The formula (or equation) is a relation between the interest and the other three quantities that are related to it, the principal, the rate of interest and the period. This formula is a mathematical model and its relation which denotes a few real-life situation.

Mathematical models are used to calculate so many real-life situations like:

Satellite launching.

Predicting the arrival of the monsoon.

Controlling pollution payable to vehicles.

Reducing traffic jams in large cities.

Here we will see the process of constructing mathematical models, which is called mathematical modelling.

## Stages in Mathematical modelling:

The stages involved in mathematical modelling are formulation, solution, interpretation and validation.

### 1. Formulation:

We analyse the problem and see which factors have a major influence on the solution to the problem. These are known as the relevant factors. We neglect  the other factors like the nature of the route, driving speed etc. otherwise, the problem would have been much  difficult to calculate. The factors that we ignore are the irrelevant factors.

We then describe the problem mathematically, in the form of one or more mathematical equations.

### 2. Solution:

We find the solution of the problem by solving the mathematical equations obtained in step 1 using some suitable method.

### 3. Interpretation:

What the solution obtained in step 2 means in the context of the original word problem.

## Example for mathematical modelling:

Luke travelled 432 kilometers on 48 liters of petrol in his car. He has to go by his car to a place which is 180 km away. How much petrol does he need?

Solution:

Stage 1: Formulation:

The more petrol we require, that is, the amount of petrol we need varies directly with the distance he travel.

Luke need Petrol for travelling 432 km = 48 liters.

Luke need Petrol for travelling 180 km =?

Mathematical description: Let

X = distance luke travel

Y = petrol luke needed

Y varies directly with x

So,                     y = k x, where k is constant.

Luke able to travel 432 kms with 48 liters of petrol.

Hence,              y = 48, x = 432.

So                      k = `y/x`  =  `48/432`   = `1/9` .

Since                 y = k x,

Therefore,         y = `1/9`

Stage 2: Solution:

We have to calculate the petrol and  we must travel 180 kms;

So, we have to find out the value of y when x = 180. put x = 180 in (1). Now

Y=  `180/9` = 20.

Stage 3: Interpretation:

Because y = 20, Luke have to travel 180 kms. So he need 20 liters of petrol.