# Worked Examples: Type-II

*Example-1*
12 |
47 |
____ |

30 |
3 |
17 |

5 |
38 |
8 |

*Solution*First, arrange the numbers in the ascending order and then take the difference.

You can observe that the differences are 2, 3, 4, 5, 13, 8 and 9.

If the difference 13 is split into 2 parts as 6 and 7, then you observe that the differences are 2, 3, 4, 5, 6, 7, 8 and 9, which are in arithmetic progression with common difference 1.

So, the missing number is 17 + 6 = 23.

Answer: 23

*Example-2*
2 |
19 |
13 |

7 |
11 |
3 |

23 |
5 |
____ |

*Solution*In this example, the numbers are consecutive prime numbers in a jumbled order. The missing prime number in the series is 17.

Answer: 17

*Example-3*
0 |
80 |
24 |

____ |
48 |
35 |

15 |
63 |
3 |

*Solution*Arrange the given numbers in the ascending order.

You can notice that the differences are 3, 12, 9, 11, 13, 15 and 17. However, between the second and third numbers, the difference is 12. Split this as 5 + 7. The missing number is 3 + 5 = 8 (Verification: 8 + 7 = 15 which is the next number).

Answer: 8

You can also solve the above problem by applying the algebraic formula, n^{2} â€“ 1.

1^{2} â€“ 1 = 0; 2^{2} â€“ 1 = 3

4^{2} â€“ 1 = 15; 5^{2} â€“ 1 = 24

6^{2} â€“ 1 = 35; 7^{2} â€“ 1 = 48

8^{2} â€“ 1 = 63; 9^{2} â€“ 1 = 80

The missing number is 3^{2} â€“ 1 = 8

Answer: 8

*Example-4*
2 |
9 |
1 |

3 |
4 |
____ |

1 |
27 |
1 |

*Solution*Arrange the numbers in the ascending order.

1 1 1 2 3 4 9 27

On careful observation, you notice that the formula followed by the numbers is n, n^{2}, n^{3}.

That is, 1 1^{2} 1^{3} 2 2^{2} 3 3^{2} 3^{3}

^{}

The missing number is 2^{3} = 8.

Answer: 8

*Example-5*
10 |
8 |
12 |

9 |
7 |
11 |

8 |
6 |
____ |

*Solution*Arrange the numbers in the ascending order.

6 7 8 8 9 10 11 12

Observe that the first three numbers 6, 7 and 8 are consecutive natural numbers. The next three numbers 8, 9 and 10 are also consecutive natural numbers. Further, 8 has appeared in both these sets of three numbers. By following the same pattern, the next three numbers should be 10, 11, 12. So, the missing number is 10.

Answer: 10

*Example-6*
20 |
12 |
42 |

6 |
30 |
2 |

____ |
90 |
72 |

*Solution*Arrange the numbers in the ascending order.

2 |
6 |
12 |
20 |
30 |
42 |
72 |
90 |

1 |
2 |
3 |
4 |
5 |
6 |
8 |
9 |

2 |
3 |
4 |
5 |
6 |
7 |
9 |
10 |

Formula: n^{2} + n (or) n^{2} â€“ n

The missing number is 7^{2} + 7 (or 8^{2} â€“ 8) = 56

Answer: 56