Inverse Proportion
Two quantities may change in such a manner that if one quantity increases, the other quantity decreases and vice versa. For example, as the number of workers increases, time taken to finish the job decreases. Similarly, if we increase the speed, the time taken to cover a given distance decreases. To understand this, let us look into the following situation.
Let us consider example. A school wants to spend ₹ 6000 on mathematics textbooks. How many books could be bought at ₹ 40 each? Clearly 150 books can be bought. If the price of a textbook is more than ₹ 40, then the number of books which could be purchased with the same amount of money would be less than 150. Observe the following table.
| Price of each book (in ) | Number of books that can be bought |
|---|---|
| 40 | 150 |
| 50 | 120 |
| 60 | 100 |
| 75 | 80 |
| 80 | 75 |
| 100 | 60 |
If
We say that x and y are in inverse proportion.
Do this
In the below drawing panel, draw a certain number of row and columns (straight lines) such that the total number of divisions within the rows and columns is equal to 48.
In how many ways can this be done?
| Number of Rows (R) | Count | Number of Columns (C) | Count |
|---|---|---|---|
| (R1) | 2 | (C1) | 24 |
| (R2) | 3 | (C2) | 16 |
| (R3) | 4 | (C3) | 12 |
| (R4) | 6 | (C4) | 8 |
| (R5) | 8 | (C5) | 6 |
What do you observe? As R increases, C decreases.
For 2 rows there will be 24 columns. For 3 rows there are 16 columns, also for 8 rows there are 6 columns. By putting these values in table we get:
Example 4
If 15 workers can build a wall in 48 hours, how many workers will be required to do the same work in 30 hours?
Solution
We have the following table.
| Number of hours | Number of workers |
|---|---|
| 48 | 15 |
| 30 | y |
Example 5
There are 100 students in a hostel. Food provision for them is for 20 days. How long will these provisions last, if 25 more students join the group?
Solution:
We have the following table.
| Number of students | Number of days |
|---|---|
| 100 | 20 |
| 125 | y |
Note that more the number of students, the sooner would the provisions exhaust. Therefore, this is a case of inverse proportion.
- So, 100 × 20 = 125 × y
- which gives the value of y
- y =
- Thus, the provisions will last for 16 days, if 25 more students join the hostel.
Do this
- Take a sheet of paper. Fold it as shown in the figure. Count the number of parts and the area of a part in each case.
Tabulate your observations and discuss with your friends. Is it a case of inverse proportion? Why?
| Number of parts | Area of each part |
|---|---|
| 1 | area of the paper |
| 2 | |
| 4 | |
| 8 | |
| 16 |