Understanding the Number Analogy Pattern

The question presents a number analogy in the format A : B :: C : D :: E : F. This means that the relationship between A and B is the same as the relationship between C and D, and this same relationship should also exist between E and F.

We are given the following pairs:

• 1444 ∶ 38
• 1225 ∶ 35
• ? ∶ 47

Our goal is to identify the relationship between the numbers in the first two pairs and then apply it to the third pair to find the missing number.

Analyzing the Relationship in the Pairs

Let's examine the first pair: 1444 and 38.

We can try to find a mathematical relationship between these two numbers. If we consider squaring the second number, we get:

$\text{38}^2 = 38 \times 38$

Let's calculate this:

$\text{38}^2 = (40 - 2)^2 = 40^2 - 2 \times 40 \times 2 + 2^2 = 1600 - 160 + 4 = 1440 + 4 = 1444$

So, the first number (1444) is the square of the second number (38). Alternatively, the second number (38) is the square root of the first number (1444), i.e., $\sqrt{1444} = 38$.

Now, let's check if this relationship holds true for the second pair: 1225 and 35.

Let's square the second number (35):

$\text{35}^2 = 35 \times 35$

We can calculate this:

$\text{35}^2 = (30 + 5)^2 = 30^2 + 2 \times 30 \times 5 + 5^2 = 900 + 300 + 25 = 1200 + 25 = 1225$

Indeed, the third number (1225) is the square of the fourth number (35). This confirms that the relationship is that the first number in each pair is the square of the second number, or the second number is the square root of the first.

Step-by-Step Solution for the Analogy

Based on the pattern found, the missing number (?) should be the square of the number 47.

We need to calculate $47^2$.

1. Identify the pattern: The first number is the square of the second number in each pair.
2. Apply the pattern to the third pair: The missing number (?) must be the square of 47.
3. Calculate the square of 47:

$\text{47}^2 = 47 \times 47$

Calculation:

$\text{47}^2 = (50 - 3)^2 = 50^2 - 2 \times 50 \times 3 + 3^2 = 2500 - 300 + 9 = 2200 + 9 = 2209$

So, the missing number is 2209.

Checking the Options and Final Answer

Now let's look at the given options to see which one matches our calculated missing number:

1. 2162
2. 1369
3. 1024
4. 2209

Our calculated missing number is 2209, which matches option 4.

Final Answer Determination

The missing number in the analogy is 2209, following the established pattern where the first number is the square of the second number in each pair.

Revision Table: Key Analogy Concepts

Additional Information: Perfect Squares

A perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a perfect square because $9 = 3 \times 3 = 3^2$.

In this analogy, 1444, 1225, and 2209 are perfect squares. Knowing common perfect squares or being able to quickly calculate squares helps in solving such analogy problems.

• $35^2 = 1225$
• $38^2 = 1444$
• $47^2 = 2209$

Recognizing these numbers as squares can quickly reveal the underlying pattern in the analogy.