Direct proportionalities from daily life
Direct proportions play a big role in our everyday lives. Here we look at a spectrum of illustrative examples.
In this step you will learn about
- the key definitions for direct proportionality
- direct proportions arising in situations involving speed, money, fuel efficiency of your car, and the role of .
What is a direct proportionality?
If and are two variable quantities that satisfy the relation
for some fixed number , then we say that is directly proportional to . The number is called the constant of proportionality. One important way of thinking about this constant is that it tells us how much increases by if we increase by exactly .
We will write
to express that is directly proportional to . Note that this symbol however loses an important piece of information: the exact value of the constant of proportionality.
Distance is proportional to time when walking.
If we walk at a steady pace, say of km/hour, then we have a basic example of a direct proportionality
since distance (km) = time (hr).
The constant of proportionality, in this case km/hr is called the velocity or speed. However an Olympic race-walker will go much faster than this–say twice as fast at km/hr.
Q1 (E): Describe the relationship between distance and time for an Olympic race walker using this terminology.
Q2 (M): Distance is directly proportional to time. But time is also directly proportional to distance! What is the constant of proportionality in this relationship for Olympic race walkers?
Time in hours and minutes
Perhaps the most compelling use of direct proportionality is the relationship between hours and minutes:
The relation between hours and minutes is so ubiquitous that most people know by rote that 15 minutes is one quarter of an hour. This information is contained within the above equation, which we can see by setting
And how many minutes are in half an hour? The answer is
Q3 (E): How many minutes are in a day?
Distance your car can travel vs amount of gas in your tank.
This is a different kind of proportionality also involving distance–now between how far you can travel in your car and the amount of gas in your tank.
If you have zero gas, you can go nowhere. Let’s say you drive a Holden Commodore – a popular Australian car with a tank capacity of litres. How far you get depends on whether you are driving in the city or on a freeway, and whether you have the air conditioner on or off etc. But on average, let’s say you can go km on a litre (L) of gas. Then the relevant proportionality is
with the constant of proportionality the number km/L, which is a measure of the fuel efficiency of the car.
Q4 (E): Can you drive a Holden Commodore with a full tank of gas from Broken Hill to Dubbo without stopping?
Diameter vs circumference of a circle
“The Earth seen from Apollo 17” Public domain/Wikimedia Commons
Direct proportionalities are also very important in mathematics. For example, it is an important fact that the circumference of a circle is directly proportional to its diameter . In fact
where the constant of proportionality is the famous constant , roughly
So if you double the diameter, you double the circumference. Another implication is that if you increase the Diameter by unit, then the circumference increases by units. This is independent of the actual size of the circle, which can be surprising.
Q5 (M): If we have a rope tied tight around the equator of the earth, and we want to add some amount or rope so that we can stretch it to be uniformly half a metre off the ground, then how much length to the rope must we add?
A1. One answer would be that . However this could describe anyone! A more meaningful answer is
Better still if you include units of distance and time.
A2. We can rearrange into
So the constant of proportionality is .
A3. The relationship between days and hours is another example of a direct proportionality:
Now we combine the direct proportionality for minutes and hours, with the direct proportionality for hours and days:
We get a new direct proportionality:
So there are minutes in a day.
A4. You can expect to travel km on a single tank of gas. Since you can get from Broken Hill to Dubbo in km, then yes – you can make this trip without stopping.
A5. We are increasing the diameter by , so the circumference must increase by , that is by a bit more than metres. This is quite unintuitive to many people!
© UNSW Australia 2015