# Worked Examples

6 | 8 | (5) |

14 | 7 | (10) |

13 | 6 | (-) |

a | b | c |

Consider the numbers in the first row. Multiply the first two numbers, i.e., 6 Ã— 8, and add 2 to the product. Now, divide the result by 10. You will get the third number.

Formula: (a Ã— b + 2) Ã· 10 = c

(6 Ã— 8) + 2 = 50; 50 Ã· 10 = 5

(14 Ã— 7) + 2 = 100; 100 Ã· 10 = 10

(13 Ã— 6) + 2 = 80; 80 Ã· 10 = 8

Answer: 8

140 | 29 | (13) |

82 | 18 | (10) |

120 | 24 | (-) |

a | b | c |

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

Formula:

140 + 29 = 169;

82 + 18 = 100;

120 + 24 = 144;

Answer: 12

(111) | 128 | 94 |

(132) | 156 | 108 |

(â€“) | 87 | 165 |

The first number in each row is half the sum of the second and the third numbers.

Formula:

(128 + 94) Ã· 2 = 111

(156 + 108) Ã· 2 = 132

(87 + 165) Ã· 2 = 126

Answer: 126

25 | 81 | (196) |

16 | 49 | (121) |

64 | 100 | (-) |

The third number in each row is the square of the sum of the square roots of the first and second numbers.

Formula:

= 5; = 9; 5 + 9 = 14; 14^{2} = 196

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

= 8; = 10; 8 + 10 = 18; 18^{2} = 324

Answer: 324

7 | 5 | (36) |

9 | 1 | (25) |

12 | 4 | (-) |

The third number in each row is the square of half of the sum of the first and second numbers.

Formula:

7 + 5 = 12; 12 Ã· 2 = 6; 6^{2} = 36

9 + 1 = 10; 10 Ã· 2 = 5; 5^{2} = 25

12 + 4 = 16; 16 Ã· 2 = 8; 8^{2} = 64

Answer: 64

4 | 6 | (26) |

7 | 3 | (29) |

3 | 9 | (-) |

The third number in each row is half the sum of the squares of the first and second numbers.

Formula:

4^{2} = 16; 6^{2} = 36; 16 + 36 = 52; 52 Ã· 2 = 26

7^{2} = 49; 3^{2} = 9; 49 + 9 = 58; 58 Ã· 2 = 29

3^{2} = 9; 9^{2} = 81; 9 + 81 = 90; 90 Ã· 2 = 45

Answer: 45

36 | (23) | 15 |

29 | (39) | 37 |

47 | (-) | 53 |

The second number, in each row, is the sum of the products of the digits of the first and second numbers.

Formula: POD of a + POD of c = b

POD of 36 = 3 Ã— 6 = 18; POD of 15 = 1 Ã— 5 = 5; 18 + 5 = 23

POD of 29 = 2 Ã— 9 = 18; POD of 37 = 3 Ã— 7 = 21; 18 + 21 = 39

POD of 47 = 4 Ã— 7 = 28; POD of 53 = 5 Ã— 3 = 15; 28 + 15 = 43

Answer: 43

14 | 4 | (30) |

8 | 12 | (22) |

19 | 14 | (-) |

The third number in each row is the sum of twice the first number and half the second number.

Formula:

(14 Ã— 2) + (4 Ã· 2) = 28 + 2 = 30

(8 Ã— 2) + (12 Ã· 2) = 16 + 6 = 22

(19 Ã— 2) + (14 Ã· 2) = 38 + 7 = 45

Answer: 45

6 | 4 | (100) |

8 | 5 | (189) |

11 | 3 | (-) |

The third number in each row is the sum of the square of the first number and the cube of the second number.

Formula: a^{2} + b^{3} = c

6^{2} + 4^{3} = 36 + 64 = 100

8^{2} + 5^{3} = 64 + 125 = 189

11^{2} + 3^{3} = 121 + 27 = 148

Answer: 148

14 | 11 | (75) |

11 | 7 | (72) |

10 | 6 | (-) |

The third number in each row is the difference between the squares of the first and second numbers.

Formula: a^{2} â€“ b^{2} = c

14^{2} = 196; 11^{2} = 121; 196 â€“ 121 = 75

11^{2} = 121; 7^{2} = 49; 121 â€“ 49 = 72

10^{2} = 100; 6^{2} = 36; 100 â€“ 36 = 64

Answer: 64

8 | 6 | (72) |

5 | 12 | (90) |

12 | 9 | (-) |

The third number in each row is one and half (or 3/2) times the product of the first two numbers.

Formula:

8 Ã— 6 = 48; Ã— 48 = 72

5 Ã— 12 = 60; Ã— 60 = 90

12 Ã— 9 = 108; Ã— 108 = 162

Answer: 162

7 | 9 | (41) |

8 | 6 | (34) |

9 | 5 | (-) |

The third number in each row is the sum of twice the first number and thrice the second number.

Formula: 2a + 3b = c

(7 Ã— 2) + (9 Ã— 3) = 14 + 27 = 41

(8 Ã— 2) + (6 Ã— 3) = 16 + 18 = 34

(9 Ã— 2) + (5 Ã— 3) = 18 + 15 = 33

Answer: 33

14 | 6 | (50) |

11 | 9 | (92) |

12 | 7 | (-) |

The third number in each row is the sum of the first number plus the square of the second number.

Formula: a + b^{2} = c

14 + 6^{2} = 14 + 36 = 50

11 + 9^{2} = 11 + 81 = 92

12 + 7^{2} = 12 + 49 = 61

Answer: 61

39 | (3) | 12 |

54 | (2) | 46 |

184 | (-) | 59 |

The second number in each row is the cube root of the difference between the first and the third numbers.

Formula:

39 â€“ 12 = 27; = 3

54 â€“ 46 = 8; = 2

184 â€“ 59 = 125; = 5

Answer: 5

272 | 151 | (11) |

308 | 227 | (9) |

195 | 131 | (-) |

The third number in each row is the square root of the difference between the first and second numbers.

Formula:

272 â€“ 151 = 121; = 11

308 â€“ 227 = 81; = 9

195 â€“ 131 = 64; = 8

Answer: 8

17 | (11) | 36 |

21 | (13) | 64 |

19 | (-) | 25 |

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

Formula:

17 â€“ = 17 â€“ 6 = 11

21 â€“ = 21 â€“ 8 = 13

19 â€“ = 19 â€“ 5 = 14

Answer: 14

36 | 49 | (117) |

23 | 36 | (45) |

45 | 57 | (-) |

The third number in each row is the product of SOD (sum of digits) of the first and second numbers.

Formula: SOD of a Ã— SOD of b = c

3 + 6 = 9; 4 + 9 = 13; 9 Ã— 13 = 117

2 + 3 = 5; 3 + 6 = 9; 5 Ã— 9 = 45

4 + 5 = 9; 5 + 7 = 12; 9 Ã— 12 = 108

Answer: 108

916 | (166) | 976 |

971 | (177) | 879 |

245 | (-) | 555 |

Sum of digits of the first number and the difference between SOD of the third and first numbers are written side by side to get the second number.

Consider the numbers in the first row.

SOD of 916 = 9 + 1 + 6 = 16; SOD of 976 = 9 + 7 + 6 = 22; 22 â€“ 16 = 6

If you write 16 and 6 side by side, you get 166, which is the second number.

Consider the numbers in the second row.

SOD of 971 = 9 + 7 + 1 = 17; SOD of 879 = 8 + 7 + 9 = 24; 24 â€“ 17 = 7

On writing 17 and 7 side by side you get 177.

Consider the numbers in the third row.

SOD of 245 = 2 + 4 + 5 = 11; SOD of 555 = 5 + 5 + 5 = 15; 15 â€“ 11 = 4

On writing 11 and 4 side by side you get 114.

Answer: 114

213 | (180) | 534 |

352 | (-) | 612 |

Formula: (SOD of a + SOD of c) Ã— 10 = b

SOD of 213 = 2 + 1 + 3 = 6; SOD of 534 = 5 + 3 + 4 = 12; (12 + 6) Ã— 10 = 180

SOD of 352 = 3 + 5 + 2 = 10; SOD of 612 = 6 + 1 + 2 = 9; (10 + 9) Ã— 10 = 190

Answer: 190