How to Use Math to Solve Problems

Mathematics is an enormous topic. You probably started your mathematical journey when you learned your numbers around age three. And you will keep using math for the rest of your life. You may decide that doing simple math with a calculator takes care of your needs. Or you may continue to advanced research with a PhD -- or anything in between.

In this article we will help you learn to evaluate your need to acquire math skills. We will do this by describing five broad areas of applied mathematics in today's world of work. The links will guide you to expanded discussions of these issues.

By looking at these broad areas, you should be able to determine what types of math instruction you will want to pursue in the future to meet your professional and personal needs.

Management Science

Management science is a branch of math that is also sometimes called operations research. The nuts and bolts of management science are concerned with finding ways to make operations as productive and economic as possible.

Efficiency matters to all organizations. Management science uses math to make it possible to apply quantitative techniques (that is, techniques involving the measurement of quantity or amount) to project planning. The underlying principle of management science is to find the best method for solving a problem.

Some examples where special math techniques come into play are:

Efficient Circuits: This involves problems of developing efficient circuits when the objective is to cover each segment of the circuit only once. Some examples are parking control, mail delivery, garbage collection, street cleaning and plowing, reading electric meters and many other tasks that require the use of transportation networks.

Another version is involved when the objective is to visit set points in the circuit in a certain order, or at minimum cost. These are also called traveling salesman problems.

Planning and Scheduling: Another aspect of delivering services in a complex society involves the accurate planning of human and machine resources. Simple scheduling considers only independent tasks. Critical path scheduling requires the completion of some tasks before being able to proceed with later ones.

Linear Programming: This mathematical technique was developed to solve mixture problems. In these problems, a variety of resources are available in limited quantities. The desired outcome is to combine the resources to produce the most profit. For example, a juice manufacturer has a set amount of two different fruits available. The manufacturer can combine the fruits in many different proportions and then sell them at variable rates of profit.


Statistics is the science of gathering data, putting them in clear and usable form, and then interpreting them to draw conclusions about the world around us.

Collecting Data: Problems of defining what a population is for purposes of data collection, how the data will be collected and what experiments will be conducted to produce data all affect the statistical results of any social measurement activity.

Describing Data: Collected data have to be processed to be useful. They have to be converted to numbers and the numbers are the raw material for drawing conclusions from the data. Numerical summaries can be represented by graphs.

Probability: The result of many thousands of chance outcomes can be known with near certainty. The theory of probability describes the predictable long-run patterns of random outcomes.

Statistical Inference: Inference is the process of reaching conclusions from evidence. In statistical inference, numerical evidence is used. Formal statistical evidence requires a statement of probabilities. Statistical inference is used in polling, testing and statistical process control in manufacturing.

Social Choice

Mathematics and computers play an important role in understanding social institutions and human behavior. Mathematics is applied more and more to human decision making.

The problem in social choice is turning individual preferences for different outcomes into a single choice by the group as a whole. The voting method used to make the choice can significantly affect the outcome. Some of the different types of voting methods are majority rule, strategic, sincere, sequential and approval voting.

Weighted Voting Systems: In some voting systems "one person, one vote" applies. But there are weighted voting systems where the vote of each individual is based on a measure of influence. Shareholders who own stock would be an example of this type of voter.

Game Theory: This is an approach that brings mathematics and the scientific method to bear on the topic of conflict and cooperation. A zero-sum game is one in which the payoff to one player is the opposite of the payoff to the other. Various strategies are employed to achieve the best outcomes for each player in various types of games.

Fair Division and Apportionment: The goal of fair allocation problems is to have all participants feel that they can obtain a fair and unbiased share of the available benefits or losses in light of the limitations present in the situation.

Size and Shape

Mathematics is the study of patterns and relationships. The patterns may be numerical, geometric and symmetric.

Patterns: Patterns appear in nature which are balanced, regular, symmetric or isometric. Mathematical formulas can be used to measure these. Patterns also apply to growth and form.

Measurement: Euclidean geometry made it possible to measure the physical world strictly with visual observation. It provides the capability to estimate inaccessible distances.

Modern Science: The experimental method, quantitative approach and mathematical theory characterize modern science. Mathematical theory makes it possible to make predictions based on observed physical laws. Elliptic geometry applies in a system where there are no parallel lines. The theory of relativity revolutionized science by using a new way of thinking about events in space-time.


Computers are computational machines that were developed to perform mathematical procedures. Computing machinery made it possible to speed up the calculation of complex problems.

The computer began as an idea to help us understand the meaning of mathematical proof. Mathematics continues to grow and change. Today, computers not only help us to do calculations, they're changing the notion of mathematical proof.

Computer Algorithms: A computational procedure or finite number of definite and effective steps that terminates on every input and produces some output.

Computer Program: A sequence of instructions in a language that a computer can interpret.

Computer Code: Computers use codes to represent the specific type of information being processed. Codes are mechanisms for representing information. Because computers are built from two-state electronic components, they use two-state (binary) codes to represent information.