Understanding Number Relationships: Finding the Analogous Pair

This question asks us to identify an option where the pair of numbers shares the same relationship as the given pair, 11 : 132. To solve this, we first need to figure out the mathematical relationship connecting the numbers in the given pair.

Analyzing the Given Number Pair: 11 : 132

Let's look at the numbers 11 and 132. How are they related?

• Is it addition? $11 + X = 132$. $X = 132 - 11 = 121$. This doesn't seem like a simple, common relationship.
• Is it multiplication? $11 \times X = 132$. To find X, we divide 132 by 11.
  $$ X = \frac{132}{11} $$ $$ X = 12 $$ So, the relationship is $11 \times 12 = 132$.

Let's observe the multiplier, 12. It is one more than the first number, 11 ($12 = 11 + 1$). This suggests a possible pattern: the second number is the first number multiplied by (the first number + 1). If the first number is $n$, the second number is $n \times (n+1)$.

Let's verify this pattern with the given pair:

• For $n=11$: $n \times (n+1) = 11 \times (11+1) = 11 \times 12 = 132$. This matches the given pair 11 : 132.

So, the established relationship is $n : n \times (n+1)$.

Evaluating the Options to Find the Same Relationship

Now, we apply this relationship $n : n \times (n+1)$ to each of the given options to see which one fits the pattern.

Option 1: 6 : 48

• Here, the first number is $n=6$.
• According to our pattern, the second number should be $n \times (n+1) = 6 \times (6+1) = 6 \times 7 = 42$.
• The given second number is 48.
• Since 42 is not equal to 48, this option does not share the same relationship.

Option 2: 8 : 72

• Here, the first number is $n=8$.
• According to our pattern, the second number should be $n \times (n+1) = 8 \times (8+1) = 8 \times 9 = 72$.
• The given second number is 72.
• Since 72 is equal to 72, this option shares the same relationship.

Option 3: 7 : 61

• Here, the first number is $n=7$.
• According to our pattern, the second number should be $n \times (n+1) = 7 \times (7+1) = 7 \times 8 = 56$.
• The given second number is 61.
• Since 56 is not equal to 61, this option does not share the same relationship.

Option 4: 9 : 93

• Here, the first number is $n=9$.
• According to our pattern, the second number should be $n \times (n+1) = 9 \times (9+1) = 9 \times 10 = 90$.
• The given second number is 93.
• Since 90 is not equal to 93, this option does not share the same relationship.

Conclusion

Only Option 2, 8 : 72, follows the established relationship $n : n \times (n+1)$, where the second number is the first number multiplied by one more than the first number.

The option that shares the same number relationship as 11 : 132 is 8 : 72.

Revision Table: Number Relationship Concepts

Additional Information: Solving Analogous Pair Problems

Solving number analogy problems often involves checking common mathematical relationships. Here are some types of relationships to look for:

• Arithmetic Operations: Constant addition, subtraction, multiplication, or division. Example: $n : n+c$ or $n : c \times n$.
• Powers and Roots: Squaring, cubing, square roots, cube roots. Example: $n : n^2$ or $n : \sqrt{n}$.
• Based on $n$: Relationships involving $n+1$, $n-1$, $2n$, $n/2$, $n^2+1$, $n^2-1$, $n(n+1)$, etc. Example: $n : n^2+1$.
• Prime Numbers: Relationships based on prime numbers. Example: $n : \text{next prime number after } n$.
• Digit Properties: Relationships based on the sum of digits, product of digits, etc. Example: $n : \text{sum of digits of } n$.

When tackling these problems, it's best to systematically test out common relationships until you find one that fits the given pair and then check the options.