Constant of Proportionality
- Introduction to Constant of Proportionality
- What is the Constant of Proportionality?
- The Significance of Constant of Proportionality
- How to Determine the Constant of Proportionality?
- Representing Direct Variation in a Table
- Representing Direct Variation in a Graph
- Solved Examples
- Practice Problems
- Frequently Asked Question
Introduction to Constant of Proportionality
The constant of proportionality is a crucial concept in mathematics that defines the relationship between two variables that change together. It represents the factor by which one quantity changes in proportion to another. This constant allows us to quantify and understand the precise connection between the two variables, whether the two variables are directly or indirectly proportional.
What is the Constant of Proportionality?
The constant of proportionality defines the unchanging ratio between two proportional quantities. When two varying quantities are proportional, their ratio or product remains constant. This constant varies depending on the type of proportion: direct variation and inverse variation.
Direct Variation:
In the direct variation, represented by the equation \( y = kx \), as one quantity increases the other also increases at a constant rate. For instance, the cost per item is directly proportional to the number of items purchased. As we increase the count of the same item being purchased, the total cost of the items also increases at a constant rate.
Inverse variation:
In inverse variation, represented by \( y = \frac{k}{x} \), as one quantity increases, the other decreases, and vice versa. For example, the speed of a moving vehicle inversely varies with the time taken to cover a certain distance. As you increase the speed of the vehicle, the time taken to cover a certain distance decreases.
In both cases, \( k \) remains constant, known as the coefficient of proportionality or the constant of proportionality. It is also referred to as the unit rate, indicating the rate of change between the two quantities.
How to Determine the Constant of Proportionality?
`1`. Identify the relationship type: Begin by recognizing whether the relationship between the two quantities is direct or inverse variation. In direct variation, both quantities change in the same direction, while in inverse variation, they change in opposite directions.
`2`. Establish the proportional equation: Express the relationship between the two quantities using the appropriate equation. For direct variation, use the form \( y = kx \), where \( y \) and \( x \) are the quantities, and \( k \) is the constant of proportionality. For inverse variation, utilize \( y = \frac{k}{x} \), where \( y \) and \( x \) are the quantities, and \( k \) is the constant of proportionality.
`3`. Substitute given values: Plug the provided values for \( x \) and \( y \) into the proportional equation obtained in step `2`. Ensure that the values correspond to a specific data point within the given relationship.
`4`. Solve for the constant of proportionality: Calculate the value of the constant \( k \) by solving the equation derived in step `3` for \( k \). This value represents the constant of proportionality for the given relationship.
Example `1`: Determine the constant of proportionality if \( y = 15 \) when \( x = 5 \), and \( y \) is directly proportional to \( x \).
Solution:
As \( y \) is directly proportional to \( x \), we can represent the relationship between `x` and `y` as \( y = kx \).
Given \( y = 15 \) when \( x = 5 \), substitute these values into the equation:
\(15 = k \times 5\)
To find \( k \):
\(k = \frac{15}{5} = 3\)
Thus, the constant of proportionality is `3`.
Example `2`: Determine the constant of proportionality if \( y = 20 \) when \( x = 10 \), and \( y \) is inversely proportional to \( x \).
Solution:
To find the constant of proportionality when \( y \) is inversely proportional to \( x \), we use the formula \( y = \frac{k}{x} \), where \( k \) is the constant of proportionality.
Given that \( y = 20 \) when \( x = 10 \), we can substitute these values into the equation:
\( 20 = \frac{k}{10} \)
Now, to solve for \( k \), we multiply both sides of the equation by `10`:
\( k = 20 \times 10 = 200 \)
So, the constant of proportionality \( k \) is `200`.
Representing Direct Variation in a Table
When representing direct variation in a table, you list the values of two related quantities, usually \( x \) and \( y \), where one quantity is directly proportional to the other. Here's an example:
In this table, as \( x \) increases, \( y \) also increases, and they do so at a constant rate. This indicates direct variation between \( x \) and \( y \).
Finding the constant of proportionality from a table:
To find the constant of proportionality (\( k \)) from a table representing direct variation, you can choose any two corresponding values of \( x \) and \( y \) and then divide \( y \) by \( x \). Here's how:
`1`. Choose two pairs of values from the table, such as \( (x_1, y_1) \) and \( (x_2, y_2) \).
`2`. Calculate the ratio of \( y \) to \( x \) for each pair: \( \frac{y_1}{x_1} \) and \( \frac{y_2}{x_2} \).
`3`. Since \( y \) is directly proportional to \( x \), these ratios should be equal.
`4`. Set up an equation using one of the pairs: \( \frac{y_1}{x_1} = k \), where \( k \) is the constant of proportionality.
`5`. Solve for \( k \) by substituting the values of \( y_1 \) and \( x_1 \) from the chosen pair.
Let's take an example using the table above:
Choose the pair \( (1, 5) \) and \( (2, 10) \).
\( k = \frac{y_1}{x_1} = \frac{5}{1} = 5 \)
Thus, the constant of proportionality (\( k \)) is `5`. This means that \( y \) is `5` times \( x \) in this direct variation relationship.
Representing Direct Variation in a Graph
When representing direct variation in a graph, you typically plot the values of two related quantities, usually denoted as \( x \) and \( y \), on a coordinate plane. In direct variation, as one variable increases, the other variable also increases at a constant rate.
Here's how to represent direct variation in a graph:
`1`. Plot the points on a coordinate plane using the values of \( x \) and \( y \) from your data.
`2`. Connect the points with a straight line. In direct variation, the graph will always be a straight line passing through the origin `(0,0)`.
For example, consider the direct variation relationship \( y = 2x \). If we choose values of \( x \) such as `1`, `2`, `3`, and `4`, corresponding values of \( y \) would be `2`, `4`, `6`, and `8` respectively. Plotting these points `(1, 2)`, `(2, 4)`, `(3, 6)` and `(4, 8)` and connecting them with a straight line will result in a graph representing direct variation.
Finding the constant of proportionality from a graph:
To find the constant of proportionality (\( k \)) from a graph representing direct variation, you can choose any point on the line and then determine the ratio of \( y \) to \( x \) at that point. Here's how:
`1`. Select a point \( (x_1, y_1) \) on the graph.
`2`. Calculate the ratio of \( y \) to \( x \) for that point: \( \frac{y_1}{x_1} \).
`3`. This ratio represents the constant of proportionality (\( k \)).
Since the graph is a straight line passing through the origin in direct variation, any point on the line can be used to find \( k \).
For example, consider the graph of \( y = 2x \).
If we select the point `(1, 2)`, the ratio \( \frac{y_1}{x_1} \) would be \( \frac{2}{1} = 2 \). Thus, the constant of proportionality (\( k \)) is `2`. This means that \( y \) is twice \( x \) in this direct variation relationship.
The Significance of Constant of Proportionality
The constant of proportionality is a fundamental concept in mathematics with broad applications. It enables us to understand the relationship between varying quantities and their rates of change. For instance, consider the scenario of adjusting ingredient ratios in a recipe. Maintaining proportional relationships ensures the desired taste and consistency in the final dish. By identifying the constant of proportionality, we can accurately adjust the quantities of ingredients to achieve the desired outcome.
Moreover, the constant of proportionality plays a crucial role in various real-life scenarios. For instance, when designing a scaled model of a car, maintaining proportional dimensions is essential for accuracy. By understanding the constant of proportionality, engineers can ensure that the scaled model accurately represents the original car's proportions, allowing for precise analysis and design modifications.
Additionally, the constant of proportionality is vital in financial contexts, such as determining price markups and discounts. For businesses, understanding the constant of proportionality allows for effective pricing strategies, ensuring profitability while remaining competitive in the market.
Solved Examples
Q`1`. \( y \) varies directly with \( x \) and \( y = 45 \) when \( x = 9 \). Using the constant of proportionality, determine the value of \( y \) when \( x = 150 \).
Solution:
Start by expressing the equation of the constant of proportionality as \( y = kx \). Substitute the given values \( x = 9 \) and \( y = 45 \), then solve for \( k \):
\( 45 = k \times 9 \implies k = \frac{45}{9} = 5 \)
Now, the equation becomes \( y = 5x \). Substitute \( x = 150 \) to find \( y \):
\( y = 5 \times 150 = 750 \)
Therefore, when \( x = 150 \), the value of \( y \) is `750`.
Q`2`. Emily takes `10` days to reduce `2` kilograms of her weight by exercising for `40` minutes each day. If she increases her exercise duration to `1` hour and `20` minutes daily, how many days will it take her to achieve the same weight loss?
Solution:
In this scenario, exercise time and days needed to achieve weight loss are inversely proportional. Let \( m \) represent minutes and \( d \) represent days. We have \( m = \frac{k}{d} \).
Given \( m_1 = 40 \), \( m_2 = 80 \), and \( d_1 = 10 \), we need to find \( d_2 \).
Start by finding the constant \( k \):
\( k = 40 \times 10 = 400 \)
Now, apply the constant to find \( d_2 \):
\( 80 = \frac{400}{d_2} \)
\( d_2 = \frac{400}{80} = 5 \)
Therefore, if Emily exercises for `1` hour and `20` minutes each day, it will take her only `5` days to lose `20` kilograms of weight.
Q`3`. Examine the following table. Do the variables display any type of proportionality? If so, what is the constant of proportionality?
Solution:
To determine if there's proportionality, we utilize the equation \( y = kx \), where \( k \) represents the constant of proportionality.
We compute the ratios for each pair of \( x \) and \( y \):
For \( x = 4 \), \( y = 8 \), \( k = \frac{8}{4} = 2 \).
For \( x = 10 \), \( y = 20 \), \( k = \frac{20}{10} = 2 \).
For \( x = 6 \), \( y = 12 \), \( k = \frac{12}{6} = 2 \).
For \( x = 15 \), \( y = 30 \), \( k = \frac{30}{15} = 2 \).
Since all the ratios yield the same constant \( k = 2 \), the variables exhibit direct proportionality with a constant of proportionality \( k = 2 \).
Practice Problems
Q`1`. If \( y \) varies directly with \( x \) and \( y = 20 \) when \( x = 5 \), what is the value of \( y \) when \( x = 12 \)?
- `48`
- `24`
- `40`
- `32`
Answer: a
Q`2`. The time taken to complete a task is inversely proportional to the number of people working on it. If `6` people can complete the task in 4 hours, how long will it take for `10` people to complete the same task?
- `2.4` hours
- `2.5` hours
- `1.5` hours
- `3` hours
Answer: a
Q`3`. A car travels `240` miles in `3` hours at a constant speed. What distance will it travel in `5` hours at the same speed?
- `320` miles
- `400` miles
- `200` miles
- `360` miles
Answer: b
Q`4`. If the weight of an object is directly proportional to its volume, and the weight is `15` kilograms when the volume is `5` cubic meters, what will be the weight when the volume is `12` cubic meters?
- `36` kilograms
- `24` kilograms
- `30` kilograms
- `45` kilograms
Answer: a
Q`5`. The number of pages a printer can print is directly proportional to the time taken to print. If it takes `6` hours to print `300` pages, how long will it take to print `500` pages?
- `14` hours
- `12` hours
- `10` hours
- `8` hours
Answer: c
Frequently Asked Questions
Q`1`. What is the constant of proportionality?
Answer: The constant of proportionality is a fixed value that represents the ratio between two proportional quantities. It indicates how one quantity changes about another in a proportional relationship.
Q`2`. How do you find the constant of proportionality?
Answer: To find the constant of proportionality between two directly proportional values, you can use the formula \( k = \frac{y}{x} \), where \( y \) and \( x \) are corresponding values of the two quantities in the proportional relationship.
Q`3`. What does it mean if the constant of proportionality is zero?
Answer: If the constant of proportionality is zero, it indicates that there is no proportional relationship between the two quantities. This means that changes in one quantity do not affect the other quantity in any predictable manner.
Q`4`. Can the constant of proportionality be negative?
Answer: Yes, the constant of proportionality can be negative. A negative constant of proportionality indicates an inverse proportionality, where an increase in one quantity leads to a decrease in the other quantity, and vice versa.
Q`5`. How is the constant of proportionality useful in real life?
Answer: The constant of proportionality is used in various real-life scenarios such as scaling models, adjusting recipe ingredients, calculating rates of change, and determining pricing strategies. It helps in understanding and predicting the relationship between different quantities.