Understanding the Number Analogy Problem

This question is all about finding a hidden pattern or rule that connects pairs of numbers. We see a sequence like this: 4 : 32 :: 7 : ? :: 12 : 192. Our job is to figure out the missing number (represented by '?'). We do this by finding the rule that links the first number (4) to the second number (32), and also links the fifth number (12) to the sixth number (192). Once we find that rule, we apply it to the third number (7) to find the unknown fourth number.

Analyzing the Given Number Pairs

First, let's look closely at the pairs we already know (4 to 32, and 12 to 192) to find the secret rule connecting them.

• Looking at the first pair (4 : 32):

  How is 4 related to 32? Let's try multiplication.

  • $32 \div 4 = 8$. So, $32 = 4 \times 8$.

  Now, how can we get the multiplier 8 from the number 4?

  • Possibility 1: Multiply 4 by 2 ($4 \times 2 = 8$).
  • Possibility 2: Add 4 to itself ($4 + 4 = 8$).
• Looking at the third pair (12 : 192):

  Let's see how 12 is related to 192.

  • $192 \div 12 = 16$. So, $192 = 12 \times 16$.

Identifying the Consistent Pattern

We need a single rule that works for both pairs. Let's test the possibilities we found for getting the multiplier.

• Testing Possibility 1 (Multiplier = First Number $\times$ 2):
  • Pair 1: $4 \times (4 \times 2) = 4 \times 8 = 32$. (Works)
  • Pair 3: $12 \times (12 \times 2) = 12 \times 24 = 288$. (Does not equal 192)

  So, this rule doesn't work for both pairs.

• Testing Possibility 2 (Multiplier = First Number + 4):

  Let's check if the rule is: Second Number = First Number $\times$ (First Number + 4).

  • Pair 1: $4 \times (4 + 4) = 4 \times 8 = 32$. (Works)
  • Pair 3: $12 \times (12 + 4) = 12 \times 16 = 192$. (Works)

  This rule works perfectly for both given pairs!

The pattern is confirmed: The second number in each pair is found by multiplying the first number by the sum of the first number and 4. We can write this rule as: $M = N \times (N+4)$, where $N$ is the first number and $M$ is the second number.

Applying the Pattern to Find the Missing Number

Now we use this rule ($M = N \times (N+4)$) for the middle pair: 7 : ?.

• In this case, the first number ($N$) is 7.
• Substitute $N=7$ into our rule: $M = 7 \times (7 + 4)$.
• First, calculate the part in the parentheses: $7 + 4 = 11$.
• Now, perform the multiplication: $M = 7 \times 11 = 77$.

So, the missing number is 77. The full sequence is 4 : 32 :: 7 : 77 :: 12 : 192.

Final Answer Verification

Let's double-check our answer by ensuring the pattern works for all parts of the analogy:

• Pair 1: $4 \times (4 + 4) = 4 \times 8 = 32$. Correct.
• Pair 2: $7 \times (7 + 4) = 7 \times 11 = 77$. Correct.
• Pair 3: $12 \times (12 + 4) = 12 \times 16 = 192$. Correct.

The pattern consistently produces the correct numbers. The calculated missing number is 77. Looking at the options provided, 77 corresponds to option 4.