Finding the Number Pair Analogy

The question asks us to identify the number pair that shares the same relationship as the pair 28 : 63. To solve this, we first need to analyze the relationship between 28 and 63.

Analyzing the Given Number Pair: 28 : 63

Let's find the greatest common divisor (GCD) of 28 and 63.

• Factors of 28 are 1, 2, 4, 7, 14, 28.
• Factors of 63 are 1, 3, 7, 9, 21, 63.

The greatest common divisor of 28 and 63 is 7.

Now, let's divide both numbers in the pair by their GCD:

$$ \frac{28}{7} = 4 $$

$$ \frac{63}{7} = 9 $$

The ratio of the two numbers in the pair 28 : 63 is 4 : 9. So, the relationship is that the two numbers are in the ratio 4:9, or one number is (4/9) times the other, or both numbers are multiples of some number, with the base factors being 4 and 9.

We need to find an option pair where the two numbers have the same ratio of 4 : 9.

Evaluating the Options

Let's check each option pair to see if their ratio is 4 : 9.

Option 1: 48 : 98

Find the GCD of 48 and 98.

• Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
• Factors of 98: 1, 2, 7, 14, 49, 98

GCD(48, 98) = 2.

Divide both numbers by their GCD:

$$ \frac{48}{2} = 24 $$

$$ \frac{98}{2} = 49 $$

The ratio is 24 : 49. This is not 4 : 9.

Option 2: 16 : 36

Find the GCD of 16 and 36.

• Factors of 16: 1, 2, 4, 8, 16
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

GCD(16, 36) = 4.

Divide both numbers by their GCD:

$$ \frac{16}{4} = 4 $$

$$ \frac{36}{4} = 9 $$

The ratio is 4 : 9. This matches the ratio of the original pair 28 : 63.

Option 3: 36 : 78

Find the GCD of 36 and 78.

• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Factors of 78: 1, 2, 3, 6, 13, 26, 39, 78

GCD(36, 78) = 6.

Divide both numbers by their GCD:

$$ \frac{36}{6} = 6 $$

$$ \frac{78}{6} = 13 $$

The ratio is 6 : 13. This is not 4 : 9.

Option 4: 12 : 28

Find the GCD of 12 and 28.

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 28: 1, 2, 4, 7, 14, 28

GCD(12, 28) = 4.

Divide both numbers by their GCD:

$$ \frac{12}{4} = 3 $$

$$ \frac{28}{4} = 7 $$

The ratio is 3 : 7. This is not 4 : 9.

Conclusion

The only number pair that has the same 4 : 9 ratio as 28 : 63 is 16 : 36.

Revision Table: Number Analogy Basics

Additional Information: Solving Number Analogies

Solving number analogy questions often involves identifying the mathematical relationship between the numbers in the given pair. Common relationships include:

• Arithmetic operations (addition, subtraction, multiplication, division)
• Powers or roots (squares, cubes, square roots, cube roots)
• Sequences (e.g., consecutive numbers, prime numbers, odd/even numbers)
• Specific factors or multiples
• Ratios
• Combinations of the above

A good strategy is to:

1. Look for obvious relationships like addition, subtraction, multiplication, or division.
2. Consider powers or roots.
3. Calculate the ratio of the numbers and simplify it using GCD. This often reveals the core relationship, as seen in the 28 : 63 example.
4. Check for sequential relationships or properties of the numbers (prime, composite, even, odd).
5. Apply the identified relationship to each option pair.
6. Select the option that fits the established rule.

Practicing different types of number analogy problems helps in quickly recognizing common patterns and relationships.