## Shortcuts in Reasoning Competitive Exams – Coded Inequalities

Shortcuts in Reasoning Quantitative Aptitude English

#### INTRODUCTION

As we know, 3 x 3=9

Now, we can say that the result of multiplication between 3 and 3 is equal to 9.

Therefore, 3 x 3 = 9 is a case of equality. But when we multiply 3 x 4, we get 12 as a result of this multiplication. It does mean that 3 x 4 ≠9 As 3 x 4, is not equal to 9, it is a case of inequality.

When, we come to know that one thing is not equal to another:

**There can be only two possibilities: –**

- One thing is greater than another thing.

(or) - One thing is less than the another thing.

**When, we denote (1) and (2) mathematically, then we will write.**

- One thing > another thing.

(or) - One thing < another thing,

where ‘>’ denotes ‘greater than’, and ‘<’ denotes ‘less than’.

Hence, you can write,

3 x 4>9

4 x 1 <9

(3 x 4 > 9) means ‘Product of 3 and 4 is greater than 9’.

(4 x l < 9) means ‘Product of 4 and 1 is less than 9’

Sometimes we come across two numbers where, we do not know the exact state of inequality between them.

**Let us see :**

m ≥ n means m is either greater than or equal to n.

m ≤ n means n is either less or equal to m.

Hence, we can summarise the signs to be used in inequalities as below:

**‘=’**denots equal to**‘>’**denots greater than**‘≥’**denots greater than or equal to**‘<’**denots less than**‘≤’**denots less than or equal to

**CHAIN OF INEQUALITIES**

Sometimes two or more inequalities are combined together to create a single inequality having three or more terms. Such combination is called chain of inequalities.

**Note :**

If you see the given problem format (Example). You will find that your primarily task is to combine two or more inequalities to create a single inequality.

**Conditions for Combining Two Inequalities**

**Condition I: **

Two inequalities will be combined if and only if they have a common term.

**Condition II: **Two inequalities will be combined if and only if the common term is greater than (or ‘greater’ than or equal to’) one and less than (or ‘less than or equal to’) the other.

**Example:** 14 > 13, 13 > 12 can be easily combined as ‘14> 13 > 12’.

**Coded Inequalities**

Here,

Clearly, 14 >13 and 13 > 12 have common term 13 and this common term is greater than 12 and less than 14. Hence, 14 > 13 and 13 > 12 have been combined into 14 > 13 > 12 as per the conditions I and II.

**Example:** 17< 19,and 19<20 can be easily combined as 17<19<20.

Here,

Clearly, 17<19 and 19<20 have common term 19 and this common term is greater than 17 and less than 20. Hence, 17 < 19 and 19 < 20 have been combined into 17 < 19 < 20 as per the conditions I and II. Now, let us see some examples of inequalities which can not be combined.

**Some such examples are given below:**

**14>12,19 > 18****18<20,22<25****100>99,80>77****100 <115,118 < 119**

Clearly, (1), (2), (3) and (4) can not be combined as they do not have any common term and therefore, they do not follow condition I and condition II.

**How to Derive Conclusions from a Combined Inequalities? **

To derive conclusion from a combined inequality, you have to eliminate the common term.

**For example,**

- If we have m> l >n

then, our conclusion is

**m > n** - When, we have

m< l<n

then, our conclusion is

**m < n** - When, we have ‘≥’ signs in the combined inequalities then you have to think a little bit more. Let us consider the combined inequality given below:

**m>l>n**

Here, m is either greater than l or equal to l. - Hence, the minimum value for m is equal to l.

But l is always greater than n. Therefore, m is always greater than n.

.’. Our conclusion is

**m > n** - When, we have the following inequalities:-

m>l≥n

In this case, m is always greater than l. and l is either greater than n or equal to it. When l is greater than n; m will obviously be greater than n; Even when l is equal to n; m will be greater than n as m is always greater than L

Our conclusion is**m > n** - When, we have combine inequality

**m≥l ≥ n**Here, m is either greater than l or equal to l.

When m is greater than we have m > l > n, which gives the conclusion

**m > n ———— (A)**When m is equal to l we have m = l ≥ n, which gives the conclusion

**m ≥ n ———— (B)**Combining (A) and (B), we have the final conclusion as

**m ≥ n**

From (a), (b), (c), (d) and (e), we get a rule for deriving conclusions from a combined inequality, we may say it **‘Golden Rule’.**

**GOLDEN RULE**

The conclusion inequality will have an ’≥’ sign or a ‘≤’ sign if and only if both the signs in the combined inequality are ‘≥’ or ‘≤’ sign

Clearly, in (a), (b), (c), (d) and (e) only one inequality (e) (m≥l≤n) has ‘≥’ as its both the sign.

**Remember**

- If m > n, then n < m must be true.
- If m < n, then n > m must be true.
- If m ≥n, then n ≤ m must be true.
- If m ≤ n, then n ≥ m must be true.

**EITHER CHOICE RULES**

- When your derived conclusion is of the type m ≥ n (or m ≤n) then check if the two conclusions are m > n and m = n (or, m < n and m = n). If yes, choice “either follows” is true.
- If neither of the given conclusions seems correct. Then try to check if the given conclusions form a complementary pair. Given conclusions form a complementary pair in the 4 cases given below:-

- m ≥ n and m < n
- m > n and m ≤ n
- m ≤ n and m > n
- m < n and m ≥ n

In such case, the choice “either follows” is correct.

**Shortcut Approach Steps for Solving Problems **

**Step I: **Decode the given symbols like . @, $. δ, #, *,etc.

**Step II:** Take one conclusion at a time and make an idea that which statements are relevant for evaluating it.

**Step III: **Use conditions I and II and the ‘Golden Rule’ to combine the relevant statements and derive a conclusion from it. They are:

**Condition I:** There must be a common term.

**Condition II:** The common term must be less than or equal to one term and greater than or equal to another.

**GOLDEN RULE:**

The conclusion — inequality is obtained by letting the common term be eliminated and it has a ‘≥’ or a ‘≤’ sign if and only if both the inequalities in 2nd step had a ‘≥’ or a ‘ ≤’ sign. In all other cases, there will be a ‘>’ or a ‘<’ sign in the conclusion. After performing the above mentioned three steps, if a conclusion is established and verified, it is well and good. But if does not happen so, then you have to perform 4 more new steps given below:

**New Step I:** Check if the given conclusion directly follows from anyone single statement.

**New Step II:** Check if the conclusion — inequality you get is essentially as same as the given conclusion but written differently.

**New Step III:** Check if the derived conclusion follows ‘Either choice Rule I’.

**New Step IV:** If neither of the conclusions has been proved correct till now, then check ‘Either choice Rule II’.

**EXAMPLE :** **In the following question,**

The symbols ©, @, =,* and $ are used with the following meanings:

**P © Q means ‘P is greater than Q’; **

**P @ Q means ‘P is greater than or equal to Q’; **

**P = Q means ‘P is equal to Q’; **

**P * Q means ‘P is smaller than Q’; **

**P $ Q means ‘P is either smaller than or equal to Q’.**

Now in each of following questions, assuming the given statements to be true, find which of the two conclusions I and II given below them is/are definitely true.

**Give answer: **

- if only conclusion I is true;
- if only conclusion II is true;
- if either I or II is true;
- if neither I nor II is true.
- if both I and II are true.

**Statements :** P © T, M $ K, T = K

**Conclusions :** I. T © M II. T=M

**Solution:** Given statements:

P>T, M≤ K, T = K.

T = K,K>M

=> T>M

=>T>M or T = M Complementary,

=>T©M or T = M-‘pair

So, either I or II is true.

**DIRECT INEQUALITY**

In this type of questions, direct relation between two or more than two elements are given in a meaningful inequality. Candidates are required to establish the relation between elements with the help of used signs between the elements.

**EXAMPLE :**

Which of the following symbols should replace the question mark in the given expression in order to make the expressions.

‘I > L’ as well as ‘M ≥K’ definitely true?

I > J ≥ K ? L < N=M

- (a) >
- (b) <
- (c) ≤
- (d) =
- (e) Either < or <

**Solution: **On putting sign (=) in place of question mark (?)

I> J ≥K = L ≤ N = M

=> meansI>L and M>K

**Remember**

Inequality depends upon combining more than two element with a common term. Now observe the below diagram thoroughly

**Accordance to this diagram Definite Conclusion**

**> = —>> • < = —»<****≥ = —>≥ • ≤ = —»<****≥>—>> • ≤< —»<****< = ≤—»< • > = ≥—>**

**Indefinite Conclusion**

- >< —> No relation • ≥≤ —» No relation
- >≤—> No relation • ≥< —»No relation

**Shortcut Approach**

**Case 1. < OR >**

Two signs opposite to each other will make the conclusion wrong But again. if the signs are in same manner that will not make it wrong.

If A>D then

A < C = False, C > A = False.

But

If E>F>G>H then E >G = True,

F > H = True, E > H = True.

**Statement:** A<D<C<E<B

**Conclusions:**

- C > B —>False
- A < E —> False
- D > B —> False

In simple way, whenever these two sign comes in opposite direction the answer will be false.

**Shortcut Approach**

**Case 2. < OR >**

Two signs opposite to each other will make the conclusion wrong But again if the signs are same then it will be true.

**EXAMPLE**

If A>B A< C = False, C > A= False

But

If A>B>C then A >C = True.

C< A = True.

**Statement:** B>D<A>F> C

**Conclusions:**

- A > C —> True
- B < F —> False
- D> C —> False

**Shortcut Approach**

**Case 3: Sets Priority**

- 1st Priority :< or >
- 2nd Priority: ≤ or ≥
- 3rd Priority: =

**Statement:** P≥R>Q=T≥S

**Conclusions:**

- P > Q —> False
- P>Q —> True
- Q > S —>True

**case 4:**

When it occurs to you that the statement ‘of order is opposite just change the Isign into similar opposite direction. |Then change the sign into similar opposite /corresponding / alternative direction.

If A >B >F >C than F < A —>True

**Example**

[.’. A>B>F= F<B< A]

**Statements :** A>B>F>C;D>E>C

**Conclusions:**

- C <A—>True
- C>A —> False

**Shortcut Approach**

**Case 5.** **> or < and ≥ or ≤**

Whenever there is two conclusions which are false then check for these two symbols (> or < and ≥ or ≤). In most of case where two conclusions are false and these two similar signs are not there respectively then that statement can call it as either or but should check there variable it should same.

**Either Or :**

**Note:**

First thing need to check whether in conclusion any two or more conclusions are wrong then if it is there then check whether the two variables are same. If It happens then write it as ‘Either or’ but after checking their symbols.

**Rules:**

- Both conclusion should False
- Should have same Predicate or Variable
- Check the symbols

If above condition are satisfied then write it as ‘Either Or’ Other wise leave it.

**Note :**

If Rule 3 is satisfied than the conclusions are called’Either Or’.

**EXAMPLE**

**B. Neither Nor**

First thing you need to check whether in your conclusion any 2 or more conclusions are wrong then write it as ‘Neither Nor’ but before checking their symbols.

**Rules:**

- Both conclusion should False
- Check the symbols

If both the rules are satisfied then write it as ” Neither Nor’ other wise leave it.

**EXAMPLE**

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