It proceeds from unknown to known. ‘Analysis’ means ‘breaking up’ . In this method we break up the unknown problem into simpler parts and then see how these can be recombined to find the solution. So we start with what is to be found out and then think of further steps or possibilities the may connect the unknown built the known and find out the desired result.


Example: if  a/b=c/d, prove that ac-2b2/b=c2-2bd/d

Solution start from unknown i.e. to solve this problem we have to start from

ac-2b2/b=c2-2bd/d  and get the answer a/b=c/d


if a2+b2=7ab prove that 2log (a+b) = 2log3+loga+logb


To prove this using analytic method, begin from the unknown.

The unknown is 2log (a+b) = 2log3+loga+logb

Now, 2log (a+b) = 2log 3+ log a+ log b is true

If log (a+b)2 = log 3+ log a + log b is true

If log (a+b)2 = log 9 + log ab is true

If log (a+b)2 = log 9ab is true

If (a+b)2 = 9ab is true

if a2+b2=7ab which is known and true

Thus if a2+b2= 7ab prove that 2log (a+b) = 2log3+loga+logb



  • It is logical method. It leaves no doubts and convinces the learner.
  • It helps student in understanding and strengthen the urge to discover facts.
  • Each steps has the its reason and justification. So , no fixed steps is required.



  • It is lengthy method
  • It is difficult to acquire efficiency and speed
  • It may not be applicable to all topics equally well.


Synthetic Method

It is opposite of analytic.


Example: if  a/b=c/d, prove that ac-2b2/b=c2-2bd/d

Solution start from unknown i.e. to solve this problem we have to start from

a/b=c/d and get the answer ac-2b2/b=c2-2bd/d


Merit Synthetic Method

  • It is short method, it glorify memory, it suit the teachers and it follow the same process as given in the text book.

Demerits Synthetic Method

  •    It is not a psychological method.
  •  There is a scope for forgetting.
  • It makes the students passive listeners and encourages cramming.
  • In this method confidence is generally lacking in the student.
  • There is no scope of discovery.
  • The recall of each step cannot be possible for every child.


Difference between Analytic Method & Synthetic Method

No. Analytic Method Synthetic Method
1 Analysis means breaking up into simpler elements. Synthesis means building up separate element and their combination
2 It proceeds from the unknown to the known facts. It proceeds from the known to the unknown facts.
3 It is a method of discovery. It is a method of present­ation of discovered facts.
4 It is a process of thinking (exploration). It is a product of thought.
5 It is lengthy and laborious. It is short and concise.
6 It pulls apart or breaks up the statement under solution. It puts together or synthesizes known facts.
7 It can be rediscovered. Once forgotten, it cannot be recalled.
8 It is slow, round-about and involves trial and error. It is quick, straight forward and does without trail and error.
9 It answers satisfactorily and question that may arise in the mind of pupil. It does not satisfy doubts and questions arising in the mind of the leaner.
10 It is a general method; it is a method for the thinker and discoverer. It is a special device; it is a method for the crammer.
11 The students can recall and reconstruct easily any step if forgotten. It is not that easy to recall or reconstruct any forgotten step.
12 It develops originality. It develops memory.
13 It is informal. It is informal.
14 It is formational. It is simply informational.
15 It is based on heuristic lines. There is no heuristic approach in it.
16 It is fore-runner of synthetic. It is the follower of analysis.

Conclusion: Since analysis is a lengthy method, it needs the help of Synthesis for the removal of doubts. Synthesis is the complement of the analysis method. both the method are interdependent. Teachers should offer help for the analytic form of the solution and that synthetic work should be left for the students.

Source: The Teaching of mathematics by KULBIR SINGH SIDHU (Sterling Publisher Pvt Ltd)

Published by rkdskool

I am working in the field of education for more than 15 years. I teach Math. Presently I'm Working as the vice principal in reputed School.

Leave a Reply

%d bloggers like this: