Under this chapter, some numbers are given in the examination in one or more Figures or in the form of matrix or equations, which have been arranged according to a certain rule.

Generally, these problems are based on a particular rule, so candidates are carefully observing the rules of systemization of numbers to find out what should be the appropriate number in place of the missing post (?)

**⇒ Now**, please follow the examples of different types below for clarifying the types of the questions and the above-mentioned facts under this chapter, which is very helpful in solving other questions related to this chapter easily.

**Type 1 **

**Example:** What figure would come in place of the question mark (?) in the figure below?

(a) 35 (b) 47 (c) 39 (d) 45

**Solution: **(c) After carefully observing the above figure It is found That the middle number in each shape is equal to the sum of the product of the digits opposite to each other.

3 × 4 + 5 × 5 = 12 + 25 = 37

4 × 4 + 7 × 5 = 16 + 35 = 51

Similarly, 3 × 3 + 5 × 6 = 9 + 30 = 39

So in the given third shape, the number in place of (?) would be 39.

**Type 2 **

**Example:** What would be the number in place of the question mark (?) in the matrix given below?

(a) 4 (b) 6 (c) 8 (d) 10

**Solution:** (c) In the first two columns of the given matrix, we see that the after diving the multiplication of first and third numbers of each column by the second number we get the fourth number,

**Type 3 **

**Example:** What number will replace the question mark (?) in the following question?

(a) 47 (b) 64 (c) 81 (d) 121

Solution:(c) After observing number on each side of each streak in the figure above, we find that (6)^{2} = 36, (7)^{ 2} = 49, (4)^{2} = 16

Similarly (9)^{2} = 81 So the questionmark (?) is replaced by the appropriate number 81.

**Type 4**

**Example:** In the given figure, what would be the appropriate number instead of the question mark (?)?

(a) 20 (b) 27 (c) 26 (d) 22

**Solution:** (d) After carefully observing the given figure, we find that the number inside the given circular shape is the sum of the square root of the four numbers outside the shape, such as

So instead of the question mark (?), The appropriate number would be 22.

**Type 5 **

**Example:** In the figure given below, what would be the appropriate number instead of the question mark (?)?

(a) 21 (b) 27 (c) 31 (d) 37

** Solution: **(b) Observe the above figure carefully On observation we find that the number between the digits located on both angles in each shape is the difference of the square of the number on both angles, such as

In Figure I, (9) ^{ 2} – (5) ^{ 2} = 81-25 = 56

(9) ^{ 2} – (8) ^{ 2} = 81-64 = 17

(8) ^{ 2 }– (5) ^{ 2} = 64 – 25 = 39

Then in shape II , (11) ^{ 2 }– (9) ^{ 2} = 121 – 81 = 40

(11) ^{ 2} – (8) ^{ 2} = 121-64 = 57

(9) ^{ 2} – (8) ^{ 2 }= 81-64 = 17

Similar in shape III, (7) ^{ 2} – (3) ^{ 2} = 49 – 9 = 40

(6) ^{ 2 }– (3) ^{ 2 }= 36 – 9 = 27

(7) ^{ 2} – (6) ^{ 2} = 49-36 = 13

Hence the appropriate number ’27’ will be in place of the question mark (?).

**Type 6 **

**Example:** In the given figure, what would be the appropriate number in place of the question mark (?)?

(a) 39 (b) 42 (c) 48 (d) 56

**Solution:** (b) After carefully observing the given figure we find that the number located in each corner of the square shape will be thrice of the difference between the two numbers used in the semi-circular shape like –

(30-12) × 3 = 18 × 3 = 54

(30-13) × 3 = 17 × 3 = 51

(27-12) × 3 = 15 × 3 = 45

Similarly, (27 – 13) × 3 = 1.4 × 3 = 42

Hence the appropriate number = 42 instead of the question mark (?)

**Type 7 **

**Example:** In the figure given below, which number would be replaced by the question mark (?)?

(a) 8 (b) 11 (c) 6 (d) 7

** Solution: **(c) In the above figure, we see that the numbers used in the shape of the triangle in each shape, obtained by multiplying the number used in first two square shapes, after that subtract the number used in the third square shape, Which is expressed as follows

3 × 4 – 5 = 12 – 5 – 7

7 × 3- 9 = 21 – 9 – 12

Similarly, 4 × 2 – 2 = 8 – 2 = 6

So the appropriate number of question marks (?) Will be ‘6’.

**Type 8 **

**Example:** If 9 × 3 + 8 = 24,10 × 2 + 7 = 35 and 80 × 40 + 3 = 6, then 12 × 4 + 3 =?

(a) 7 (b) 9 (c) 12 (d) 16

** Solution: **(b) After carefully observing the given equation, we find that by solving the mathematical symbol ‘×’ in ‘÷’ and solving the mathematical symbol ‘+’ in ‘× The intended value has been received. It has been solved as follows

9 × 3 + 8 = 9 ÷ 3 × 8 = 24

10 × 2 + 7 = 10 ÷ 2 × 7 = 35

80 × 40 + 3 = 80 ÷ 40 × 3 = 6

Similarly, 12 × 4 + 3 = 12 ÷ 4 × 3 = 9

So the appropriate number ‘9’ in place of the question mark (?)

**Type 9 **

**Example:** In the figure given below, what number will replace the question mark (?)?

(a) 161 (b) 157 (c) 167 (d) 162

**Solution: **(c) After carefully observing the given figure, we find that the number in the bottom of each figure is calculated by multiplying the middle number with the sum of the two numbers at the top, like

(8 + 9) × 6 + 6 = 17 × 6 + 6 = 102 + 6 = 108

(7 + 8) × 5 + 6 = 15 × 5 + 6 = 75 + 6 = 81

Similarly, (12 + 11) × 7 + 6 = 23 × 7 + 6 = 161 + 6 = 167

So the intended number at the question mark (?) will be 167.

**Type 10 **

**Example:** Select the appropriate letter in place of the question mark (?) in the matrix given below.

(a) JI (b) HS (c) KT (d) AD

** Solution: **(b) On the study of given matrix we find that each letter used in it is contrary to each other in the English alphabet, along with the first letter is in ascending order in each column and the second letter is in the descending order.

So the appropriate letter group ‘HS’ will be in place of the question mark (?).

**Type 11 **

**Example:** With the help of the options given below, select the appropriate number to replace the question mark (?).

FED × 3 = 1629

BCD × 4 = 492

BEF × 1 =?

(a) 451 (b) 145 (c) 514 (d) 415

** Solution:** (b)

Hence the appropriate number in place of the question mark (?) is ‘145’.

**Type 12 **

**Example:** In the matrix given below, select the appropriate letter-number that comes in place of the question mark (?) in the first row, with the given options.

(a) NP_{24} (b) QT_{40} (c) NP_{40} (d)PO_{68 }

** Solution: **(c) The first letter in each line is the letter located three places ahead of the English alphabet, while the second letter is also the letter located three places ahead. While the second number in each row is already 2 more, and the third number will be twice the second number.

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