Understanding Number Analogies

A number analogy question asks you to find the relationship between two given numbers and then apply that same relationship to a third number to find a fourth, missing number. The format is typically A : B :: C : D, where you need to find D, given A, B, and C.

In this question, we have the analogy 5 : 375 :: 9 : ?. We need to determine how 5 is related to 375 and then apply that same logic to find the number related to 9.

Analyzing the Relationship between 5 and 375

Let's look for a pattern connecting 5 and 375. We can consider various mathematical operations:

• Addition or Subtraction: $375 - 5 = 370$. If we add 370 to 9, we get $9 + 370 = 379$. This is not one of the options.
• Multiplication: $375 \div 5 = 75$. If we multiply 9 by 75, we get $9 \times 75 = 675$. This is not one of the options.
• Powers: Consider powers of 5. $5^1 = 5$, $5^2 = 25$, $5^3 = 125$.
• Combining powers and multiplication: Notice that 375 is related to $5^3$. Let's see if 375 is a multiple of 125. $375 \div 125 = 3$.

This suggests a possible relationship: The second number is obtained by multiplying the cube of the first number by 3.

Let's verify this relationship with the first pair (5 and 375):

First number = 5

Cube of the first number = $5^3 = 5 \times 5 \times 5 = 25 \times 5 = 125$

Multiply the cube by 3 = $3 \times 125 = 375$

This matches the given second number, 375. So, the relationship is indeed: Second Number = $3 \times (\text{First Number})^3$.

Applying the Pattern to Find the Missing Number

Now we apply the same relationship to the third number, which is 9, to find the missing fourth number.

Third number = 9

Cube of the third number = $9^3 = 9 \times 9 \times 9 = 81 \times 9$

Let's calculate $81 \times 9$:

So, $9^3 = 729$.

Now, multiply the cube by 3, according to the pattern:

Missing Number = $3 \times 9^3 = 3 \times 729$

Let's calculate $3 \times 729$:

The missing number is 2187.

Comparing with Options

Let's check the options provided:

1. 1985
2. 2187
3. 727
4. 465

Our calculated number, 2187, matches Option 2.

Conclusion

The relationship between the numbers in the analogy 5 : 375 :: 9 : ? is that the second number is 3 times the cube of the first number. Applying this rule, $3 \times 5^3 = 3 \times 125 = 375$, and $3 \times 9^3 = 3 \times 729 = 2187$. Therefore, the missing number is 2187.

Revision Table: Number Analogy

Additional Information: Finding Patterns in Analogies

Number analogy questions test your ability to recognize numerical patterns. Common patterns include:

• Arithmetic operations (addition, subtraction, multiplication, division)
• Powers and roots (squares, cubes, square roots, cube roots)
• Combinations of operations (e.g., multiply and add, square and subtract)
• Digit-based operations (sum of digits, product of digits)
• Sequences (prime numbers, odd/even numbers, specific series)

When tackling such questions, it's helpful to start with simple operations and move towards more complex ones. Calculating squares and cubes of the given numbers can often reveal the underlying pattern quickly.