# Direct proportionalities from daily life

- the key definitions for direct proportionality
- direct proportions arising in situations involving speed, money, fuel efficiency of your car, and the role of (normalsize{pi}).

## What is a direct proportionality?

If (normalsize{x}) and (normalsize{y}) are two variable quantities that satisfy the relation [Large{y=kx}] for some fixed number (normalsize{k}), then we say that (normalsize{y}) is**directly proportional**to (normalsize{x}). The number (normalsize{k}) is called the

**constant of proportionality**. One important way of thinking about this constant (normalsize{k}) is that it tells us how much (normalsize{y}) increases by if we increase (normalsize{x}) by exactly (normalsize{1}). We will write [Large{y propto x}] to express that (normalsize{y}) is directly proportional to (normalsize{x}). 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 (normalsize{6}) km/hour, then we have a basic example of a direct proportionality [Large{operatorname{distance}proptooperatorname{time}}] since distance (km) = (6 times) time (hr). The constant of proportionality, in this case (normalsize{6}) km/hr is called the**velocity**or

**speed**. However an Olympic race-walker will go much faster than this–say twice as fast at (normalsize{12}) 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

^{Clock by OpenClipartVectors Pixabay / CC0 Public Domain}

Q3(E): How many minutes are in a day?

## Distance your car can travel vs amount of gas in your tank.

^{© “Only another 1205kms to go” by Tasmin Slater/Flickr CC BY SA 3.0}

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}

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?

## Answers

[Large{ mbox{distance} = 12 times mbox{time} }.]A1.One answer would be that ({normalsize mbox{distance} propto mbox{time}}). However this could describe anyone! A more meaningful answer isBetter still if you include units of distance and time.[{ Large mbox{time} = frac{1}{12} times mbox{distance} }.]A2.We can rearrange (normalsize{ mbox{distance} = 12 times mbox{time} }) intoSo the constant of proportionality is (frac{1}{12}).[{Large mbox{time (hr)} = 24 times mbox{time (days)} .}]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:[{Large mbox{time (min)} = 60 times mbox{time (hr) } = 60 times left(24 times mbox{time (days)} right) }.]We get a new direct proportionality:[{Large mbox{time (min)} = 1440 times mbox{time (days)} .}]So there are ({normalsize 1440}) minutes in a day.A4.You can expect to travel ({normalsize 73 times 12 = 876}) km on a single tank of gas. Since you can get from Broken Hill to Dubbo in ({normalsize 799 – 49 = 750}) km, then yes – you can make this trip without stopping.A5.We are increasing the diameter by (normalsize{1}), so the circumference must increase by (normalsize{pi}), that is by a bit more than (normalsize{3}) metres. This is quite unintuitive to many people!

#### Maths for Humans: Linear, Quadratic & Inverse Relations

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