The mathematics is a subject that has vast applications. There are two main subdivisions in mathematics - pure and applied. Pure mathematics is concerned about totally abstract concepts, such as - abstract algebra, geometry, analysis, number theory etc. On the other hand, the applied mathematics is the practical application of mathematics. The applied mathematics usually requires broad knowledge and deep understanding of mathematical concepts. The training in mathematical fundamentals is also required in order to analyze, model and calculate real-life problems. |

The applied mathematics refers to a professional specialty though which the study and formulation of mathematical tools are being worked to solve practical problems. Hence, applied mathematics is defined to be the combination of specialized knowledge and mathematical science. This branch is actually the practical and real life application of development of mathematical theories. Thus, applied mathematics is said to be closely related with pure mathematics research.

The variety of topics are covered by applied mathematics. Here, we are going to classify them into two - topics that are commonly used by us in daily life and topics that are used by applied mathematicians. The commonly used fields of applied mathematics may include common branches of mathematics, such as - arithmetic, algebra, vector algebra, geometric applications, calculus, trigonometry etc.

While the topics that need specific knowledge and understanding and is used by applied mathematicians are listed below:

**1)**

**Dynamical Systems**that uses differential equations, partial differential equation and nonlinear partial differential equations.

**2)**

**Computing**being utilized in data structures, algorithm methods, scientific computing, engineering, geometry, numerical analysis, graphics etc.

**3) Mathematical physics**which deals with application of mathematics in physics problems used in classical mechanics, relativity, quantum theory, wave theory, string theory etc.

**4)**

**Signal processing and**

**Information theory**involves storage, filtering, reconstruction, collection of information from compression and noise.

**5) Probability and Statistics**

**6)**

**Operations research**which studies about the usage of mathematics tools in decision-making algorithm for the purpose of optimization and improvement of performance in real-life situations.

**7) Game theory**that deals with the models for formalized incentive structures called games. It is utilized in economics, political science, biology, military strategy and social psychology.

**The examples based on applied mathematics are very complicated, thus here we are discussing simple and common problems related to it:**

**Example 1:**In a survey on 150 students, 60 were found to have access to internet at home, while other had not. If from them, a student was chosen at random. What would be the probability that he would not have access to the internet at home.

**Solution:**Total number of students = 150

Number of students having access to internet = 60

Number of students not having access to internet = 150 - 60 = 90

The probability that the chosen student does not have access to internet = $\frac{90}{150}$

= $\frac{3}{5}$ = 0.6

**Example**

**2:**From the top of tall building, a ball is dropped. It was noticed that the ball falls 16 feet in first second, while 48 feet in two seconds, and 80 feet in three seconds and so on. Calculate the distance covered by the ball in eighth second.

**Solution:**Here, the distance covered by the ball forms an arithmetic series in the following way:

16, 48, 80, ...

First term a = 16

Common difference d = 32

We are required to find 8$^{th}$ (n) term of this series. We shall use the following formula:

T$_{n}$ = a + (n - 1)d

T$_{8}$ = 16 + (8 - 1)32

= 16 + (7)32 = 240