Number Analogy Solution: 16 to 144, 28 to What?

This question asks us to find the number that relates to 28 in the same way that 144 relates to 16. This type of problem is known as a number analogy. We need to discover the pattern or rule connecting the first pair of numbers (16 and 144) and then apply that same rule to the third number (28) to find the missing fourth number.

Identifying the Pattern between 16 and 144

Let's look closely at the numbers 16 and 144. We need to find a mathematical relationship between them. Possible relationships could involve addition, subtraction, multiplication, division, squares, cubes, or a combination of these operations.

• Is it addition/subtraction? $144 - 16 = 128$. Adding 128 to 28 gives $28 + 128 = 156$. This is not in the options.
• Is it multiplication? $16 \times 9 = 144$. So, 144 is 16 multiplied by 9.

Now, let's consider how 9 might be related to 16. Could 9 be derived from 16 using a simple rule? Let's try some common operations:

• $16 / 2 = 8$. If we add 1 to 8, we get 9. So, the multiplier could be $(\text{First Number} / 2) + 1$.

Let's test this rule with the first pair:

Multiplier for 16 = $(16 / 2) + 1 = 8 + 1 = 9$.

Applying this multiplier: $16 \times 9 = 144$.

This rule seems to fit the relationship between 16 and 144 perfectly. The pattern is: The second number is obtained by multiplying the first number by (half of the first number plus one).

Mathematically, the relationship is: $\text{Second Number} = \text{First Number} \times \left( \frac{\text{First Number}}{2} + 1 \right)$.

Applying the Pattern to Find the Missing Number for 28

Now we apply the same rule to the number 28 to find the missing fourth number in the analogy 28 : ?.

Let the missing number be $X$. According to our pattern:

$X = 28 \times \left( \frac{28}{2} + 1 \right)$

First, calculate the value inside the parentheses:

$\frac{28}{2} = 14$

$14 + 1 = 15$

So, the multiplier for 28 is 15.

Now, multiply 28 by 15 to find $X$:

$X = 28 \times 15$

Let's calculate $28 \times 15$:

$28 \times 15 = 28 \times (10 + 5)$

$= (28 \times 10) + (28 \times 5)$

$= 280 + 140$

$= 420$

So, the missing number is 420.

Checking the Options

We found the missing number to be 420. Let's check the given options:

1. 420
2. 364
3. 263
4. 544

Our calculated number, 420, matches the first option.

Conclusion for the Number Analogy

The relationship 16 : 144 follows the rule where the second number is the first number multiplied by (half of the first number plus one). Applying this rule to 28 gives us 420. Therefore, the correct analogy is 16 : 144 :: 28 : 420.

Revision Table: Number Analogy

Additional Information: Solving Number Series and Analogies

Number analogy and series problems are common in logical reasoning tests. They require you to identify the underlying pattern. Here are some common patterns to look for:

• Arithmetic Progression: Adding or subtracting a constant value.
• Geometric Progression: Multiplying or dividing by a constant value.
• Differences/Ratios: Looking at the differences or ratios between consecutive terms. These differences/ratios might form their own pattern (e.g., an arithmetic or geometric progression).
• Squares and Cubes: Numbers might be squares or cubes, or related to squares/cubes ($\text{n}^2$, $\text{n}^2 \pm \text{k}$, $\text{n}^3$, $\text{n}^3 \pm \text{k}$).
• Product/Sum of Digits: The next number might be related to the sum or product of the digits of the previous number.
• Alternating Patterns: Different rules might apply to alternate numbers in the series or pairs in the analogy.
• Combination of Operations: As seen in this problem, a combination of arithmetic operations can form the pattern.

Solving these problems often involves trial and error, trying different common patterns until one fits consistently.