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Worked Examples

Example-1
Rule:
  1. Divide by 3 and add the square of the quotient to the number.
  2. Add the number to its square.
  3. Add half of the number to twice the number.
  4. Square the number and subtract 2.
  5. Add one-third of the number to twice the number.
1. 15 25 35 55
2. 79 223 439 727
3. 21 35 49 63
4. 90 240 462 756
5. 18 40 70 108
Solution
Consider rule B, which states ‘Add the number to its square’.
 
So, the numbers are of the form (n2 + n).
 
Try to find out the set of numbers which is based on the above rule (n2 + n).
 
The fourth set of numbers (90, 240, 462, 756) is based on the rule n2 + n.
 
90 = 81 + 9 = (92 + 9)
 
240 = 225 + 15 = (152 + 15)
 
462 = 441 + 21 = (212 + 21)
 
756 = 729 + 27 = (272 + 27)
 
Thus, rule B applies to the fourth set of numbers.
 
Here, the basic set of numbers is (9, 15, 21, 27). By performing operations on this set of numbers as suggested in rule B, we get the numbers in set 4 i.e., (90, 240, 462, 756).
 
Now, examine as to whether the remaining four sets of numbers can also be derived from the basic set of numbers (9, 15, 21, 27).
 
Check rule A which states ‘Divide the number by 3 and add the square of the quotient to the number’.
 
9 ÷ 3 = 3 32 = 9 9 + 9 = 18
15 ÷ 3 = 5 52 = 25 25 + 15 = 40
21 ÷ 3 = 7 72 = 49 49 + 21 = 70
27 ÷ 3 = 9 92 = 81 81 + 27 = 108
 
Hence, fifth set of numbers (18, 40, 70, 108) is based on rule A.
 
Check rule E.
 
‘Add one-third of the number to twice the number.’
 
9 ÷ 3 = 3 9 × 2 = 18 18 + 3 = 21
15 ÷ 3 = 5 15 × 2 = 30 30 + 5 = 35
21 ÷ 3 = 7 21 × 2 = 42 42 + 7 = 49
27 ÷ 3 = 9 27 × 2 = 54 54 + 9 = 63
 
The third set of numbers (21, 35, 49, 63) is based on rule E.
 
Check rule D.
 
‘Square the number and subtract 2’
 
92 - 2 = 81 – 2 = 79
 
152 - 2 = 225 – 2 = 223
 
212 - 2 = 441 – 2 = 439
 
272 - 2 = 729 – 2 = 727
 
The second set of numbers (79, 223, 439, 727) is based on rule D.
 
So, the first set of numbers (15, 25, 35, 55) is based on rule C.
 
Check whether the first set of numbers can be obtained from the basic set of numbers (9, 15, 21, 27) by applying rule C. No, it is not possible. However, we can get the first set of numbers by applying rule C, on a different set of numbers (6, 10, 14, 22), as worked out here under:

Rule:
 
‘Add half of the number to twice the number.’
 
(6 × 2) + (6 ÷ 2) = 12 + 3 = 15
 
(10 × 2) + (10 ÷ 2) = 20 + 5 = 25
 
(14 × 2) + (14 ÷ 2) = 28 + 7 = 35
 
(22 × 2) + (22 ÷ 2) = 44 + 11 = 55

Answer:
  1. (C)
  2. (D)
  3. (E)
  4. (B)
  5. (A)
 
All the problems need not be based on the set of magic numbers. In the absence of magic numbers, start the analysis with the rules such as n2 – 2, n3 + 1, n2 + n, etc.
 
If there is a rule such as ‘square a number and subtract two’, add two to the given set of numbers and verify whether it is of the form n2.
 
Suppose a rule states ‘cube a number and add one to the result’. In that case, subtract one from the numbers in the given set and check whether it is a perfect cube.
 
Example-2
Rules:
  1. Divide the number by 2 and square the quotient.
  2. Multiply the number by 3 and divide the product by 2.
  3. Add 16 to the number and divide the sum by 2.
  4. Divide the numbers by 2 and add the square of the quotient to the number.
  5. Square the number and divide by half of the number.
1. 8 12 14 15
2. 8 16 32 64
3. 8 12 18 27
4. 8 16 64 1024
5. 8 24 168 7224
Solution
This example is slightly different from the earlier example. In this case, operations as suggested in the rules are not performed on a separate set of numbers, but on the numbers appearing in the given sets only.

Consider rule A:
 
‘Divide the number by 2 and square the quotient.’
 
Let us start with 8.
 
Description: 88875.png 42 = 16
 
Description: 88879.png; 82 = 64
 
Description: 88893.png; 322 = 1024
 
(8, 16, 64, 1024) are the numbers given in set number 4. So, set 4 is based on rule A.

Consider rule B:
 
‘Multiply the number by 3 and divide the product by 2.’
 
8 × 3 = 24; Description: 88918.png
 
12 × 3 = 36; Description: 88923.png
 
18 × 3 = 54; Description: 88933.png
 
So, the third set i.e., (8, 12, 18, 27) is based on rule B.

Consider rule C:
 
‘Add 16 to the number and divide the sum by 2.’
 
Start with first number 8.
 
8 + 16 = 24; Description: 88945.png
 
12 + 16 = 28; Description: 88961.png
 
14 + 16 = 30; Description: 88978.png
 
So, the first set i.e., (8, 12, 14, 15) is based on rule C.

Consider rule D:
 
‘Divide the number by 2 and add the square of the quotient to the number.
 
Start with first number 8.
 
Description: 88995.png; 42 = 16, 8 + 16 = 24
 
Description: 89008.png; 122 = 144, 24 + 144 = 168
 
Description: 89019.png; 842 = 7056, 168 + 7056 = 7224
 
So, the fifth set i.e., (3, 24, 168, 7224) is based on rule D.

Consider rule E:
 
‘Square the number and divide by half of the number.’
 
Start with first number 8.
 
Description: 89073.png
 
Description: 89083.png
 
Description: 89092.png
 
The second set i.e., (8, 16, 32, 64) is based on rule E.

Answer:
  1. (C)
  2. (E)
  3. (B)
  4. (A)
  5. (D)




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