Ratio, rate and proportion

Fig.1. European road sign warning for a steep upward hill ahead. The sign indicates a slope of 10%. What do you think this means?

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.

a b = 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:

80 100 = literate adults 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.

400 kcal 100 g = 424 kcal 106 g = 800 kcal 200 g = 4 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:

E m = 400 100 E m × m = 400 100 × m E = 4 × m

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:

90 3 = 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:

d 5 = 90 3 d 5 × 5 = 90 3 × 5 d = 30×5 = 150 km

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:

5×30 5 = 3×30 3 = 30 km/h

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.

Examples

Example 1:

A recipe for cupcakes calls for 250 grams of butter to make 9 cupcakes. How much butter do you need to make 12 cupcakes?

Let x represent the quantity of butter you need. Then you solve the proportion:

250 9 = x 12 9×x = 250×12 x = 250×12 9 = 333.33 grams

You need 333.33 grams of butter for 12 cupcakes.


Example 2:

Suppose the ratio of married to unmarried persons in a group is 3:5. What is the ratio of married persons to all persons in this group?

3 3+5 = 3 8

The ratio of married persons to all persons in this group is 3:8.

Does this say anything about the actual number of married or unmarried persons in the group?

No! In the group could be 3 married and 5 unmarried persons or 6 married and 10 unmarried persons or...


Example 3:

Someone earning an hourly wage works 3 hours and gets paid $36.45. What is the wage rate per hour for this employee?

Wages are typically directly proportional.

36.45 dollars 3 hours = 12.15 dollars 1 hour = 12.15 dollars hour

So, the wage rate is $12.15 per hour. How much earns the employee after 8 hours work?

12.15 dollars hour × 8 hour = 97.20 dollars

It is a directly proportional relationship: earnings = 12.15 × hours .

Or:

p 8 = 36.45 3 p 8 × 8 = 36.45 3 × 8 p = 36.45×8 3 = 97.20 dollars

The payment after 8 hours work will be $97.20.


Example 4:
A B C D E

Triangles ABC and ADE in the figure above are similar. This means that all the lengths of corresponding sides are proportional and thus are in the same ratio. That is:

AB AD = AC AE = BC DE

In the above equation numerators and denominators represent lengths of corresponding sides. Thus AB represents the length of the side between vertices A and B.

Let AB = 90, AD = 36 and BC = 108.17. What length does DE have?

90 36 = 108.17 DE 90 36 × 36 90 × DE = 108.17 DE × 36 90 × DE DE = 108.17×36 90 43.27

Do you think it is also possible to calculate lengths AC and AE from what is given? Can you draw two figures that meet the given properties, but with different lengths AC and AE?


Example 5:

Five hundred individuals of a community of twelve hundred fifty people are unemployed. What is the unemployment rate in this community?

The unemployment rate is:

500 1250 = 4×125 10×125 = 4 10

Four out of ten are unemployed in this community. What percentage is that?

Do you think it was reasonable to assume that a relationship like this is directly proportional, that is, do you think the unemployment rate is the same for any number of (random) individuals from a community?