Understanding Number Analogies and Finding the Relationship

The question asks us to find the relationship between pairs of numbers and apply that same relationship to a third pair. We are given two pairs: 68 : 19 and 76 : 21. We need to find the number that relates to 164 in the same way.

Let's analyze the relationship between the first pair of numbers: 68 and 19.

We need to find a mathematical operation or series of operations that transforms 68 into 19. Let's try simple arithmetic operations:

• Subtracting a number: \(68 - x = 19 \implies x = 68 - 19 = 49\). If we subtract 49 from the first number, we get the second. Let's test this on the second pair: \(76 - 49 = 27\). This is not 21, so subtraction is not the rule.
• Division: Let's try dividing 68 by a small number and see if it gets us close to 19. \(68 \div 2 = 34\), \(68 \div 3 \approx 22.67\), \(68 \div 4 = 17\). The result 17 is close to 19.

If we divide 68 by 4, we get 17. To get from 17 to 19, we can add 2. So, the potential rule for the first pair is: \(\text{First number} \div 4 + 2 = \text{Second number}\). Let's check this rule with the first pair:

\(68 \div 4 + 2 = 17 + 2 = 19\). This matches the first pair.

Now, let's test this rule on the second pair: 76 and 21.

Applying the same rule:

\(76 \div 4 + 2 = 19 + 2 = 21\). This also matches the second pair.

Since the rule \(\text{First number} \div 4 + 2 = \text{Second number}\) works for both given pairs, it is likely the correct relationship.

Now, we need to apply this rule to the third number, 164, to find the missing number in the analogy 164 : ?.

Using the established rule:

\(164 \div 4 + 2 = ?\)

First, perform the division:

\(164 \div 4 = 41\)

Now, add 2 to the result:

\(41 + 2 = 43\)

So, the missing number is 43.

Let's verify this with the given options:

1. 45
2. 39
3. 43
4. 41

The calculated number, 43, is present in the options.

Step-by-Step Solution

Here are the steps followed to solve the number analogy:

1. Analyze the relationship between the first pair (68 : 19).
2. Hypothesize a rule that transforms the first number into the second. The rule \(68 \div 4 + 2 = 19\) is found.
3. Test the hypothesized rule on the second pair (76 : 21). The rule \(76 \div 4 + 2 = 21\) holds true.
4. Confirm the rule: Divide the first number by 4 and add 2 to get the second number.
5. Apply the confirmed rule to the third number (164) to find the missing number.
6. Calculate \(164 \div 4 + 2\).
7. \(164 \div 4 = 41\).
8. \(41 + 2 = 43\).
9. The missing number is 43.

Summary of Relationships

Applying the operation to the third pair gives \(164 \div 4 + 2 = 41 + 2 = 43\).

Conclusion

Based on the established pattern, the number related to 164 in the same way is 43.

Revision Table: Key Concepts in Number Analogies

Additional Information: Common Number Analogy Patterns

Number analogies can involve various types of relationships. Some common patterns include:

• Basic arithmetic operations (addition, subtraction, multiplication, division).
• Operations involving squares, cubes, or roots.
• Operations involving digits of the number (sum of digits, product of digits, reversing digits, etc.).
• Combinations of operations.
• Prime numbers, composite numbers, or sequences.
• Specific number series or properties.

Solving number analogies requires careful observation, trial and error, and logical reasoning to identify the pattern that consistently applies across the given pairs.