Directly Proportional Relationships

The best way to show and explain direct proportional relationships is by graphing two sets of related quantities. If the relation is proportional, the graph will form a straight line that passes through the origin.

Note: Different countries around the world use different money (or different currencies). People and businesses often need to buy and sell things in a different country so they need to exchange their currency for another. To help with this, each currency has a relationship with each other currency around the world. For currencies this relationship is known as the exchange rate.

Graphing Proportional Relationships - Example 1

Let us use the relationship between U.S. Dollars and U.K. Pounds to illustrate this. The exchange rate used in this example is 0.69 U.S. Dollars per 1 U.K. Pound. (Note that this, and all currency exchange rates, change all the time).

1 USD = 0.69 UKP
USD
UKP
0 0
100 69
200 138
300 207
400 276

 

graph showing relationship between US Dollars and UK Pounds with an exchange rate of 0.69 USD to 1 UKP

The table of values and their graph show above a straight line that passes through the origin. This indicates that the relationship between the two currencies is in direct proportion. Think about what this means in real terms – if you have ten times more dollars than another person, when you both exchange your money, you will still have ten times more money. Notice also that the graph passes through the origin; this makes sense as if you have no dollars you will get no pounds!

We can express these relationships algebraically as well as graphically.

U.S. Dollars = 0.69 x U.K. Pounds or, using conventional algebraic terms, where y represents U.S. Dollars and x represents U.K. Pounds, as
y = 0.69x
All directly proportional relationships can be expressed in the form y = mx
m represents the slope (or steepness of the line) when the relationship is graphed.

Unit rate as slope

In the example above the unit rate is the exchange rate which is 0.69 US Dollars per UK Pound. Note this relationship in algebraic terms, y = 0.69x meaning the slope equals 0.69. The slope of a the line that represents a directly proportional relationship equates to the unit rate. There is more on the slope of a line here.

Graphing Proportional Relationships - Example 2

Another common example of directly proportional relationships is that between time and distance when travelling at a constant speed. The graph below shows the relationship between distance and time for a vehicle travelling at a constant speed of 30 miles per hour. Note that this means the unit rate is 30 miles per hour.

30 miles/ hour
Time
Distance
0 0
1 30
2 60
3 90
4 120

 

graph showing relationship between time and distance for an object travelling at 30 miles per hour.

Notice below a similar graph. In this example the vehicle is travelling at a constant speed of 50 miles per hour. The slope of the graph is steeper. The steepness of the slope for directly proportional relationships increases as the value of the constant m (y = mx) increases. In our two speed examples, the change in steepness of slope represents the change in speed or the change in the unit rate.

50 miles/ hour
Time
Distance
0 0
1 50
2 100
3 150
4 200

graph showing relationship between time and distance for an object travelling at 50 miles per hour.

Graphing Other Linear Relationships

Directly proportional relationships always pass through the origin (0,0). There are other linear relationships that do not pass through the origin. The example below is for a taxi fare that has a standing charge (a fee payable no matter how far is travelled) and a cost per mile.

$5 + $4/ mile
Distance
Cost
0 5
1 9
2 13
3 17
4 21
5 25

 

graph showing relationship between distance and cost for a taxi fare with a standing charge of $5 and a rate of $4 per mile

This relationship is expressed algebraically as follows:
y = 4x + 5

Notice where the line intercepts the y-axis in the above example. This point is known as the y-intercept and there is more on this here.

Worksheets

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