Understanding the Analogy Pattern

The question asks us to find the relationship between the pairs of numbers in the analogy and apply the same relationship to the last pair to find the missing term. The analogy is given as: 32 ∶ 8 ∶∶ 72 ∶ 12 ∶∶ 128 ∶ ?

Let's break down the analogy into pairs:

• Pair 1: 32 and 8
• Pair 2: 72 and 12
• Pair 3: 128 and the missing term (?)

We need to discover the rule that connects the first number to the second number in Pair 1 and Pair 2, and then use that rule to find the missing number in Pair 3.

Analyzing the Relationship in Pair 1 (32 ∶ 8)

Let's look at the numbers 32 and 8. How can we get from 32 to 8, or from 8 to 32, using a simple mathematical operation or a combination of operations?

• If we divide 32 by 4, we get 8 ($32 \div 4 = 8$).
• If we multiply 8 by 4, we get 32 ($8 \times 4 = 32$).

Now let's look at Pair 2.

Analyzing the Relationship in Pair 2 (72 ∶ 12)

Let's look at the numbers 72 and 12.

• If we divide 72 by 6, we get 12 ($72 \div 6 = 12$).
• If we multiply 12 by 6, we get 72 ($12 \times 6 = 72$).

The simple division or multiplication factors (4 and 6) are different. This suggests the relationship is not a fixed factor. Let's look for a pattern involving the numbers themselves.

Consider the second number in each pair. In Pair 1, it's 8. In Pair 2, it's 12. The corresponding first numbers are 32 and 72.

Let's explore squaring the second number:

• For Pair 1: $8^2 = 8 \times 8 = 64$. How is 64 related to 32? 32 is half of 64 ($32 = 64 \div 2$).
• This suggests a possible rule: First Term = $(\text{Second Term})^2 \div 2$.

Let's test this rule with Pair 2:

• For Pair 2: $12^2 = 12 \times 12 = 144$. Is 72 half of 144? Yes, $72 = 144 \div 2$.

The rule seems consistent for both pairs:

\(\text{First Term} = \frac{(\text{Second Term})^2}{2}\)

We can also express this rule to find the second term if we know the first term:

Multiply both sides by 2: \(2 \times \text{First Term} = (\text{Second Term})^2\)

Take the square root of both sides: \(\text{Second Term} = \sqrt{2 \times \text{First Term}}\)

Calculating the Missing Term in Pair 3 (128 ∶ ?)

Now we apply the discovered relationship to the third pair: 128 and the missing term. Let the missing term be \(X\).

Using the rule \(\text{Second Term} = \sqrt{2 \times \text{First Term}}\), we have:

\(X = \sqrt{2 \times 128}\)

First, calculate \(2 \times 128\):

\(2 \times 128 = 256\)

Now, find the square root of 256:

\(X = \sqrt{256}\)

The square root of 256 is 16, because \(16 \times 16 = 256\).

\(X = 16\)

So, the missing term is 16.

Verifying the Answer with the Pattern

Let's check if the relationship \(\text{First Term} = \frac{(\text{Second Term})^2}{2}\) holds for the third pair (128 and 16):

\(128 = \frac{16^2}{2}\)

\(128 = \frac{16 \times 16}{2}\)

\(128 = \frac{256}{2}\)

\(128 = 128\)

The relationship holds true for the third pair with the missing term being 16.

Comparing with Options

The calculated missing term is 16. Let's check the given options:

• Option 1: 18
• Option 2: 16
• Option 3: 14
• Option 4: 15

Our calculated answer, 16, matches Option 2.

Conclusion

Based on the consistent relationship observed in the first two pairs of the analogy, the missing term related to 128 is 16.

Revision Table: Analogy Pattern

Additional Information: Solving Number Analogies

Number analogies are common in reasoning and aptitude tests. They require you to find the logical rule or pattern that connects the numbers in the given pairs. Here are some common types of patterns to look for:

• Arithmetic Operations: Addition, subtraction, multiplication, division.
• Powers and Roots: Squares, cubes, square roots, cube roots.
• Series: Arithmetic progression, geometric progression.
• Digit Operations: Sum of digits, product of digits, reversing digits.
• Combination of Operations: Applying multiple steps (e.g., multiply and add, square and divide).
• Divisibility Rules: Checking if one number is a factor or multiple of another.

Solving number analogies often involves trial and error. Start by looking for the simplest relationships (like basic arithmetic) and then move to more complex ones (like squares, roots, or combinations) if the simple ones don't fit all pairs.