Category : 11th Class

Symbols Relation

Learning Objectives

• Introduction
• Case – 1st
• Case – 2nd
• Case – 3rd

 

Introduction

In this section, question pattern is based on basic fundamentals of simple mathematical operations. It is divided into four types. Problems in this type of reasoning questions may be on the symbols used in basic mathematical operations, such as:

Additon:            \[\mathbf{(+)}\]

Subtraction:       \[\mathbf{(-)}\]

Multiplication:    \[\mathbf{(\times )}\]

Division:            \[\mathbf{(\div )}\]

Also (>, <, =) 'greater than' less than' and 'equal to etc.

 

Case - 1^st

Basic BODMAS rule is applied to solve simple mathematical operations.

B          =         Brackets [first solve big bracket, followed middle and small]

O          =         Of

D          =         Division

M         =         Multiplication

A          =         Addition

S          =         Subtraction

Note: This chapter will also help the students to solve the problems of quantitative aptitude along with that of the reasoning.

• Example:

\[(64-14)\div 5+10-2\times 3\]

\[=30-\left( 2\times 6+15\div 3 \right)=12+5=17\]

Now, \[30-17+8\times 3\div 6=30-17+8\times \frac{1}{2}=30-17+4=17\]

\[=10+24=34\]

 

1. If + means\[\mathbf{\div ,-}\]means \[\mathbf{\times ,\div }\] means + and \[\mathbf{\times }\] means -, then the value of \[\mathbf{36\times 12+4\div 6+2-3}\] when simplified is

(a) 12                                                    (b) 38

(c) 42                                                    (d) 56

(e) None of these

Ans.     (c)

Explanation: Option (c) is correct. Using proper signs in the given expression,

We get \[36-12\div 4+6\div 2\times 3=36-3+3\times 3=36-3+9=42.\]

 

2. If P denotes\[\mathbf{\div }\], Q denotes\[\mathbf{\times }\], R denotes + and S denotes -, then 18Q12 P4 R5 S6 =?

(a) 46                                                    (b) 53

(c) 64                                                    (d) 75

(e) None of these

Ans.     (b)

Explanation: Option (b) is correct. Using correct symbols, we get

\[18\times 12\div 4+5-6=18\times 3+5-6=54+5-6=53\]

 

Commonly Asked Questions

 

Direction: If \['+'\text{ }is\text{ }X\text{  }\!\!'\!\!\text{ }-'\text{ }is\text{ }'+'\,\,'\times '~is\,\,'\div '\,and\,\,'\div '\text{ }is\,\,'-'\] then answer the following questions.

 

1. \[\mathbf{9\div 5-4+3\times 2}\]

(a) 2                                                      (b) -9

(c) -3                                                     (d) 10

(e) None of these

Ans.     (d)

Explanation: \[9-5+4\times 3\div 2=9-5+4\times \frac{3}{2}=9-5+6=15-5=10.\]

 

2. \[\mathbf{6+7\times 3-8\div 20=?}\]

(a) -3                                                    (b) 7

(c) 2                                                     (d) 1

(e) None of these

Ans.     (c)

Explanation: \[6\times 7\div 3+8-20\]

\[6\times \frac{7}{3}+8-20=14+8-20=2\]

 

3. \[\mathbf{3\times 2+4-2\div 9=?}\]

(a) -1                                                     (b) 1

(c) -2                                                     (d) 3

(e) None of these

Ans.     (a)

Explanation: \[3\div 2\times 4+2-9\]

\[=\frac{3}{2}\times 4+2-9=6+2-9=-1\]

 

4. \[\mathbf{6-9+8\times 3\div 20=?}\]

(a) -2                                                     (b) 6

(c) 10                                                    (d) 12

(e) None of these

Ans.     (c)

Explanation: \[6+9\times 8\div 3-20=6+24-20=10\]

 

5. \[\mathbf{5\times 4-6\div 3+1=?}\]

(a) 5                                                      (b) 4

(c) -I                                                     (d) 2

(e) None of these

Ans. (e)

 

6. If \[\mathbf{'+'}\] means \[\mathbf{'\times ','-'}\] means\[\mathbf{'\div ','\times '}\]means \[\mathbf{'-'}\] and\[\mathbf{'\div '}\], means\[\mathbf{'+'}\], then what will be the value of\[\mathbf{12\div 48-8\times 4+4=}\]?

(a) 8                                                      (b) 4

(c) 20                                                    (d) 2

(e) None of these

Ans.     (d)

Explanation: \[12+48\div 8-4\times 4=12+6-4\times 4=12+6-16=18-6=2\]

 

Case - 2^nd

Interchange of Signs and Numbers: In this type, interrelated signs and numbers are interchanged of corresponds also.

 

• Example:

Directions: In each of the following questions, an equation becomes incorrect due to the interchange of two signs. One of the four alternations under it specifies the interchange of signs in the equation, which when made will make the equation correct. Find the correct alternative.

 

 

1. \[\mathbf{12\div 2-6\times 3+8=16}\]

(a) \[\div and+\]                                     (b) \[\times and+\]

(c) \[-and+\]                                           (d) \[\div and\times \]

(e) None of these

Ans.     (c)

Explanation: Option (c) is correct. On interchanging - and +, we get:

Given expression \[=12\div 2+6\times 3-8\]

\[=6+6\times 3-8\]

\[=6+18-8=16\]

 

2. \[\mathbf{9+5\div 4\times 3-6=12}\]

(a) \[+and\times \]                             (b) \[\div and-\]

(c) \[+and-\]                                           (d) \[\div and+\]

(e) None of these

Ans.     (b)

Explanation: Option (b) is correct. On interchanging - and -, we get:

Given expression \[=9+5-4\times 3\div 6\]

\[=9+5-4\times 3\div 6\]

\[=9+5-4\times \frac{1}{2}=9+5-2=12\]

           

Commonly Asked Questions

Direction: Correct the following equations by interchanging the two signs:

 

1. \[\mathbf{16+4\div 2-21\times 7=21}\]

(a) \[+and-\]                                          (b) \[+and\times \]

(c) \[-and\div \]                                   (d) \[\times and\div \]

(e) \[-and\times \]

Ans.     (d)

Explanation: \[16+4\times 2-21\div 7=16+8-3=21\]

 

2. \[\mathbf{5\times 15\div 7-20+4=77}\]

(a) \[-and+\]                                         (b) \[\times and\div \]

(c) \[+and\div \]                                 (d) \[+and\times \]

(e) None of these

Ans.     (c)

Explanation: \[5\times 15+7-20\div 4=75+7-5=77\]

3. \[\mathbf{4\times 2+6\div 2-12=2}\]

(a) \[\div and\times \]

(b) \[+and-\]

(c) \[\times and+\]

(d) \[\div and-\]

(e) None of these

Ans.     (a)

Explanation: \[4\div 2+6\times 2-12\]

\[=2+12-12=2\]

Direction: If the given interchanges are made in signs and number, which one of the four equations would be correct?

 

 

4. Given interchanges: signs \[\mathbf{'+' and '-';}\] numbers '5' and '8'.

(a) \[82-35+55=2\]                     (b) \[82-35+55=102\]

(c) \[85-38+85=132\]                (d) \[52-35+55=72\]

(e) None of these

Ans.     (a)

Explanation: \[52+38-88\]

\[=90-88=2\]

 

5. Given interchanges: signs \[\mathbf{'-'}\] and \[\mathbf{'\div '}\]; numbers '2' and '6'.

(a) \[32-12\div 6=30\]                           (b) \[36\div 12-2=1\]

(c) \[32-16\div 6=1\]                              (d) \[36-12\div 6=0\]

(e) None of these

Ans.     (d)

Explanation: \[32\div 16-6=2-2=0\]

 

6. Given interchanges: signs \[\mathbf{'+'}\]and\[\mathbf{'\times '}\]; numbers ‘3’ and ‘7’

(a) \[23+17\times 73=1241\]

(b) \[37+73\times 12=112\]

(c) \[23\times 17+37=428\]

(d) \[23+17x\,\,73=388\]

(e) None of these

Ans.     (d)

Explanation: \[27\times 13+37=351+37=388\]

 

Direction: In each of the following questions/ the two expressions on either side of the sign (=) will have the same value if two terms on either side or on the same side are interchanged. The correct terms to be interchanged have been given as one of the four alternatives under the expressions. Find the correct alternative in each case.                            _

 

7. \[\mathbf{5+3\times 6-4\div 2=6\times 3 10\div 2+7}\]

(a) 4, 7                                                  (b) 5, 7

(c) 6, 4                                                  (d) 6, 10

(e) None of these                                                              

Ans.     (c)

Explanation: On interchanging 6 and 4 on LH.S, we get the statement as:

\[5+3\times 4-6\div 2=4\times 3-10\div 2+7\]or \[5+12-3=12-5+7\text{ }or\text{ }14=14,\] which is true.

 

8. \[\mathbf{7\times 2-3+8\div 4=5+6\times 2-24\div 3}\]

(a) 2, 6                                                  (b) 6, 5

(c) 3, 24                                                (d) 7, 6

(e) None of these

Ans.     (d)

Explanation: On interchanging 7 and 6 we get the statement as:

\[6\times 2-3+8-4=5+7\times 2-24\div 3\]or \[12-3+2=5+14-8\text{ }or\text{ }11=11,\] Which is true.

 

9. \[\mathbf{15+3\times 4-8\div 2=8\times 5+16\div 2-1}\]

(a) 3, 5                                                    (b) 15, 5

(c) 15, 16                                              (d) 3, 1

(e) None of these

Ans.     (a)

Explanation: On interchanging 3 and 5, we get the statement as:

\[15+5\times 4-8\div 2=8\times 3+16\div 2-1\]or \[15+20-20-4=24+8-1\text{ }or\text{ }31=31\]which is true.

 

10. \[\mathbf{6\times 3+8\div 2-1=9-8\div 4+5\times 2}\]

(a) 3, 4                                                 (b) 3, 5

(c) 6, 9                                                 (d) 9, 5

(e) None of these

Ans.     (d)

Explanation: On interchanging 9 and 5 we get the statement as:

\[6\times 3+8\div 2-1=5-8\div 4+9\times 2\text{ }or\text{ }18+4-1=5-2+18\]or\[21=21\], which is true.

 

11. \[\mathbf{8\div 2\times 5-11+9=6\times 2-5+4\div 2}\]

(a) 5, 9                                                  (b) 8, 5

(c) 9, 6                                                  (d) 11, 5

(e) None of these

Ans.     (c)

Explanation: On interchanging 9 and 6, we get the statement as:

\[8\div 2\times 5-11+6\text{ }=9\times 2-5+4\div 2\text{ }or\text{ }4\times 5-11+6=\text{ }18-5+2\text{ }or\text{ }15\text{ }=15,\]which is true.

 

Case 3^rd

In this type of questions certain relations between different sets of elements is given (in terms ''<, '>' or ‘=’), using either the real symbols or substituted symbols. The candidate is required to analyse the given statements and then decide which of the relations given as alternatives follows from the given statements.

 

• Example:

In the following questions, the symbols \[,\text{ }\phi ,\text{ }\$,\text{}%\] and # are used with the following meanings as illustrated below:

'A $ B' means A’ is not smaller than B'

'A # B' means 'A is not greater than B'

'A @ B' means 'A is neither smaller than nor equal to B'

\['A\text{ }\phi \text{ }B'\]means A is neither smaller than nor greater than B'

'A % B' means 'A is neither greater then nor equal to B'

Now, in each of the following questions, assuming the given statements to be true, find which of the three conclusions I, II and III given below them is/are definitely true and give your answer accordingly.

 

1. "Statement: H % J, J \[\phi \] N, N @ R

            Conclusions: I. R % J       II.  H @ J      III. N @ H

(a) Only I is true                                      (b) Only II is true

(c) Only III is true                                   (d) Only I and III are true

(e) None of these

Ans.     (d)

Explanation: Given statements: \[H<J,\text{ }J=N,\text{ }N>R\]

\[R<N,N=J\Rightarrow R<N=J\Rightarrow R<J\,\,i.e.\,\,R\,\,%\,\,J\]

I Relation between H and J: H < J i.e. H % J.

II Relation between N and H:

\[N=J,J>H\Rightarrow N=J>H\Rightarrow N>H\,\,i.e.\,\,NH\]

So, only I and III are true.

 

2. Statement: \[M\text{ }\text{ }J,\text{ }J\text{ }\$\text{}T,\text{}T\,\,\phi\text{}N\]

            Conclusions: I. N # J       II. T % M      III. M @ N

(a) Only land IS are true                          (b) Only li and III are true

(c) Only I and III are true                         (d) All of these

(e) None of these

Ans. (d)

Explanation: Given statement: \[M>J,J\ge T,T=N\]

1. Relation between N and J:

\[N=T,\text{ }T\le J\text{ }N=T<J\Rightarrow \text{ }N\ge J\text{ }i.e.\text{ }N\text{ }\#\text{ }J\]

1. Relation between M and N

\[T\le J,J<M\Rightarrow T\le J<M\Rightarrow T<M\,\,i.e.\,\,T\,%\,M\]

III. Relation between M and N:

\[M>J,J\ge T,T=N\Rightarrow M>J\ge T=N\Rightarrow M>N\,\,i.e.\,\,M\,\,\,\,N\]

Thus, all I, II and III are true.

 

3. Statements: \[D\,\phi \,K,K\#F,FP\]

            Conclusions: I. P @ D       II.  K # P       III.  F $ D

(a) Only I and II are true                          (b) Only II is true

(c) Only II and III are true                       (d) Only III is true

(e) None of these

Ans.     (d)

Explanation: Given statement: \[D=K,K\le F,F>P\]

1. Relation between P and D

\[P<F,F\ge K,K=D\Rightarrow P<F,F\ge D\Rightarrow P<F\ge D\Rightarrow \]No definite conclusion.

1. Relation between K and P:

\[K\le F,F>P\Rightarrow K\Rightarrow F>P\]No definite

III. Relation between F and D

\[F\ge K,K=D\Rightarrow K\,\,D\Rightarrow F\ge D\,\,I.E.\,\,f\,\,\$\,\,d\]

Thus, only III is true.

 

Commonly Asked Questions

1. Statements: \[R\text{ }\#\text{ }D,\text{ }D\text{ }\$\text{}M,\text{}M\text{}\phi\text{}N\]

            Conclusions: I. R # M      II.  N # D      III. N $ R

(a) Only I is true                                     (b) Only II is true

(c) Only III is true                                   (d) All of these

(e) None of these

Ans.     (b)

Explanation: Given statement: \[R\le D,D\ge M,M=N\]

1. Relation between R and M.

\[R\le D,D\ge M\Rightarrow R\le D\ge M\Rightarrow \]No indefinite conclusion.

1. Relation between N and D.

\[N=M,M\le D\Rightarrow N=M\le D\Rightarrow N\le D\,\,i.e.\,\,N\text{  }\!\!\#\!\!\text{ }\,\,D\]

III. Relation between N and R:

\[N=M,M\le D,D\ge R\Rightarrow N=M\le D\ge R\Rightarrow \]No definite conclusion.

Thus, only II is true.

 

2. Statements: \[K\,\,\phi \,\,P,PQ,Q\,\,\$\,\,R\]

            Conclusions: I. K @ R      II.  R % P      III. Q % K

(a) Only I and II are true                          (b) Only II and III are true

(c) Only III is true                                         (d) All of these

(e) None of these

Ans.     (d)

Given statement: \[K=P,P>Q,Q\ge R\]

1. Relation between K and R:

\[K=P,P>Q,Q\ge R\Rightarrow K=P>Q\ge R\Rightarrow K>R\text{ }i.e.\text{ }K\text{ }\text{ }R\]

1. Relation between R and P:

\[R\le Q,Q<P\Rightarrow R\le Q<P\Rightarrow R<P\,\,i.e.\,\,KR.\]

III. Relation between Q and K:

\[Q<P,P=K\Rightarrow Q<P=K\Rightarrow Q<K\text{ }i.e.\text{ }Q\,%\,K.\]

Thus, all I, II and III are true.

3. Statements: K # N, N $ T, T % J

            Conclusions: I. J @ N       II.  K @ T      III. T @ K

(a) Only I and II are true                          (b) Only II and III and true

(c) Only II and III are true                       (d) Only 1^st

(e) None of these

Ans.     (e)

Explanation: Given statement: \[K\le N,\text{ }N\ge T,\text{ }T<J.\]

1. Relation between J and N:

\[J>T,T\le N\Rightarrow J>T\le N\Rightarrow \]no definite conclusion .

II & III. Relation between K and T:

\[K\le N,N\ge T\Rightarrow K\le N\ge T\Rightarrow \]no infinite conclusion

Thus, none of I, II and ill is true.

 

4. Statements: \[M\text{ }\text{ }D,\text{ }D\text{ }\phi \text{ }V.\text{ }V\text{ }\$\text{}W\]

            Conclusions: I. W @ M     II.  M % V     III. D $ W

(a) Only I and II are true                          (b) Only II and III are true

(c) Only III is true                                         (d) Only I and III are true

(e) None of these

Ans.     (c)

Explanation: Given statement: \[M>D,\text{ }D=V,\text{ }V\ge W.\]

1. Relation between W and M:

\[W\le V,V\,D,D<M\Rightarrow W<V=D<M\Rightarrow W<M\,\,.i.e.\,\,W\,\,%\,\,M.\]

1. Relation between M and V:

\[M>D,D=V\Rightarrow M>D=V\Rightarrow M>V\text{ }I.e.\text{ }M\text{ }\text{ }V.\]

III. Relation between D and W:

\[D=V,V\ge W\Rightarrow D=V\ge W\Rightarrow D\ge W\text{ }i.e.\text{ }D\text{ }\$\text{}W.\]

Thus, only III is true.

 

5. If A + E = B + D, A + B > C + E, A + D = 2B, C + E > B + D, then

(a) \[A>B>C>D>E\]                             (b) \[C>B>D>A>E\]

(c) \[C>B>A>E>D\]                              (d) \[C>A>B>D>E\]

(e) None of these

Ans.     (d)

Explanation: \[C+E>B+D,B+D=A+E\Rightarrow C+E>A+E\Rightarrow C>A\]

\[A+B>C+E,C+E,>B+D\Rightarrow A+B>B+D\Rightarrow A>D\]

\[A+D=2B,\,\,A>D\Rightarrow A>B>D\]

\[A+E=B+D,A>B\Rightarrow E<\text{ }D\]

From (i), (iii) and (iv) we get\[L\text{ }C\text{ }>\text{ }A\text{ }>\text{ }B\text{ }>\text{ }D\text{ }>\text{ }E.\]

 

6. If A + B = 2C and C + D = 2A, then

(a) \[A+C=B+D\]                                  (b) \[A+C=2D\]

(c) \[A+D=B+C\]                                  (d) \[A+C=2B\]

(e) None of these

Ans.     (a)

Explanation: Given: \[A+B=2C\]

Adding (i) and (ii), we get: \[A+B+C+D=2C+2A\Rightarrow B+D=A+C\]