Expert: Josh - 11/3/2009

Question
How can i solve mathematical problems...involving fractions , ratios and also trigonometric problems without using any calculator.?
Will u please also give me some examples.?

Answer
Hello Awais,

Since you did not mention your school level, it is rather difficult for me to address your specific needs. If you look up "fractions", "trigonometry", "examples" etc. on Yahoo or Google, you will probably find plenty of tutorials on these topics. With java applets and interactive web content, I would think they can do a better job at explaining these things to you than what I can manage in this forum.

However, one thing I would like to do is to give you a brief, unifying view of "fractions" "ratios" and "basic trigonometry". These are not separate theories which have nothing to do with one another. In fact, we can view them as all related in some way.

First, fractions e.g., 2/3, 1/4 (or more generally, A/B for some numbers A and B) tell us about the PROPORTION of A relative to B. For example, 1/4 (one quarter) may be understood as "one out of four equal portions". The same information can be expressed as a ratio, like 1:4. Once we get to the mechanics of it, the actual adding and subtraction of fractions, I recommend that you learn about methods for finding "common denominators". This is the way most people go about combining fractions without using a calculator. On this topic, look up "adding fractions", "least common denominator" or something on a search engine for many well organized websites.

The second comment about trigonometric functions, such as sin(x), cos(x) and tan(x) is that they "simply relate an angle with the ratio between any two sides of a right angle triangle". This, in essence, is how elementary trigonometric functions are related to ratios. I think this is a good point to remember before you start to learn about trigonometric identities and solving algebraic equations.

Example: (figure not drawn to scale)
C
|
|
|
A---------------B

| Suppose that the angle ABC is 30 degrees. In trigonometry, it is well known that
| sin(30)=1/2. This simply tells us that if we measure the sides |AC| and |BC|,
| the ratio between the opposite side to the 30 degree angle (namely, |AC|) and
| the hypotheuse (longest side in a right angle triangle |BC|) is exactly a half.
| i.e., |AC|/|BC|=0.5.

The problem solving aspect, the routine steps to take etc., are best learned through practice and seeing worked examples -- either from a textbook or from a website -- at a level appropriate to your needs. I encourage you to take this personal journey and discover these things for yourself.