Some study material for Verbal & NonVerbal Reasoning on Mathematics Operation that can help in your competitive exams

1. Slide 1 of 17
Type - 1
Sol. (b).
Using the correct symbols, we have :
Given expression = 16 + 4 × 5 − 72 ÷ 8
= 16 + 20 − 9 = 36 − 9 = 27.
3. If $ means +, # means −, @ means × and * means ÷,
then what is the value of 16 $ 4 @ 5 # 72 * 8 ?
(a) 25 (b) 27 (c) 28 (d) 36
(e) None of these
Sol. (a).
Using the correct symbols, we have :
4. If ÷ means ×, × means +, + means − and − means ÷,
find the value of 16 × 3 + 5 − 2 ÷ 4.
(a) 9 (b) 10 (c) 19
(d) None of these
Sol. (e).
Using the correct symbols, we have :
Given expression = 27 + 81 ÷ 9 − 6
= 27 + 9 − 6 = 36 − 6 = 30.
1. If ‘<’ means ‘minus’, ‘>’ means ‘plus’, ‘=’ means
‘multiplied by’ and ‘$’ means ‘divided by’ then what
would be the value of 27 > 81 $ 9 < 6 ?
(a) 6 (b) 33 (c) 36 (d) 54
(e) None of these
Sol. (d).
Using the correct symbols, we have :
Given expression = 24 × 12 + 18 ÷ 9
= 288 + 2 = 290.
2. If ‘+’ means ‘divided by’, ‘−’ means ‘added to’, ‘×’
means ‘subtracted from’ and ‘÷’ means ‘multiplied
by’, then what is the value of 24 ÷ 12 − 18 + 9 ?
(a) − 25 (b) 0.72 (c) 15.30 (d) 290
(e) None of these
Given expression 16 3 5 2 4
5
16 3 4 19 10 9.
2
= + − ÷ ×
= + − × = − =

2. Slide 2 of 17
Sol. (b).
Using the correct symbols, we have :
Given expression = (3 × 15 + 19) ÷ 8 − 6
= 64 ÷ 8 − 6 = 8 − 6 = 8 − 6 = 2.
7. If × means ÷, − means ×, ÷ means + and + means −,
then (3 − 15 ÷ 19) × 8 + 6 = ?
(a) − 1 (b) 2 (c) 4 (d) 8
Sol. (c).
Using the correct symbols, we have :
Given expression = 12 ÷ 6 − 3 × 2 + 8
= 2 − 6 + 8 = 10 − 6 = 4.
5. If + means ÷, ÷ means −, − means ×, × means +, then
12 + 6 ÷ 3 − 2 × 8 = ?
(a) − 2 (b) 2 (c) 4 (d) 8
Sol. (c).
Using the correct symbols, we have :
Given expression = 15 × 3 – 10 ÷ 5 + 5
= 45 − 2 + 5 = 50 − 2 = 48.
6. If + means −, − means ×, ÷ means + and × means ÷,
then 15 − 3 + 10 × 5 ÷ 5 = ?
(a) 5 (b) 22 (c) 48 (d) 52
Sol. (b).
Using the correct symbols, we have :
8. If × means +, + means ÷, − means × and ÷ means −,
then 8 × 7 − 8 + 40 ÷ 2 = ?
( ) ( ) ( ) ( )
2 3
1 7 8 44
5 5
a b c d
1
Given expression 8 7 8 40 2 8 7 2
5
7 37 2
6 7 .
5 5 5
= + × ÷ − = + × −
= + = =

3. Slide 3 of 17
Sol. (e).
Using the correct symbols, we have :
Given expression = 15 × 2 + 900 ÷ 90 − 100
= 30 + 10 − 100 = − 60.
9. If × means −, + means ÷, − means × and ÷ means +,
then 15 − 2 ÷ 900 + 90 × 100 = ?
(a) 190 (b) 180 (c) 90 (d) 0
(e) None of these
Sol. (a).
Using the correct symbols, we have :
10.
(a) 0 (b) 8 (c) 12 (d) 16
( )
, , ,
36 4 8 4
?
4 8 2 16 1
÷ + − ÷ × − + ×
× − ×
=
+ × + ÷
If means means means and means
then
( )36 4 8 4
Given expression
4 8 2 16 1
32 8 4 4 4
0.
32 32 1 0 1
− ÷ −
=
× − × +
÷ − −
= = =
− + +

4. Slide 4 of 17
Sol. (b).
Given expression = (23 + 45) × 12 = 68 × 12 = 816.
11. (cd × ef) × bc is equal to -
Directions :
In an imaginary language, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are substituted by a, b, c, d, e, f, g, h, i and j. And 10 is
written as ba.
(a) 684 (b) 816 (c) 916 (d) 1564
Sol. (a).
Using the proper notations in (a), we get the statement as :
4 + 5 × 9 ÷ 3 − 4 = 4 + 5 × 3 − 4 = 4 + 15 − 4 = 15.
12. If ‘−’ stands for ‘division’, ‘+’ for ‘multiplication’, ‘÷’
for ‘subtraction’ and ‘×’ for ‘addition’, then which one
of the following equations is correct ?
(a) 4 × 5 + 9 − 3 ÷ 4 = 15
(b) 4 × 5 × 9 + 3 ÷ 4 = 11
(c) 4 − 5 ÷ 9 + 3 × 4 = 20
(d) 4 ÷ 5 + 9 − 3 + 4 = 18

5. Slide 5 of 17
Sol. (c).
Using the proper notations in (c), we get the statement as :
18 ÷ 3 − 6 + 8 + 4 = 12 or 6 − 6 + 8 + 4 = 12
or 12 = 12, which is true.
13. (a) 18 C 3 D 6 B 8 C 4 G 12
(b) 18 A 3 E 6 B 8 G 4 B 12
(c) 18 C 3 G 6 B 8 B 4 D 12
(d) 18 F 3 B 6 E 8 G 4 E 12
Directions :
In each of the following questions, some symbols are represented by letters as shown below :
Now, identify the correct expression in each case.
Sol. (d).
Using the proper notations in (d), we get the statement as :
15 ÷ 5 < 8 ÷ 4 + 6 ÷ 3 or 3 < 2 + 2
or 3 < 4, which is true.
14. (a) 15 B 5 G 8 B 4 G 6 F 3
(b) 15 C 15 B 8 F 4 B 6 C 3
(c) 15 A 5 E 8 C 4 B 6 E 3
(d) 15 C 5 F 8 C 4 B 6 C 3
+ − × ÷ = > <
B G E C D A F

6. Slide 6 of 17
Sol. (c).
Using the proper notations in (c), we get the statement as :
8 − 4 ÷ 2 < 6 + 3 or 6 < 9, which is true.
15. (a) 6 + 3 > 8 = 4 + 2 < 1 (b) 4 > 6 + 2 × 32 + 4 < 1
(c) 8 < 4 + 2 = 6 > 3 (d) 14 + 7 > 3 = 6 + 3 > 2
Directions :
If > denotes +, < denotes −, + denotes ÷, ^ denotes ×, − denotes =, × denotes > and = denotes <, choose the correct
statement in each of the following questions.

7. Slide 7 of 17
Sol. (c).
On interchanging − and ÷ and 4 and 8 in (c), we get the
equation as :
8 − 4 ÷ 2 = 6 or 8 − 2 = 6 or 6 = 6, which is true.
1. Given interchanges :
Signs − and ÷ and numbers 4 and 8.
(a) 6 − 8 ÷ 4 = − 1 (b) 8 − 6 ÷ 4 = 1
(c) 4 ÷ 8 − 2 = 6 (d) 4 − 8 ÷ 6 = 2
Sol. (c).
On interchanging + and × and 4 and 5 in (c), we get the
equation as :
4 + 5 × 20 = 104 or 104 = 104, which is true.
2. Given interchanges :
Signs + and × and numbers 4 and 5.
(a) 5 × 4 + 20 = 40 (b) 5 × 4 + 20 = 85
(c) 5 × 4 + 20 = 104 (d) 5 × 4 + 20 = 95
Directions :
In each of the following questions, if the given interchanges are made in signs and numbers, which one of the four
equations would be correct ?
Sol. (b).
On interchanging + and − and 4 and 8 in (b), we get the
equation as :
8 + 4 − 12 = 0 or 12 − 12 = 0 or 0 = 0, which is true.
3. Given interchanges :
Signs + and − and numbers 4 and 8.
(a) 4 ÷ 8 − 12 = 16 (b) 4 − 8 + 12 = 0
(c) 8 ÷ 4 − 12 = 24 (d) 8 − 4 ÷ 12 = 8
Sol. (b).
On interchanging − and × and 3 and 6 in (b), we get the
equation as :
6 × 3 − 8 = 10 or 18 − 8 = 10 or 10 = 10, which is true.
4. Given interchanges :
Signs − and × and numbers 3 and 6.
(a) 6 − 3 × 2 = 9 (b) 3 − 6 × = 10
(c) 6 × 3 − 4 = 15 (d) 3 × 6 − 4 = 33
Type - 2

8. Slide 8 of 17
Sol. (b).
On interchanging − and ÷, we get the equation as :
5. 5 + 3 × 8 − 12 ÷4 = 3
(a) + and − (b) − and ÷ (c) + and × (d) + and ÷
Directions :
In each of the following questions, the given equation becomes correct due to the interchange of two signs. One of the
four alternatives under it specifies the interchange of signs in the equation which when made will make the equation
correct. Find the correct alternative.
Sol. (a).
On interchanging ÷ and ×, we get :
Given expression = 5 + 6 × 3 − 12 ÷ 2
= 5 + 6 × 3 − 6 = 5 + 18 − 6 = 17.
6. 5 + 6 ÷ 3 − 12 × 2 = 17
(a) ÷ and × (b) + and × (c) + and ÷ (d) + and −
2
5 3 8 12 4 3 or 5 3 4 3
3
or 3 = 3, which is true.
+ × ÷ − = + × − =
Sol. (a).
On interchanging × and +, we get :
Given expression = 2 + 3 × 6 − 12 ÷ 4
= 2 + 3 × 6 − 3 = 2 + 18 − 3 = 17.
7. 2 × 3 + 6 − 12 ÷ 4 = 17
(a) × and + (b) + and − (c) + and ÷ (d) − and ÷
Sol. (b).
On interchanging − and ÷, we get :
Given expression = 16 ÷ 8 − 4 + 5 × 2
= 2 − 4 + 5 × 2 = 2 − 4 + 10 = 8.
8. 16 − 8 ÷ 4 + 5 × 2 = 8
(a) ÷ and × (b) − and ÷ (c) ÷ and + (d) − and ×

9. Slide 9 of 17
Sol. (c).
On interchanging 6 and 4 on L.H.S., we get the statement as :
5 + 3 × 4 − 6 ÷ 2 = 4 × 3 − 10 ÷ 2 + 7
or 5 + 12 − 3 = 12 − 5 + 7
or 14 = 14, which is true.
9. 5 + 3 × 6 − 4 ÷ 2 = 4 × 3 − 10 ÷ 2 + 7
(a) 4, 7 (b) 5, 7 (c) 6, 4 (d) 6, 10
Directions :
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.
Sol. (d).
On interchanging 7 and 6, we get the statement as :
6 × 2 − 3 + 8 ÷ 4 = 5 + 7 × 2 − 24 ÷ 3
or 12 − 3 + 2 = 5 + 14 − 8
or 11 = 11, which is true.
10. 7 × 2 − 3 + 8 ÷ 4 = 5 + 6 × 2 − 24 ÷ 3
(a) 2, 6 (b) 6, 5 (c) 3, 24 (d) 7, 6
Sol. (a).
On interchanging 3 and 5, we get the statement as :
15 + 5 × 4 − 8 ÷ 2 = 8 × 3 + 16 ÷ 2 − 1
or 15 + 20 − 4 = 24 + 8 − 1
or 31 = 31, which is true.
11. 15 + 3 × 4 − 8 ÷ 2 = 8 × 5 + 16 ÷ 2 − 1
(a) 3, 5 (b) 15, 5 (c) 15, 16 (d) 3, 1
Sol. (d).
On interchanging 9 and 5 on R.H.S., we get the statement as :
6 × 3 + 8 ÷ 2 − 1 = 5 − 8 ÷ 4 + 9 × 2
or 18 + 4 − 1 = 5 − 2 + 18
or 21 = 21, which is true.
12. 6 × 3 + 8 ÷ 2 − 1 = 9 − 8 ÷ 4 + 5 × 2
(a) 3, 4 (b) 3, 5 (c) 6, 9 (d) 9, 5

10. Slide 10 of 17
Sol. (c).
On interchanging + and × and 4 and 6, we get the equation as
:
4 + 6 × 2 = 16 or 4 + 12 = 16
or 16 = 16, which is true.
13. 6 × 4 + 2 = 16
(a) + and ×, 2 and 4 (b) + and ×, 2 and 6
(c) + and ×, 4 and 6 (d) None of these
Directions :
In each of the following questions, which one of the four interchanges in signs and numbers would make the given
equation correct ?
Sol. (a).
On interchanging + and ÷ and 2 and 3, we get the equation as :
(2 + 4) ÷ 3 = 2 or 6 ÷ 3 = 2
or 2 = 2, which is true.
14. (3 ÷ 4) + 2 = 2
(a) + and ÷, 2 and 3 (b) + and ÷, 2 and 4
(c) + and ÷, 3 and 4 (d) No interchange, 3 and 4
Sol. (b).
Using the proper operations given in (b), we get the given expression as :
200 + 100 − 300 + 200 ÷ 10 × 2 − 40 = 300 − 300 + 20 × 2 − 40 = 0 + 40 − 40 = 0.
15. Which of the following meanings of the arithmetical signs will yield the value ‘zero’ for the expression given below ?
200× 100 + 300 × 200 − 10 ÷ 2 + 40
(a) + means −, − means ×, × means ÷, ÷ means + (b) + means −, − means ÷, × means +, ÷ means ×
(c) + means ×, − means −, × means ÷, ÷ means + (d) + means ÷, − means +, × means −, ÷ means ×
(e) None of these

11. Slide 11 of 17
Type - 3
Sol. (b).
3. If A + B > C + D, B + E = 2C and C + D > B + E, it
necessarily follows that -
(a) A + B > 2E (b) A + B > 2C
(c) A > C (d) A + B > 2D
Sol. (c).
1. If A > B, B > C and C > D, then which of the following
conclusions is definitely wrong ?
(a) A > D (b) A > C (c) D > A (d) B > D
Sol. (b).
2. If A + D = B + C, A + E = C + D, 2C < A + E and 2A >
B + D, then
(a) A > B > C > D > E (b) B > A > D > C > E
(c) D > B > C > A > E (d) B > C > D > E > A
A B, B C, C D A B C D
A D D
> > > ⇒ > > >
⇒ > ⇒ >
( )
A.
So, is false.c
( )
( )
( )
( )
( ) ( ) ( ) ( )
2C < A+E, A+E = C+D 2C < C+D C < D .....
A+D = B+C, C < D A < B .....
2A > B+D, A < B A > D .....
A+E = C+D, A > D E < C .....
From , , and , we get : B > A > D > C > E.
i
ii
iii
iv
i ii iii iv
⇒ ⇒
⇒
⇒
⇒
A + B > C + D, C + D > B + E, B + E = 2C
A + B > 2C.⇒

12. Slide 12 of 17
Sol. (d).
4. 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
Sol. (a).
Given A + B = 2C ….(i) and C + D = 2A ….(ii)
Adding (i) and (ii), we get : A + B + C + D = 2C + 2A ⇒ B + D = A + C.
5. 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
( )
( )
( )
( )
( ) ( ) ( )
C + E > B + D, B + D = A + E C + E > A + E C > A ....
A + B > C + E, C + E > B + D A + B > B + D A > D ....
A + D = 2B, A > D A > B > D ....
A + E = B + D, A > B E > D ....
From , , and
i
ii
iii
iv
i ii iii
⇒ ⇒
⇒ ⇒
⇒
⇒
( ) , we get : C > A > B > D> E.iv

13. Slide 13 of 17
Sol. (b).
With usual notations, we have :
6. a − b − c implies
(a) a − b + c (b) b + a − c (c) c × b + a (d) b + a ÷ c
Directions :
It being given that :
∆ denotes ‘equal to’ ;  denotes ‘not equal to’ ; + denotes ‘greater than’ ; − denotes ‘less than’ ; × denotes ‘not greater
than’ ; ÷ denotes ‘not less than’.
Sol. (c).
With usual notations, we have :
7. a + b − c implies
(a) b − c − a (b) c − b + a (c) c + b − a (d) c × b ÷ a
( )
( )
( )
, which is false.
, which is true.
a a b c a b c
b a b c b a c
c a b c c
< < ⇒ < >
< < ⇒ > <
< < ⇒ >
( )
, which is false.
b a
d a b c b a
>
< < ⇒ > < , which is false.c
( )
( )
( )
( )
, which is false.
, which is false.
, which is true.
a a b c b c a
b a b c c b a
c a b c c b a
d a b c c
> < ⇒ < <
> < ⇒ < >
> < ⇒ > <
> < ⇒ >b < , which is false.a

14. Slide 14 of 17
Sol. (b).
With usual notations, we have :
8. a × b ÷ c implies
(a) a − b + c (b) c × b ÷ a (c) a  b  c(d) b ÷ a ÷ c
Sol. (d).
With usual notations, we have :
9. a + b + c does not imply
(a) b − a + c (b) c − b − a (c) c − a + b (d) b − a − c
( )a a >b <
( )
, which is not true.
c a b c
b a
⇒ < >
>b < c c⇒ >b <
( )
, which is true.
a
c a >b <
( )
, which is not true.
c a b c
d a
⇒ ≠ ≠
>b < c b⇒ <a < , which is not true.c
( )
( )
( )
( )
, which is false.
, which is false.
, which is false.
, which is true.
a a b c b a c
b a b c c b a
c a b c c a b
d a b c b a c
> > < >
> > < <
> > < >
> > < <

15. Slide 15 of 17
Sol. (c).
With usual notations, we have :
10. If a © b × c, it implies that
(a) a © b φ c (b) a ∆ b © c (c) a × c + b (d) c × b × a
Directions :
In these questions, some symbols have been used for some mathematical operations as indicated below :
× for ‘greater than’ ; © for ‘not less than’ ; ÷ for ‘not equal to’ ; φ for ‘equal to’ ; + for ‘not greater than’ ; ∆ for ‘less than’.
Using these symbols, choose the correct alternative in each of the following questions.
Sol. (b).
With usual notations, we have :
11. If a × b ∆ c, if follows that
(a) c + b © a (b) a © b + c (c) b © a × c (d) a φ c ∆ b
( )a a < b c a< ⇒ <
( )
, which is false.
b c
b a
=
< b c a b< ⇒ < <
( )
, which is false.
c
c a < b c a c< ⇒ > >
( )
, which is true.
b
d a < , which is false.b c c b a< ⇒ > >
( ) a a b c c> < ⇒ >b <
( )
, which is false.
a
b a b c a> < ⇒ <b >
( )
, which is true.
c
c a b c b> < ⇒ <
( )
, which is false.
, which is false.
a c
d a b c a c b
>
> < ⇒ = <

16. Slide 16 of 17
Sol. (b).
12. a α 2b and 2b θ r
(a) a η r (b) a α r (c) a β r (d) a γ r
Directions :
If α means ‘greater than’, β means ‘equal to’ , θ means ‘not less than’, γ means ‘less than’ , δ means ‘not equal to’ and η
means ‘not greater than’, then which of the four alternatives could be α correct or proper inference in each of the
following ?
Sol. (c).
(2x δ y) and (y β 3z)
⇒ 2x ≠ y and y = 3z
⇒ 2x ≠ 3z i.e. 2x δ 3z.
13. 2x δ y and y β 3z
(a) y δ 6x (b) 2x η 3z (c) 2x δ 3z (d) 3z η 3y
( 2 ) and (2 ) 2 and 2a b b r a b bα θ ⇒ > <
2 and 2 . .
r
a b b r a r i e a rα⇒ > ≤ ⇒ >

17. Slide 17 of 17
Sol. (b).
a ∆ b and b + c
⇒ a > b and b is a little more than c.
⇒ a > c
⇒ c > a i.e. c % a.
14. If a ∆ b and b + c, then
(a) a % c (b) c % a (c) c + a (d) Can’t say
Directions :
In the following questions, ∆ means ‘is greater than’, % means ‘is lesser than’,  means ‘is equal to’, = means ‘is not
equal to’, + mean ‘is a little more than’ , × means ‘is a little less than’.
Choose the correct alternative in each of the following questions.
Sol. (c).
c = a and a = b
⇒ c ≠ a and a ≠ b
⇒ b ≠ a i.e. b = a.
15. If c = a and a = b, then
(a) b ∆ a (b) c  a (c) b = a (d) Can’t say