Understanding the Analogy Question

This question asks us to find a relationship between pairs of numbers and apply that same relationship to a third pair to find a missing number. The structure given is 16 ∶ 144 ∶∶ 22 ∶ ? ∶∶ 14 ∶ 112. This structure implies that the relationship between the first two numbers (16 and 144) is the same as the relationship between the third and fourth numbers (22 and ?) and also the same as the relationship between the fifth and sixth numbers (14 and 112).

Analyzing the Relationship in the Pairs

Let's look at the first pair of numbers: 16 and 144. We need to find a rule that connects 16 to 144.

• If we divide 144 by 16, we get \$ 144 \div 16 = 9 \$. So, \$ 16 \times 9 = 144 \$. The multiplier is 9.

Now let's look at the third given pair of numbers: 14 and 112. We need to find the rule connecting 14 to 112.

• If we divide 112 by 14, we get \$ 112 \div 14 = 8 \$. So, \$ 14 \times 8 = 112 \$. The multiplier is 8.

The multipliers (9 and 8) are different, which means the relationship isn't just multiplying by a fixed number. There must be a rule that relates the first number of each pair to the multiplier used to get the second number.

Identifying the Pattern or Rule

Let's look at the first numbers (16 and 14) and their corresponding multipliers (9 and 8):

• For 16, the multiplier is 9.
• For 14, the multiplier is 8.

Can we find a connection between the first number and its multiplier? Notice that 9 is related to 16, and 8 is related to 14. Let's try simple arithmetic operations.

• Could the multiplier be related to half of the first number? Half of 16 is 8. If we add 1 to 8, we get 9. So, \$(16 \div 2) + 1 = 8 + 1 = 9\$. This matches the multiplier for 16.
• Let's test this rule with the second pair (14 and 112). Half of 14 is 7. If we add 1 to 7, we get 8. So, \$(14 \div 2) + 1 = 7 + 1 = 8\$. This matches the multiplier for 14.

It seems the rule is: Second Term = First Term × ((First Term ÷ 2) + 1).

Applying the Rule to Find the Missing Term

Now we apply this established rule to the second pair: 22 ∶ ?

• The first term is 22.
• Using the rule, the multiplier is \$(22 \div 2) + 1 = 11 + 1 = 12\$.
• The missing second term is the first term multiplied by the multiplier: \$ 22 \times 12 \$.

Let's calculate \$ 22 \times 12 \$.

\$ 22 \times 12 = 22 \times (10 + 2) = (22 \times 10) + (22 \times 2) = 220 + 44 = 264 \$.

So, the missing term is 264.

Comparing with Options

The calculated missing term is 264. Let's check the given options:

• Option 1: 264
• Option 2: 244
• Option 3: 246
• Option 4: 266

Our calculated value, 264, matches Option 1.

Conclusion

The relationship between the numbers in each pair follows the rule: Second Term = First Term × ((First Term ÷ 2) + 1). Applying this rule to the pair 22 : ? gives us the missing term as 264.

Revision Table: Analogy Rules

Additional Information: Solving Reasoning Analogy Questions

Solving numerical analogy questions requires sharp observation and logical thinking. Here are some common approaches and tips:

• Basic Operations: Look for relationships involving addition, subtraction, multiplication, division.
• Squares and Cubes: The second term might be the square or cube of the first term, or related to them (\$ n^2 \pm k \$, \$ n^3 \pm k \$).
• Prime Numbers: The terms might be prime numbers or related to prime numbers.
• Digit Operations: Sometimes the relationship involves operations on the digits of the numbers (e.g., sum of digits, product of digits).
• Sequence/Pattern: The multipliers or the differences between terms might follow a pattern (arithmetic progression, geometric progression, etc.) across different pairs, as seen in this question.
• Checking All Pairs: Always test the identified rule on all given pairs to ensure it is consistent before applying it to find the missing term.
• Options Analysis: Sometimes looking at the options can give clues about the type of relationship involved.

Practice with different types of numerical analogy problems helps in quickly identifying potential patterns and rules.