# Variation

Two quantities A and B are said to be varying with each other if there exists some relationship between A and B such that the change in A and B is uniform and guided by some rule.

**Some typical examples of variation**

- Area (A) of a circle = Ï€ R
^{2}, where R is the radius of the circle. - At a constant temperature, pressure is inversely proportional to the volume.
- If the speed of any vehicle is constant, then the distance traversed is proportional to the time taken to cover the distance.

# Direct Proportion

If A varies directly to B, then A is said to be in direct proportion to B.
It is written asâ€”A Î± B
It can be understood with the typical example of percentage relating to expenses, consumption and price of the article.
If the price of a article is constant, then Consumption Î± Expenses.
â‡’ Consumption = K. Expenses, where K is proportionality constant.
If we increase consumption by 20%, then the expenses will also increase by 20%.
At a constant price, if a graph is drawn between consumption and expenses by taking them at X-axis and Y-axis respectively, then this graph will be a straight line.

This is what we mean to say with direct proportion.

# Inverse Proportion

If A varies inversely to B, then A is said to be in inverse proportion to B.
It is written asâ€”A Î± 1/B
It can be understood with a time-speed-distance example, where if the distance is constant, then speed Î± 1/time.
Assuming the distance between New Delhi to Patna is 1000 km then consider the following table:

It can be seen here that the multiplier of time is reciprocal of the multiplier of speed.
For any fixed distance, if we draw a graph between speed and time by taking them at X-axis and Y-axis respectively, then this graph will be a curve.

Thus, when the speed is minimum, the time is maximum and when the speed is maximum, the time is minimum.

Example

The height of a tree varies as the square root of its age (between 5 and 17 years). When the age of a tree is 9 years, its height is 4 feet. What will be the height of the tree at the age of 16?

Solution

Let us assume the height of the tree is H and its age is A years.
So, H Î± âˆšA, or, H = K Ã— âˆšA
Now, 4 = K Ã— âˆš9
â‡’ K = 4/3
So, height at the age of 16 years = H = K Ã— âˆšA = 4/3 Ã— 4 = 16/3 = 5 feet 4 inches.

# Direct Relation

A is directly related to B and as B changes, A also changes but not proportionally.

It is written as âˆ’ A = C + K. B, where C and K are constants.

One classical example of direct relation can be seen as the telephone connection in a house. In a telephone connection we pay some money as the rent along with the phone charges according to the rate and number of calls made. So, the total bill = rental + K (number of calls).

If we draw a graph between the number of calls and the total bill by taking them at X-axis and Y-axis respectively, this graph will be a straight line in the following way:

It is written as âˆ’ A = C + K. B, where C and K are constants.

One classical example of direct relation can be seen as the telephone connection in a house. In a telephone connection we pay some money as the rent along with the phone charges according to the rate and number of calls made. So, the total bill = rental + K (number of calls).

If we draw a graph between the number of calls and the total bill by taking them at X-axis and Y-axis respectively, this graph will be a straight line in the following way:

Example

Total expenses at a hostel is partly fixed and partly variable. When the number of students is 20, total expense is Rs 15,000 and when the number of students is 30, total expense is Rs 20,000. What will be the expense when the number of students is 40?

Solution

Expenses = F + K. V; where F is the fixed cost and V is the number of students.
Rs 15,000 = F + K.20 ......(1)
Rs 20,000 = F + K.30......(2)
Solving (1) and (2),
Rs 5,000 = 10. K â‡’ K = Rs 500
So, F = Rs 5000
So, F + 40 K = Rs 5,000 + 40 Ã— 500 = Rs 25,000