# 8th Class Mathematics Missing Character Missing Character

## Missing Character

Category : 8th Class

Introduction

In this chapter you need to find a missing character with the help of given clues. This will help to develop the ability of identifying missing numbers/ characters of the students.

MISSING CHARACTER

In such type of questions, a figure, a set of figures, an arrangement or a matrix is given, each of which bears certain characteristics, and be it numbers, letters or a group / combination of letters / numbers, following a certain pattern. The candidate has to find a missing character in the figure out of the given options.

Let us develop the ability to identify missing character with the help of following examples.

EXAMPLE 1:

 11 3 49 5 19 ? 7 13 100

(a) 96                                     (b) 120

(c) 144                                   (d) 100

Sol.        (c)  In a row the third term is the square of the average of the first two numbers.

$\therefore$   $?={{\left( \frac{5+19}{2} \right)}^{2}}={{12}^{2}}=144.$

EXAMPLE 2:

 1 4 9 ? 1 2 3 4 2 4 6 ?

(a) 16, 8                                (b) 49, 7

(c) 36, 4                                (d) 25, 5

Sol.        (a).

1st row: ${{1}^{2}},{{2}^{2}},{{3}^{2}},{{4}^{2}},$

Third row: 2,4,6,8

$\therefore$ The missing numbers =16, 8

EXAMPLE 3:

 3C 24D 8E 7I 21K 3M 4D ? 7J

(a) 11E                                  (b) 28G

(c) 351                                   (d) 48F

Sol.        (b).

In the first row, letters are consecutive CDE .In the 2nd row, letters are one step forward I-K-M In the third row, the letters are + 2 forward i.e

D-G -J.

Number is the product of the two numbers.

Hence, $4\times 7=28$

EXAMPLE 4:

(a) 117 1G                            (b) 100

(c) 78                                     (d) 63

Sol.        (c).

In the first figure $9\times 10-4\times 8=58$

$\therefore$ The missing figure $=15\times 10-9\times 8=78$

EXAMPLE 5:

(a) 82                                     (b) 100

(c) 68                                     (d) 64

Sol.        (b)

Required number$=\left( 2+8 \right)2=100$.

EXAMPLE 6:

(a) 47                                      (b) 45

(c) 37                                     (d) 35

Sol.        (c)

Fig  $1:3\times 3+6\times 5=39$

Fig  $2:4\times 4+5\times 7=37$

$\therefore$ So? $=3\times 4+5\times 5=37$

EXAMPLE 7:

Find the number in place of question mark (?) in the following matrix

 3 5 7 9 11 13 8 26 4 8 82 ? 170

(a) 121                                  (b) 120

(c) 119                                   (d) 111

Sol.        (b).

The numbers are according to the rule ${{n}^{2}}\pm 1.$

i.e, ${{3}^{2}}-1,{{5}^{2}}+1,{{7}^{2}}-1,{{9}^{2}}+{{1.11}^{2}}-1$and ${{13}^{2}}+1$

$\therefore$ the missing number is 120.

EXAMPLE 8:

Find the missing number

(a) 52                                     (b) 36

(c) 117                                   (d) 81

Sol.        (b).

The rule is: In the figure: $5\times 4=20,5+4=9$

In the second figure: $3\times 8=24$ and $3+8=11$

$\therefore$ In the third figure $9\times 4=36.$

EXAMPLE 9:

Insert the missing letter

 B G N D J R G N ?

(a) U                                      (b) V

(c) W                                     (d) X

Sol.        (c).

In 1st column, the rule is +1, +2; in column two, the rule is + 2, + 3 and so in column three, the rule is +3, + 4.

So the letter is 4 letters from R i.e, W.

EXAMPLE 10:

Find the missing term in the second figure out of the given option given below the figure.

(a) 15                                     (b) 19

(c) 27                                     (d) 89

Sol.        (d).

In first figure central number $=5\times 15+6\times 3$

In the second figure central number $=7\times 5+9\times 6=89.$.

EXAMPLE 11:

 1 2 3 11 7 5 120 45 ?

(a) 19                                     (b) 17

(c) 16                                     (d) 15

Sol.        (c).

In each column, the third number is the difference of the squares of the other two numbers.

$\therefore$  $?={{5}^{2}}-{{3}^{2}}=25-9=16.$

EXAMPLE 12:

(a) 60                                     (b) 50

(c) 25                                     (d) 40

Sol.        (c).

First figure $\Rightarrow$ $\sqrt{4\times 9}=6;$

second figure  $\Rightarrow$ $\sqrt{9\times 16}=12$

$\therefore$ $\sqrt{16\times ?}=20$i.e., $16\times ?=400$or $?=25$

EXAMPLE 13:

(a) 6                                       (b) 8

(c) 10                                     (d) 14

Sol.        (c)

Fig   $1:10-4=6;18-10=8:18-4=14$

Fig  $~2:14-8=6:22-14=8:22-8=14$

Fig   $3:11-5=6:15-11=4:15-5=10$

Miscellaneous Solved Examples

DIRECTION: Find the missing numbers around the circle from the choice given below:

EXAMPLE 1:

(a)  5                      (b) 6

(c)  7                                      (d) 8

Sol.        (c)

Take the difference of opposite numbers and then take its average, i.e. $\frac{(85-81)+(32-24)}{2}=6$ etc.

EXAMPLE 2:

What number should replace the question mark?

Sol.        41

$7\times 8-15=41$

EXAMPLE 3:

(a) 95                     (b) 105

(c) 125                  (d) 130

Sol.        (a)

Fourth root of the numbers at the periphery added together, i.e. $1+43+15+22=81,\sqrt[4]{81}=3$ etc.

EXAMPLE 4:

 1 $\frac{1}{2}$ $\frac{1}{3}$ 2 1 ? 3 $\frac{3}{2}$ 1

(a) 1/3                                   (b) 1/2

(c) 2/3                                   (d) 1

Sol.        (c)

Numbers in the 1st and 3rd rows are $\frac{1}{1},\frac{1}{2},\frac{1}{3},\frac{3}{1},\frac{3}{2},$

$\frac{3}{3}$ Numbers in the second row should be $\frac{2}{1},\frac{2}{2},\frac{2}{3}$

EXAMPLE 5:

What number should replace the question mark?

 8 9 3 69 7 5 6 29 4 7 9 19 9 8 4 ?

Sol.        68

$(8\times 9)-3=69;(7\times 5)-6=29$

$(4\times 7)-9=19;(9\times 8)-4=68$

EXAMPLE 6:

What number should replace the question mark?

 1st 2nd 3rd 4th 26 27 29 25 28 29 31 36 30 31 37 49 32 33 ? 64 34 35 43 81

Sol.        41

The first column has even numbers starting at 26.

The second column has odd numbers starting at 27.

The third column has prime numbers starting at 29.

The fourth column has square numbers starting at 25.

EXAMPLE 7:

What number should replace the question mark?

Sol.        92

$12+6+7=25.$Reverse = 52

$17+9+9=35$. Reverse =53

$7+4+8=19$. Reverse=91

$11+13+5=29.$Reverse=92

EXAMPLE 8:

What number should replace the question mark?

Sol.        31

Start at 1 and work clockwise to each segment, adding 3,6, 9, 12, 15, 18.

EXAMPLE 9:

What number should replace the question mark?

Sol.        9

Looking across, rows having two numbers total 14, rows having three numbers total 18, and rows having four numbers total 22.

EXAMPLE 10:

What number should replace the question mark?

Sol.        9

Look across to some sections in each of the three figures : 4,

6, 8 (increase by 2); 9, 7,5 (reduce by 2);

8,5,2 (reduce by 3); 1,5,9 (increase by 4).

EXAMPLE 11:

What number should replace the question mark?

Sol.        212

7 x 3 = 21, 1 + 1 = 2

EXAMPLE 12:

What number should replace the question mark?

Sol.        32.

Multiply by 3 and add 2 to obtain the pairings, so,  $10\times 3=30,30+2=32$

The other such pairs are:

3/11,7/23,20/62,2/8,16/50, and 15/47

EXAMPLE 13:

What number should replace the question mark?

Sol.        46.

Beginning at 17, go across the row and then follow the direction of the arrows alliterating between +9 and-5.

EXAMPLE 14:

What number should replace the question mark?

Sol.        171.

1st circle

$17\times 4=68;13\times 4=52;19\times 4=76$

II nd circle

$7\times 6=42;11\times 6=66;16\times 6=96$

III rd circle

$21\times 9=189;14\times 9=126;19\times 9=171$

EXAMPLE 15:

What number should replace the question mark?

Sol.        130.

$+18,+20,+22,+24$

EXAMPLE 16:

What number should replace the question mark?

Sol.        12.

$10+11+13+5=39;39-3=13$

$17+5+9+17=48;48-3=16$

$19+6+7+10=42\text{ };42-3=\text{ }14$

$9+15+6+6=36;36-3=12$

EXAMPLE 17:

What number should replace the question mark?

Sol.        42.

$(5\times 8)-7=33$

$(6\times 7)-6=36$

$(7\times 5)-4=31$

$(8\times 7)-4=42$

EXAMPLE 18:

Find the weight which should be placed at the question mark for the scales to balance.

Sol.        3.

5 x 9 = 45                             6 x 6 = 36

3 x 5 = 15                             3 x 8 = 24

60                                           60

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