Understanding Number Analogy Questions

Number analogy questions test your ability to find the relationship between a pair of numbers and apply that same relationship to another number to find a missing value. The format is typically A : B :: C : D, where A is related to B in the same way that C is related to D.

Analyzing the Given Analogy: 7 : 56 :: 13 : ?

We are given the first pair 7 : 56 and the third number 13. We need to find the number that is related to 13 in the same way that 56 is related to 7.

Step 1: Find the relationship between the first pair (7 and 56)

Let's look for common mathematical relationships between 7 and 56.

• Is 56 a multiple of 7? Yes, $56 \div 7 = 8$. So, $7 \times 8 = 56$.
• Is there a relationship involving squares or cubes? $7^2 = 49$, which is close to 56 ($49+7=56$). So, $7^2 + 7 = 56$. This is also $7 \times (7+1) = 7 \times 8 = 56$.

Both $7 \times 8 = 56$ and $7 \times (7+1) = 56$ describe the relationship between 7 and 56. The second one ($x \times (x+1)$) is a common pattern in such analogies where the multiplier is one more than the first number itself. However, the first relationship ($x \times 8$) suggests a constant multiplier of 8.

Step 2: Apply the relationship to the third number (13)

Let's test both potential patterns:

1. Pattern 1: Constant Multiplier
   The relationship is $7 \times 8 = 56$. If we assume the multiplier is a constant 8, we apply it to 13:
   $13 \times 8 = ?$
2. Pattern 2: $x \times (x+1)$
   The relationship is $x \times (x+1)$. For $x=7$, we get $7 \times (7+1) = 7 \times 8 = 56$. If we apply this pattern to $x=13$:
   $13 \times (13+1) = 13 \times 14 = ?$

Step 3: Calculate the result and check options

Let's calculate the results for both patterns:

• Using Pattern 1 (Constant Multiplier 8):
  $13 \times 8 = 104$
• Using Pattern 2 ($x \times (x+1)$):
  $13 \times 14 = 182$

Now, let's compare these results with the given options:

The result from Pattern 1 (Constant Multiplier 8), which is 104, matches Option 4. The result from Pattern 2 (182) is not among the options.

Therefore, the most likely pattern based on the available options is that the second number is obtained by multiplying the first number by 8 (which was derived from the first pair 7:56 as 56/7=8).

Conclusion

The relationship between 7 and 56 is $7 \times 8 = 56$. Applying the same relationship to 13, we get $13 \times 8 = 104$. The number that should replace the question mark is 104.

Revision Table: Number Analogy Patterns

Additional Information on Solving Analogies

Solving number analogy questions requires identifying the underlying mathematical or logical rule connecting the pair of numbers. It could be arithmetic operations, squares, cubes, patterns based on digits, prime numbers, or a combination of these. It's important to look for the simplest relationship first and then explore more complex ones if the simple ones don't fit the options. Practice with various types of number series and patterns helps in quickly recognizing the relationship.

• Always check if the relationship involves a constant value (addition, subtraction, multiplication, division).
• Look for relationships involving squares, cubes, square roots, or cube roots.
• Consider patterns like $n \times (n+1)$, $n \times (n-1)$, $n^2+1$, $n^2-1$, etc.
• Sometimes, the relationship involves the sum or product of digits.
• Check for prime or composite number sequences.
• Once a pattern is found, apply it strictly to the second pair of numbers.
• Verify if the result matches one of the given options. If not, re-evaluate the relationship for the first pair.