# Worked Examples: Type-I

*Example-1*
7 |
6 |
21 |

8 |
5 |
20 |

12 |
4 |
____ |

a |
b |
c |

*Solution*In this problem, we observe that in each row the last number is half the product of the first two numbers.

Formula:

7 Ã— 6 = 42; 42 Ã· 2 = 21

8 Ã— 5 = 40; 40 Ã· 2 = 20

12 Ã— 4 = 48; 48 Ã· 2 = 24

Answer: 24

*Example-2*
7 |
5 |
6 |
a |

6 |
3 |
4 |
b |

26 |
16 |
____ |
c |

*Solution*In this example, consider the numbers in each column. The last number in each vertical column is twice the sum of the first two numbers.

Formula: (a + b) Ã— 2 = c

2 (7 + 6) = 2 (13) = 26

2 (5 + 3) = 2 (8) = 16

2 (6 + 4) = 2 (10) = 20

Answer: 20

*Example-3*
24 |
6 |
8 |

12 |
4 |
____ |

16 |
2 |
16 |

*Solution*Divide the first number of each row by the second number and multiply the quotient by 2 to get the third number.

Formula: a Ã· b Ã— 2 = c

24 Ã· 6 Ã— 2 = 8

12 Ã· 4 Ã— 2 = 6

16 Ã· 2 Ã— 2 = 16

Answer: 6

*Example-4*
7 |
4 |
121 |

5 |
8 |
169 |

8 |
6 |
____ |

*Solution*The third number in each row is the square of the sum of the first two numbers.

Formula: (a + b)^{2} = c

7 + 4 = 11; 11^{2} = 121

5 + 8 = 13; 13^{2} = 169

8 + 6 = 14; 14^{2 }= 196

Answer: 196

*Example-5*
25 |
12 |
49 |

81 |
13 |
16 |

36 |
14 |
____ |

*Solution*Observe the numbers in each row. The middle number in each row is the sum of the square roots of the first number and the last number.

The same relation can be written in the following form also.

(12 â€“ )^{2} = (12 â€“ 5)^{2} = 7^{2} = 49

(13 â€“ )^{2} = (13 â€“ 9)^{2} = 4^{2} = 16

(14 â€“ )^{2} = (14 â€“ 6)^{2} = 8^{2} = 64

Answer: 64

*Example-6*
7 |
6 |
8 |

9 |
8 |
4 |

10 |
4 |
____ |

*Solution*In this example, the sum of the numbers in each horizontal row is 21.

7 + 6 + 8 = 21

9 + 8 + 4 = 21

10 + 4 + 7 = 21

Answer: 7

*Example-7*
1 |
2 |
4 |
6 |

1 |
4 |
16 |
36 |

3 |
5 |
7 |
8 |

8 |
24 |
____ |
63 |

*Solution*The numbers in the second row are formed by squaring the numbers in the first row. The numbers in the fourth row are formed by squaring the numbers in the third row and subtracting 1.

1^{2} = 1; 2^{2} = 4; 4^{2} = 16; 6^{2} = 36 â†’ 1^{st} and 2^{nd} row numbers.

3^{2} â€“ 1 = 8; 5^{2} â€“ 1 = 24; 7^{2} â€“ 1 = 48; 8^{2} â€“ 1 = 63 â†’ 3^{rd} and 4^{th} row numbers.

Answer: 48

*Example-8*
32 |
35 |
39 |

13 |
17 |
22 |

41 |
46 |
____ |

*Solution*In the first row, the numbers are increased by 3 and 4. In the second row, they are increased by 4 and 5.

Hence, in the third row they must increase by 5 and 6.

32 |
+ |
3 |
= |
35; |
35 |
+ |
4 |
= |
39 |

13 |
+ |
4 |
= |
17; |
17 |
+ |
5 |
= |
22 |

41 |
+ |
5 |
= |
46; |
46 |
+ |
6 |
= |
52 |

Answer: 52

*Example-9*
4 |
3 |
50 |

6 |
5 |
122 |

7 |
9 |
____ |

*Solution*Add the first and second number of each row, square the sum, and add 1 to it. You get the last number in the row.

Formula: (a + b)^{2} + 1 = c

4 + 3 = 7; 7^{2} = 49; 49 + 1 = 50

6 + 5 = 11; 11^{2} = 121; 121 + 1 = 122

7 + 9 = 16; 16^{2} = 256; 256 + 1 = 257

Answer: 257

*Example-10*
5 |
7 |
74 |

8 |
4 |
80 |

6 |
9 |
____ |

*Solution*The third number of each row is the sum of the squares of the first and second numbers.

Formula: a^{2} + b^{2} = c

5^{2} + 7^{2} = 25 + 49 = 74

8^{2} + 4^{2} = 64 + 16 = 80

6^{2} + 9^{2} = 36 + 81 = 117

Answer: 117

*Example-11*
6 |
9 |
135 |

5 |
7 |
76 |

8 |
11 |
____ |

*Solution*The third number in each row is the difference between the cube of the first number and the square of the second number.

Formula: a^{3} â€“ b^{2} = c

6^{3} = 216; 9^{2} = 81; 216 â€“ 81 = 135

5^{3} = 125; 7^{2} = 49; 125 â€“ 49 = 76

8^{3} = 512; 11^{2} = 121; 512 â€“ 121 = 391

Answer: 391

*Example-12*
13 |
37 |
71 |
11 |

49 |
24 |
96 |
13 |

52 |
43 |
____ |
14 |

*Solution*The fourth number in each row is the square root of the sum of the first three numbers.

13 + 37 + 71 = 121; = 11

49 + 24 + 96 = 169; = 13

52 + 43 + x = 95 + x; = 14

So, 95 + x = 196 i.e., x = 196 - 95 = 101

Answer: 101