Number Pair Analysis: 144 : 400

This question asks us to identify a logical relationship between the first pair of numbers, 144 and 400. Once we determine this relationship, we need to apply it to the number 324 to find its corresponding related number from the given alternatives.

Number Pair Logic: Finding the Rule

Let's examine the relationship between 144 and 400.

One way to find the relationship is by looking at the difference between the two numbers:

$400 - 144 = 256$

This suggests that the second number in the pair is 256 greater than the first number. Let's verify if this pattern holds true for the given options.

Another possible approach involves square roots:

• The square root of 144 is $\sqrt{144} = 12$.
• The square root of 400 is $\sqrt{400} = 20$.

The difference between these square roots is $20 - 12 = 8$. This implies a potential pattern of $n^2 : (n+8)^2$. However, we must ensure the identified pattern correctly leads to the provided answer option.

The simpler rule, where the second number is obtained by adding 256 to the first, is directly verifiable and fits the structure of number pair analogies well.

Number Pair Application: 324 : ?

We are given the first number of the second pair as 324. We need to find the second number using the relationship established from the first pair (144 : 400).

Using the rule: Second Number = First Number + 256

Applying this rule to 324:

$324 + 256 = 580$

This calculation suggests that the related number pair should be 324 : 580.

Number Pair Verification: Checking Options

To confirm our finding, let's check each of the provided options against the rule: Second Number = First Number + 256.

The verification confirms that option 4, the pair 324 : 580, is the only one where the second number is exactly 256 more than the first number.

Number Pair Solution: Final Answer

The analysis reveals a consistent pattern where the second number in each pair is 256 greater than the first number. The pair 144 : 400 follows this pattern ($144 + 256 = 400$). Applying the same logic, the pair 324 : 580 also follows this pattern ($324 + 256 = 580$).

Therefore, the related number pair is 324 : 580.