Understanding the Number Analogy Problem

The question asks us to identify a pattern between the first pair of numbers (16:36) and the second pair (64:100) and then apply that same pattern to the third pair (25:?) to find the missing number.

Analyzing the Number Pairs

Let's examine the relationship between the numbers in each given pair:

• First Pair: 16 and 36
• Second Pair: 64 and 100
• Third Pair: 25 and ? (the number we need to find)

Identifying the Pattern

We need to find a consistent mathematical relationship between the first number and the second number in each pair.

Let's look at the first pair, 16:36.

• 16 is the square of 4, which can be written as $4^2$.
• 36 is the square of 6, which can be written as $6^2$.

Notice that 6 is 2 more than 4. So, the relationship appears to be: the second number is the square of (the square root of the first number plus 2).

Let's check if this pattern holds for the second pair, 64:100.

• 64 is the square of 8, which is $8^2$.
• 100 is the square of 10, which is $10^2$.

Here, 10 is indeed 2 more than 8. So, the pattern $n^2 : (n+2)^2$ is confirmed.

Calculating the Missing Number

Now, we apply the same identified pattern to the third pair, 25:?

• 25 is the square of 5, so it's $5^2$. Here, $n=5$.
• According to the pattern, the second number should be $(n+2)^2$.
• Substituting $n=5$, we get $(5+2)^2$.
• Calculating this: $(5+2)^2 = 7^2 = 49$.

Therefore, the missing number is 49.

Summary Table

Conclusion

The number that completes the analogy 16:36::64:100::25:? is 49, following the pattern where the second number is the square of the square root of the first number plus 2.