{5\times 30}{5}=\frac{3\times 30}{3}=30\left(\mathrm{km/h}\right)$$

We can write this as the linear equation `d` = 30 × `t`.

Solving proportions needs a little elementary algebra, as demonstrated above. This algebra is wrapped in the method
*cross-multiplication*. The tricky part however, is to use the correct ratios. In the example above:
a distance to a time is as a distance to a time, and not as a time to a distance.