Identify proportional relationships
Key Notes :
Proportional Relationships
- A proportional relationship is a relationship between two quantities that can be expressed as a constant ratio or rate.
- In other words, if two quantities are proportional, their ratio remains constant.
Key Characteristics of Proportional Relationships
- Constant Ratio: The ratio of the corresponding values of the two quantities is always the same.
- Graph: When graphed, a proportional relationship forms a straight line passing through the origin (0, 0).
- Equation: A proportional relationship can be represented by the equation y = kx, where k is the constant of proportionality.
Identifying Proportional Relationships
- Check the ratios: Calculate the ratios of corresponding values of the two quantities. If the ratios are constant, the relationship is proportional.
- Graph the data: Plot the data points on a graph. If the points form a straight line passing through the origin, the relationship is proportional.
- Check the equation: If the equation can be written in the form y = kx, where k is a constant, the relationship is proportional.
Example:
Consider the following table of values:
| x | y |
|---|---|
| 2 | 6 |
| 4 | 12 |
| 6 | 18 |
| 8 | 24 |
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To determine if the relationship is proportional, calculate the ratios of corresponding values:
- 6/2 = 3
- 12/4 = 3
- 18/6 = 3
- 24/8 = 3
Since the ratios are constant, the relationship is proportional.
Key Points to Remember:
- A proportional relationship has a constant ratio between the corresponding values of the two quantities.
- When graphed, a proportional relationship forms a straight line passing through the origin.
- A proportional relationship can be represented by the equation y = kx.
- To identify a proportional relationship, check the ratios, graph the data, or examine the equation.
Learn with an example
Look at this graph.
Is there a directly proportional relationship?
yes no
You can tell that the relationship is not directly proportional by looking at the graph. The graph is a straight line, but it does not pass through the origin. So, the relationship is not directly proportional.
You can also confirm that the linear relationship is not directly proportional by showing that the relationship cannot be written as y = kx, where k is a constant ratio.
First, create a chart. Use points from the graph, such as (2, 6) and (4, 9).
| Jam made (litres)(y) | 6 | 9 |
| Days(x) | 2 | 4 |
Now divide “Jam made (litres)(y)” by “Days(x)” to find the ratio (k)
| Jam made (litres)(y) | 6 | 9 |
| Days(x) | 2 | 4 |
| Ratio (k) | 3 | 2.25 |
The ratio (k) is not constant, so the relationship can’t be described by the equation y = kx, where k is a constant ratio. This means that the relationship is not directly proportional.
Look at this graph.
Is the total number of pieces Lucy knows proportional to the number of weeks she takes lessons?
yes no
You can tell that the relationship is directly proportional by looking at the graph. The graph is a straight line and it passes through the origin. So, the relationship is directly proportional.
You can also confirm that the linear relationship is directly proportional by showing that the relationship can be written as y = kx, where k is a constant ratio.
First, create a chart. Use points from the graph, such as (1, 1), (2, 2), (3, 3), and (4, 4).
| Number of Pieces learnt (y) | 1 | 2 | 3 | 4 |
| Number of weeks (x) | 1 | 2 | 3 | 4 |
Now divide “Number of Pieces learnt (y)” by “Number of weeks (x)” to find the ratio (k) .
| Number of Pieces learnt (y) | 1 | 2 | 3 | 4 |
| Number of weeks (x) | 1 | 2 | 3 | 4 |
| Ratio (k) | 1 | 1 | 1 | 1 |
The ratio is constant (k = 1), so the relationship can be described by the equation y = 1x. This equation means that the number of pieces learnt is always 1 times the number of weeks.
Because the relationship can be written as y = 1x, the relationship is directly proportional.
Look at this graph.
Is the total distance cycled proportional to the number of trips to work?
yes no
You can tell that the relationship is not directly proportional by looking at the graph. The graph is a straight line, but it does not pass through the origin. So, the relationship is not directly proportional.
You can also confirm that the linear relationship is not directly proportional by showing that the relationship cannot be written as y = kx, where k is a constant ratio.
First, create a chart. Use points from the graph, such as (1, 3) and (6, 10).
| Total distance cycled (kilimetres)(y) | 3 | 10 |
| Number of trips to work(x) | 1 | 6 |
Now divide “Total distance cycled (kilimetres)(y)” by “Number of trips to work(x)” to find the ratio (k)
| Total distance cycled (kilimetres)(y) | 3 | 10 |
| Number of trips to work(x) | 1 | 6 |
| Ratio (k) | 3 | 1.66 |
The ratio (k) is not constant, so the relationship can’t be described by the equation y = kx, where k is a constant ratio. This means that the relationship is not directly proportional.
let’s practice!
