Understanding Number Relation in Sets

The question asks us to find an option where the set of numbers follows the same pattern or relationship as the given set (4, 25, 105). To solve this number relation problem, we first need to identify the rule connecting the numbers in the original set.

Identifying the Pattern in (4, 25, 105)

Let the three numbers in the set be denoted as A, B, and C. In the given set (4, 25, 105), we have A=4, B=25, and C=105. We look for a relationship between A and B, and between B and C (or possibly A and C).

Let's examine common arithmetic and multiplicative relationships:

• Difference between B and A: $25 - 4 = 21$
• Difference between C and B: $105 - 25 = 80$

These differences don't immediately suggest a simple linear pattern across all numbers.

Let's look for multiplication and addition/subtraction patterns:

• From 4 to 25: Maybe $4 \times k_1 + c_1 = 25$. If we consider multiplying by 5, $4 \times 5 = 20$, and $20 + 5 = 25$. So, $5 \times A + 5 = B$ is a possible pattern for the first two numbers. Let's check: $5 \times 4 + 5 = 20 + 5 = 25$. This holds true for the first two numbers in the original set.
• From 25 to 105: Maybe $25 \times k_2 + c_2 = 105$. If we consider multiplying by 4, $25 \times 4 = 100$, and $100 + 5 = 105$. So, $4 \times B + 5 = C$ is a possible pattern for the second and third numbers. Let's check: $4 \times 25 + 5 = 100 + 5 = 105$. This holds true for the second and third numbers.

So, we have identified a potential pattern:

1. Second number = $5 \times$ First number $+ 5$ ($B = 5A + 5$)
2. Third number = $4 \times$ Second number $+ 5$ ($C = 4B + 5$)

Let's double-check this pattern with the original set (4, 25, 105):

• For the first two numbers: $B = 5 \times 4 + 5 = 20 + 5 = 25$. This is correct.
• For the second and third numbers: $C = 4 \times 25 + 5 = 100 + 5 = 105$. This is correct.

The pattern seems consistent for the given set (4, 25, 105).

Applying the Pattern to the Options

Now, we will test each option to see which set of numbers follows the pattern derived: $B = 5A + 5$ and $C = 4B + 5$.

Option 1: (26, 52, 275)

• A = 26, B = 52, C = 275
• Check the first part of the pattern: $B = 5A + 5$? $5 \times 26 + 5 = 130 + 5 = 135$. The given B is 52.
• Since $135 \neq 52$, this option does not follow the pattern.

Option 2: (12, 22, 87)

• A = 12, B = 22, C = 87
• Check the first part of the pattern: $B = 5A + 5$? $5 \times 12 + 5 = 60 + 5 = 65$. The given B is 22.
• Since $65 \neq 22$, this option does not follow the pattern.

Option 3: (34, 52, 166)

• A = 34, B = 52, C = 166
• Check the first part of the pattern: $B = 5A + 5$? $5 \times 34 + 5 = 170 + 5 = 175$. The given B is 52.
• Since $175 \neq 52$, this option does not follow the pattern.

Option 4: (14, 50, 180)

• A = 14, B = 50, C = 180
• Check the first part of the pattern: $B = 5A + 5$? $5 \times 14 + 5 = 70 + 5 = 75$. The given B is 50.
• Hold on, the pattern $B = 5A+5$ seems incorrect based on this option. Let's re-examine the options and the original set to find the true relationship.

Let's consider the possibility of a pattern like $B = k_1 A + c_1$ and $C = k_2 B + c_2$. We can use the original set (4, 25, 105) and the numbers from Option 4 (14, 50, 180) to solve for $k_1, c_1, k_2, c_2$.

Using (4, 25): $4k_1 + c_1 = 25$

Using (14, 50): $14k_1 + c_1 = 50$

Subtracting the first equation from the second:

$(14k_1 + c_1) - (4k_1 + c_1) = 50 - 25$

$10k_1 = 25$

$k_1 = \frac{25}{10} = 2.5$

Substitute $k_1 = 2.5$ into the first equation: $4(2.5) + c_1 = 25 \Rightarrow 10 + c_1 = 25 \Rightarrow c_1 = 15$.

So, the first part of the pattern is: $B = 2.5A + 15$. Let's verify:

• For (4, 25): $2.5 \times 4 + 15 = 10 + 15 = 25$. Correct.
• For (14, 50): $2.5 \times 14 + 15 = 35 + 15 = 50$. Correct.

Now let's find the relationship between B and C using the original set (25, 105) and the correct option (50, 180):

Using (25, 105): $25k_2 + c_2 = 105$

Using (50, 180): $50k_2 + c_2 = 180$

Subtracting the first equation from the second:

$(50k_2 + c_2) - (25k_2 + c_2) = 180 - 105$

$25k_2 = 75$

$k_2 = \frac{75}{25} = 3$

Substitute $k_2 = 3$ into the first equation: $25(3) + c_2 = 105 \Rightarrow 75 + c_2 = 105 \Rightarrow c_2 = 30$.

So, the second part of the pattern is: $C = 3B + 30$. Let's verify:

• For (25, 105): $3 \times 25 + 30 = 75 + 30 = 105$. Correct.
• For (50, 180): $3 \times 50 + 30 = 150 + 30 = 180$. Correct.

The correct pattern is:

1. Second number = $2.5 \times$ First number $+ 15$ ($B = 2.5A + 15$)
2. Third number = $3 \times$ Second number $+ 30$ ($C = 3B + 30$)

Let's re-check the options with this new pattern.

Option 1: (26, 52, 275)

• A = 26
• Expected B: $2.5 \times 26 + 15 = 65 + 15 = 80$. Given B is 52. $80 \neq 52$. Incorrect.

Option 2: (12, 22, 87)

• A = 12
• Expected B: $2.5 \times 12 + 15 = 30 + 15 = 45$. Given B is 22. $45 \neq 22$. Incorrect.

Option 3: (34, 52, 166)

• A = 34
• Expected B: $2.5 \times 34 + 15 = 85 + 15 = 100$. Given B is 52. $100 \neq 52$. Incorrect.

Option 4: (14, 50, 180)

• A = 14
• Check B: $2.5 \times 14 + 15 = 35 + 15 = 50$. Given B is 50. Correct.
• Check C: $3 \times B + 30 = 3 \times 50 + 30 = 150 + 30 = 180$. Given C is 180. Correct.

Option 4 (14, 50, 180) follows the same number relation pattern as the set (4, 25, 105).

Conclusion

The numbers in the set (4, 25, 105) are related by the rules: Second number = $2.5 \times$ First number $+ 15$ and Third number = $3 \times$ Second number $+ 30$. Applying these rules to the given options, only the set (14, 50, 180) satisfies both conditions.

Number Relation Problem Revision

Revisiting number relation problems requires careful observation and testing of different patterns. Common patterns include arithmetic progression, geometric progression, or a combination of multiplication/division and addition/subtraction, often with constant or changing values.

Additional Information on Number Patterns

Number analogy and number series questions are common in logical reasoning tests. They assess your ability to identify underlying mathematical relationships. Strategies include:

• Checking differences between consecutive numbers.
• Checking ratios between consecutive numbers.
• Looking for patterns involving squares, cubes, or other powers.
• Testing combinations of operations (e.g., multiply by a constant and add/subtract another constant).
• Considering relationships between non-consecutive numbers.
• If multiple options seem to fit a simple pattern, look for a more complex one that uniquely fits the original set and only one option.

Sometimes, the pattern might involve operations on the digits of the numbers, but in this case, the pattern was a direct mathematical relationship between the numbers themselves.