When two variables are directly or indirectly proportional to each other, then their relationship can be described as y = kx or y = k/x, where k determines how the two variables are related to one another. This k is known as the constant of proportionality.
1. What is Constant of Proportionality? 2. Why Do We Use The Constant of Proportionality? 3. How to Solve The Constant of Proportionality? 4. Identifying The Constant of Proportionality 5. Solved Examples 6. Practice Questions 7. FAQs on The Constant of Proportionality
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Constant of proportionality is the constant value of the ratio between two proportional quantities. Two varying quantities are said to be in a relation of proportionality when, either their ratio or their product yields a constant. The value of the constant of proportionality depends on the type of proportion between the two given quantities: Direct Variation and Inverse Variation.
- Direct Variation: The equation for direct proportionality is y = kx, which shows as x increases, y also increases at the same rate. Example: The cost per item(y) is directly proportional to the number of items(x) purchased, expressed as y ∝ x
- Inverse Variation: The equation for the indirect proportionality is y = k/x, which shows that as y increases, x decreases and vice-versa. Example:The speed of a moving vehicle (y) inversely varies as the time taken (x) to cover a certain distance, expressed as y ∝ 1/x
In both the cases, k is constant. The value of this constant is called the coefficient of proportionality. The constant of proportionality is also known as unit rate.
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If we want to draw a picture of the Taj Mahal by sitting in front of it on a piece of paper by looking at the real image in front of us, we should maintain a proportional relationship between the measures of length, height, and width of the building. We need to identify the constant of proportionality to get the desired outcome. Based on this, we can draw the monument with proportional measurements. For instance, if the height of the dome is 2 meters then in our drawing we can represent the same dome with height 2 inches. Similarly, we can draw other parts. In such scenarios, we use constant of proportionality.
Working with proportional relationships allows one to solve many real-life problems such as:
- Adjusting a recipe’s ratio of ingredients
- Quantifying chance like finding odds and probability of events
- Scaling a diagram for drafting and architectural uses
- Finding percent increase or percent decrease for price mark-ups
- Discounts on products based on unit rate
Important notes
- To check if the 2 quantities are proportional or not, we have to find the ratio of the two quantities for all the given values. If their ratios are equal, then they exhibit a proportional relationship. If all the ratios are not equal, then the relation between them is not proportional.
- If two quantities are proportional to one another, the relationship between them can be defined by y = kx, where k is the constant ratio of y-values to corresponding x-values.
- The same relationship can also be defined by the formula x=(1/k)y, where 1/k is now the constant ratio of x-values to y-values.
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