Understanding Number Analogy Problems

This question requires us to identify the relationship between numbers in given pairs and use that relationship to find a missing number. The structure provided is a number analogy:

4 : 84 :: 11 : ? :: 13 : 2379

This means that 4 is related to 84 in the same way that 11 is related to the missing term, and 13 is related to 2379.

Analyzing the Number Relationship

We need to find the mathematical pattern connecting the first number to the second number in the pairs (4, 84) and (13, 2379). Let the first number in a pair be \(n\) and the second number be related to \(n\).

Examining the Pair 4 : 84

Here, \(n=4\). We look for a relationship that transforms 4 into 84. Let's consider common mathematical operations and powers:

• What about \(4^2\)? \(4^2 = 16\). 84 is not directly related to 16 by a simple multiple or addition.
• What about \(4^3\)? \(4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64\).

The difference between 84 and 64 is \(84 - 64 = 20\). We can observe that 20 is \(5 \times 4\). So, it looks like the relationship might be \(n^3 + 5n\). Let's test this hypothesis with the next pair.

Examining the Pair 13 : 2379

Here, \(n=13\). We apply the potential pattern \(n^3 + 5n\) to 13:

• Calculate \(13^3\): \(13 \times 13 \times 13\). \(13^2 = 169\). \(13^3 = 169 \times 13\).
• \(169 \times 10 = 1690\)
• \(169 \times 3 = 507\)
• \(1690 + 507 = 2197\). So, \(13^3 = 2197\).
• Calculate \(5 \times 13 = 65\).
• According to the pattern \(n^3 + 5n\), the result should be \(2197 + 65 = 2262\).

The calculated value 2262 does not match the given value 2379. This means the pattern \(n^3 + 5n\) is incorrect.

Let's revisit the difference for \(n=13\): \(2379 - 13^3 = 2379 - 2197 = 182\). How is 182 related to 13? \(182 \div 13 = 14\). So, for \(n=13\), the relationship is \(13^3 + 14 \times 13\).

Now we have two observations for the multiplier of \(n\) that is added to \(n^3\):

• When \(n=4\), the multiplier is 5.
• When \(n=13\), the multiplier is 14.

Notice that 5 is \(4+1\) and 14 is \(13+1\). This suggests that the multiplier of \(n\) is \((n+1)\).

The consistent pattern appears to be \(n : n^3 + (n+1)n\).

Let's re-verify this pattern with the given pairs:

• For \(n=4\): \(4^3 + (4+1) \times 4 = 64 + 5 \times 4 = 64 + 20 = 84\). This matches the first pair.
• For \(n=13\): \(13^3 + (13+1) \times 13 = 2197 + 14 \times 13 = 2197 + 182 = 2379\). This matches the third pair.

The pattern \(n^3 + (n+1)n\) is confirmed.

Calculating the Missing Term for 11

Now we apply this pattern to the pair 11 : ?. Here \(n=11\). We need to calculate \(11^3 + (11+1) \times 11\).

• Calculate \(11^3\): \(11 \times 11 \times 11 = 121 \times 11 = 1331\).
• Calculate \((11+1) \times 11\): \(12 \times 11 = 132\).
• Add the results: \(1331 + 132 = 1463\).

The missing term in the analogy is 1463.

Verification with Options

Let's compare our result with the given options:

1. 1221
2. 1463
3. 1440
4. 819

Our calculated value, 1463, matches Option 2.

The number analogy follows the rule \(n : n^3 + (n+1)n\).

Revision Table: Key Concepts in Number Puzzles

Additional Information: Strategies for Number Analogy Questions

To effectively solve number analogy problems, consider the following strategies:

• Start by examining the simplest relationships: addition, subtraction, multiplication, division.
• Look at squares and cubes of the numbers, and see if the second term is close to one of these powers.
• Check for combinations of operations, like multiplying by a constant and then adding/subtracting something, or patterns involving \(n^2+n\), \(n^2-n\), \(n^3+n\), \(n^3-n\), etc.
• Sometimes the multiplier or added/subtracted value is related to the number \(n\) itself, as seen in this problem where the multiplier was \((n+1)\).
• If direct operations don't work, look at the difference between the numbers. The differences might form a pattern.
• Always test the pattern you find on all the given pairs to ensure it holds true for the entire analogy.
• Work systematically and don't be afraid to try different approaches if the first few don't lead to a consistent pattern.

Consistent practice improves your ability to spot complex patterns quickly.