Understanding Number Relationships in Sets

The question asks us to find a set of numbers that shares the same relationship between its elements as the two given sets: (8, 8, 128) and (11, 7, 162).

We need to identify a mathematical rule or pattern that connects the first two numbers in each set to the third number. The rule must apply to both given sets consistently.

Identifying the Pattern

Let the three numbers in a set be denoted by $a$, $b$, and $c$. We will examine the relationship between $a$, $b$, and $c$ in the given sets:

• Set 1: (8, 8, 128) — Here, $a=8$, $b=8$, $c=128$.
• Set 2: (11, 7, 162) — Here, $a=11$, $b=7$, $c=162$.

We look for a relationship where $c$ is derived from $a$ and $b$ using basic arithmetic operations (addition, subtraction, multiplication, division, powers, etc.) applied to the whole numbers themselves.

Let's try combining $a$ and $b$ in different ways and see if we can get $c$.

• Consider the sum $a+b$:
• Set 1: $a+b = 8+8 = 16$. How is 16 related to 128? $128 = 16 \times 8$. The multiplier is 8.
• Set 2: $a+b = 11+7 = 18$. How is 18 related to 162? $162 = 18 \times 9$. The multiplier is 9.

The multiplier changes (8 and 9). Can we express the multiplier in terms of $a$ and $b$?

• In Set 1, the multiplier is 8, which is equal to $b$ (and $a$). The rule could be $c = (a+b) \times b$.
• Let's check this rule for Set 2: $(11+7) \times 7 = 18 \times 7 = 126$. This is not 162. So, this rule is incorrect.

Let's look at the multipliers again: 8 and 9. Can 8 be derived from (8,8)? Yes, 8. Can 9 be derived from (11,7)? Maybe related to their sum or average? $(11+7)/2 = 18/2 = 9$. This matches the multiplier for Set 2.

Let's test the hypothesis that the multiplier is $\frac{a+b}{2}$. The rule would be $c = (a+b) \times \frac{a+b}{2} = \frac{(a+b)^2}{2}$.

• Check Set 1: $a=8, b=8$. $c = \frac{(8+8)^2}{2} = \frac{16^2}{2} = \frac{256}{2} = 128$. This matches the given $c$.
• Check Set 2: $a=11, b=7$. $c = \frac{(11+7)^2}{2} = \frac{18^2}{2} = \frac{324}{2} = 162$. This also matches the given $c$.

The rule $c = \frac{(a+b)^2}{2}$ works for both the given sets.

Testing the Options

Now we apply the identified rule $c = \frac{(a+b)^2}{2}$ to each of the given options to find the set that follows the same pattern.

Option 1: (7, 9, 128)

• Here, $a=7$, $b=9$.
• Let's calculate the expected third number using the rule: $c = \frac{(7+9)^2}{2}$.
• $c = \frac{16^2}{2} = \frac{256}{2} = 128$.
• The calculated third number is 128, which matches the given third number in the set.

Option 2: (9, 8, 108)

• Here, $a=9$, $b=8$.
• Calculate $c = \frac{(9+8)^2}{2}$.
• $c = \frac{17^2}{2} = \frac{289}{2} = 144.5$.
• This does not match 108. Also, the result is not a whole number, which might be a hint that the rule should produce whole numbers if the inputs are whole numbers.

Option 3: (6, 8, 124)

• Here, $a=6$, $b=8$.
• Calculate $c = \frac{(6+8)^2}{2}$.
• $c = \frac{14^2}{2} = \frac{196}{2} = 98$.
• This does not match 124.

Option 4: (7, 8, 125)

• Here, $a=7$, $b=8$.
• Calculate $c = \frac{(7+8)^2}{2}$.
• $c = \frac{15^2}{2} = \frac{225}{2} = 112.5$.
• This does not match 125 and is not a whole number.

Only Option 1 follows the identified rule $c = \frac{(a+b)^2}{2}$.

Conclusion

The set (7, 9, 128) follows the same relationship as the given sets (8, 8, 128) and (11, 7, 162).

Number Analogy Revision Table

Number analogy questions test your ability to find patterns and relationships between numbers. The relationship can involve various arithmetic operations, squares, cubes, or combinations of these.

Key strategies include:

• Look for simple arithmetic relationships (sum, difference, product, quotient).
• Consider squares or cubes of the numbers.
• Look for relationships between the sum or difference of the first two numbers and the third number.
• Test potential rules consistently across all given examples before applying them to options.

Additional Information on Number Analogies

Number analogy problems are a common type of question in reasoning and quantitative aptitude tests. They require logical thinking and numerical ability to identify the hidden rule. The rules can sometimes be complex combinations of operations.

It's important to follow the constraint that operations are performed on the whole numbers as given, not on their individual digits, unless explicitly stated otherwise.