Inverse Proportion
Inverse proportion describes a fundamental relationship between two quantities: when one quantity increases, the other decreases in such a way that the product of the two quantities remains constant. For example, if we consider the time taken to complete a job with respect to the number of workers, as more workers join a task, the time to finish decreases correspondingly. The inverse relationship can be expressed through the equation xy = k, where k is a constant.
Key Points:
- Examples of inverse proportions are highlighted through practical scenarios such as Zaheeda's travel speeds, book purchases based on price, and resource allocation.
- A table illustrates how increasing the price of books results in fewer books purchased with a fixed budget, showing an inverse relationship.
- Activities and thought experiments encourage students to identify and understand further examples of inverse proportion in everyday life.
This section builds upon the principles of direct proportion by contrasting them and enabling students to grasp the versatile nature of mathematical relationships.
Example :
There are 80 students in a dormitory. Food provisions for them is for 25 days. How long will these provisions last if 15 more students join the group?
Solution: Suppose the provisions last for \( x \) days when the number of students is 95. We have the following table:
\[ \text{Number of students} = 80 \quad \text{Number of days} = 25 \]
Note that the more the number of students, the sooner would the provisions exhaust. Therefore, this is a case of inverse proportion.
\[ 80 \times 25 = 95 \times y \]
So, \[ 2000 = 95y \quad \Rightarrow \quad y = \frac{2000}{95} \approx 21.05 \]
Thus, the provisions will last for approximately 21 days if 15 more students join the dormitory.