Learn to Solve hard MCQ algebra problems fast
Many difficult MCQ algebra problems in major competitive tests such as SSC CGL can be solved easily and quickly if you use inventive principles and methods.
Going through most of the complicated looking Algebra questions in SSC CGL level test, we know now, they look difficult but actually can be easily and quickly solved in most cases, if you know how to tackle them systematically and analytically.
Here we will explain how to use the highly powerful problem solving concepts of Free Resource Use Principle, Pattern recognition and use technique and End State Analysis Approach to reach the solution under 60 seconds with certainty when dealing with the type of problem solved here.
Let's see how.
Before going to solve an actual Algebra problem, we will quickly explain what a few of these problem solving concepts are.
When we explain solution of the problem using these special concepts, you will easily be able to understand how to use them by yourself. And only further continued intelligent use of the concepts would finally enable you to solve such problems by yourself so very fast.
Free resource use principle
This great problem solving principle is being used by many over a long time not only to solve their daily life problems but also for new inventions.
Aside: We are sure many of you also have used it without knowing that you are using a principle with this rather formal academic name.
One of the first mentions of this concept we find in literature of TRIZ, the extensive innovation system created by the great Russian scientist Genrich Altschuller.
This special problem solving concept not only helps inventions, it will now help you to solve your complicated Algebra problems.
The Free Resource Use Principle may be described for your purpose as,
Use any resource that is available free, for solving your problem and reach the solution at no extra cost and in shortest time. The free resource must also be a useful resource in its effectiveness.
Usually such resources remain unnoticed for actual problem solving. It is the job of the problem solver to identify such a free and effective resource and use it in solving her problem.
End State Analysis Approach
This is another very powerful problem solving concept that we have used time and again without previously attaching any name to it and unaware of the scope and power of the concept when precisely defined, studied and applied.
End State Analysis Approach may be stated briefly for you as,
Compare the desirable end states of your problem with the given information and analyze the similarities to know how to solve the problem in minimum number of steps with certainty.
Aside: We have already treated this concept in details. You may refer to it here at your leisure.
This name has been coined first by us as far as our knowledge goes.
Mention of similar concepts we have found to be used by leading Management Consultancy firm McKinsey as well as in TRIZ innovation system. In TRIZ a similar concept is the Ideal Final Result or IFR in short.
The concept of End State Analysis Approach is though more abstract and should cover more varieties of problems with great positive effect.
In the process of solving a problem very fast, you have to use though not only these two problem solving concepts, you will need to apply a number of other concepts. We will mention the concepts other than Free resource use principle and End State analysis approach, in passing here. Later we will present the detailed treatment of the analytical thinking process that is so effective in solving complicated problems.
The promise made so far is alluring. Let's see how to meet the promise by actually solving a complicated SSC CGL level Algebra problem practically in our mind in no time.
Problem example 1.
The value of, $\displaystyle\frac{1}{a^2 +ab + b^2}- \frac{1}{a^2 - ab + b^2} +\frac{2ab}{a^4 + a^2b^2 + b^4}$ is,
- 2
- 1
- -1
- 0
Solution
The problem sure looks complicated, especially the denominators.
But look at the choice values.
Applying Free resource use principle
The choice values are so very simple with no trace of complications! These are your free useful resources - the choice values and the knowledge that these are simple.
How to use this resource? The simplicity assures you that however complicated the denominators look, these must have some uniform pattern and commonality in them, so that we will be able to combine them easily. You will be urged to search for common pattern in them by this knowledge of simplicity in the answer choices.
Applying Problem breakdown technique
Being well aware of the great concept of Problem Breakdown Technique, you leave aside the third more complicated term for now and concentrate on the first two terms. Whatever be your approach, you must combine these two terms first.
Aside: Problem breakdown technique may be stated for you as,
Breakdown a complex problem in simpler and similar parts and attack the easiest one first.
We have used this concept earlier, though not in great details. You may refer to it here in your leisure time.
Technique of solving a simpler and similar problem, the first part
The simpler first problem that we will solve now is simplification of,
$\displaystyle\frac{1}{a^2 +ab + b^2}- \frac{1}{a^2 - ab + b^2}$
Aside: This is an application of the Technique of Solving a Simpler and Similar Problem. This is a well known technique used by Mathematicians extensively, but it can be applied for solving real life problems as well with significantly enhanced results.
The two denominator expressions look quite similar, isn't it?
Applying Pattern recognition technique
In such a case as in this situation, always look for common pattern between the two entities being compared. What are common between the two denominators? The answer you find instantly to be, $a^2 + b^2$, though these are separated by another term between them.
This is useful pattern recognition. As it usually happens, the useful patterns may not be easy to find. There is always some barrier hiding the useful pattterns. You have to look through the barrier.
And in no time you remember the very useful basic Algebraic concept of $(x + y)(x - y) = x^2 - y^2$. This is the concept important here because for combining the two terms, you have to find the LCM of the denominators. And the LCM in this case is the product of the two expressions,
$(a^2 + ab + b^2)(a^2 - ab + b^2)$.
Applying the basic algebraic concept, we transform the LCM of the two denominators to,
$(a^2 + b^2)^2 - a^2b^2$.
What about the interactions of the numerators as a result of combining the first two terms? We can easily see that the two occurrences of the term $(a^2 + b^2)$ cancel out leaving only $-2ab$. Thus, the result of combining the first two terms of the original expression turns out to be,
$\displaystyle\frac{-2ab}{(a^2 + b^2)^2 - a^2b^2}$
Aside: Pattern recognition technique is reflexive in us, it is so very intimately intertwined with all our mental processes. But it can also be nurtured and strengthened. We have already covered this concept in some detail. You may refer to it here during your leisure time.
Solving the whole problem, combining the result of the first part with the second part
Now we will solve finally the transformed whole problem as,
$\displaystyle\frac{-2ab}{(a^2 + b^2)^2 - a^2b^2} + \frac{2ab}{a^4 + a^2b^2 + b^4}$
Again searching for commonality in the two denominators you recall the other absolutely basic concept of $(x + y)^2 = x^2 + 2xy + y^2$.
Though the terms in our problem are not arranged very neatly for your easy identification of this pattern, nevertheless you see through this apparent barrier and start transforming the third denominator.
\begin{align}
(a^4+a^2b^2+b^4) & = (a^4+2a^2b^2+b^4) - a^2b^2 \\
& = (a^2+b^2)^2 - a^2b^2
\end{align}
This is exactly same as the first denominator with the numerators equal but of opposite signs.
What a relief!
Answer: Option d: 0.
Deductive reasoning binds the whole process of problem solving
At every stage, your deductive reasoning skill analyzes the present state and future prospects and helps you to select the right problem solving strategy and technique so that ultimately you achieve the most efficient solution at minimal cost.
If the powerful problem solving concepts are the flowers, deductive reasoning is the thread that connects the flowers into the garland of a beautiful solution!
In this type of problems, the key lies more often than not in usable regular common patterns in the denominators. This is direct application of pattern identification and use technique, but in a specific manner.
Once denominator complexity is resolved, numerator complexity automatically gets resolved.
End note: Though the explanation elaborated here is rather long, perhaps you can perceive the ease of actually solving this problem in well within a minute wholly in your mind and without using pen and paper.
But this is possible only if, you are quite well used to solving complicated looking many problems always using your mind only and using pressure of time.
This is a sure way to build your own repertoire of this type of problem solving strategies and techniques for solving all complex Algebra problems that you may face.
Are these special tricks? Absolutely NO.
Can this ability be attained in a very short time? We won't say, impossible. But yes, it is difficult and needs systematic and intelligent actual solving of complex problems under time pressure.
The list of Difficult algebra problem solving in a few steps quickly is available at, Quick algebra.
To go through the extended resource of powerful concepts and methods to solve difficult algebra problems, you may click on,