Solving Number Analogy Questions

Number analogy questions test your ability to find a relationship or pattern between two numbers and apply the same relationship to another number to find a missing value. The question presented is 15 : 270 :: 13 : ?

This means we need to find the pattern connecting 15 and 270, and then apply that same pattern to 13 to find the unknown number.

Finding the Pattern Between 15 and 270

Let's examine the relationship between 15 and 270. We can try common mathematical operations:

• Addition/Subtraction: $270 - 15 = 255$. If we add 255 to 13, we get $13 + 255 = 268$. This is not among the options.
• Multiplication: $270 \div 15 = 18$. If we multiply 13 by 18, we get $13 \times 18 = 234$. This is not among the options.
• Squaring or Cubing: $15^2 = 225$. $15^3 = 3375$. Neither of these directly gives 270.

Let's look closer at $15^2 = 225$. The difference between 270 and 225 is $270 - 225 = 45$. Can we find a relationship for 45 using 15? $15 \times 3 = 45$. So, one possible pattern is $n^2 + n \times 3$. Let's test this pattern with the first pair, where $n=15$: $15^2 + 15 \times 3 = 225 + 45 = 270$. This pattern works for the first pair.

Another way to express $n^2 + n \times 3$ is $n(n+3)$. Let's test this pattern with $n=15$: $15 \times (15 + 3) = 15 \times 18 = 270$. This pattern also works for the first pair.

Applying the Pattern to 13

Now we apply the pattern we found to the third number, 13. Using the pattern $n(n+3)$ with $n=13$:

Required number $= 13 \times (13 + 3)$

Required number $= 13 \times 16$

Calculating the Result

To calculate $13 \times 16$:

$13 \times 16 = (10 + 3) \times 16 = 10 \times 16 + 3 \times 16$

$160 + 48 = 208$

So, the missing number is 208.

Checking the Options

Let's look at the given options:

1. 208
2. 169
3. 144
4. 275

Our calculated value, 208, matches option 1.

Conclusion

The relationship between the first pair of numbers (15 and 270) is that the second number is the first number multiplied by (the first number plus 3), i.e., $n \times (n+3)$. Applying this same pattern to the third number (13), we get $13 \times (13 + 3) = 13 \times 16 = 208$.

Revision Table: Number Analogy

Review the key steps to solve number analogy problems:

• Identify the first pair of numbers.
• Analyze the relationship between them (arithmetic, algebraic, squares, cubes, etc.).
• Formulate a rule or pattern.
• Apply the same rule to the third number.
• Calculate the result and check against the options.

Additional Information: Types of Number Patterns

Number analogy questions often involve various types of patterns. Some common ones include:

• Addition/Subtraction series
• Multiplication/Division series
• Square or cube related patterns ($n^2$, $n^3$, $n^2 \pm k$, $n^3 \pm k$)
• Patterns involving digits of the number
• Combinations of operations
• Prime numbers, composite numbers, etc.

Practice with different types of patterns is crucial for mastering number analogy questions.