Understanding Number Set Relationships

The problem asks us to find a relationship between the numbers in the given sets and then identify which of the option sets follows the same relationship. We are told to perform operations on the whole numbers themselves, not their individual digits.

Analyzing the Given Number Sets

Let's look at the two example sets provided:

• Set 1: (122, 136, 24)
• Set 2: (222, 248, 36)

In each set, there are three numbers. We need to figure out how the third number is related to the first two numbers in a consistent way across both sets.

Let's consider common arithmetic operations. The third number is significantly smaller than the first two. This often suggests that the third number might be related to the difference between the first two, or some operation involving subtraction or division of the first two numbers.

Let's find the difference between the second and first numbers in each set:

• Set 1: \(136 - 122 = 14\)
• Set 2: \(248 - 222 = 26\)

Now, let's compare these differences (14 and 26) with the third numbers in their respective sets (24 and 36).

• For Set 1, the difference is 14 and the third number is 24.
• For Set 2, the difference is 26 and the third number is 36.

We can observe a pattern: \(14 + 10 = 24\) and \(26 + 10 = 36\). It appears the relationship is that the third number is equal to the difference between the second and first number, plus 10.

Identifying the Relationship Rule

Based on the analysis of the given sets, the relationship between the three numbers in a set \((a, b, c)\) appears to be:

\(c = (b - a) + 10\)

Let's verify this rule with the given sets:

• For (122, 136, 24): \((136 - 122) + 10 = 14 + 10 = 24\). This matches the third number.
• For (222, 248, 36): \((248 - 222) + 10 = 26 + 10 = 36\). This matches the third number.

The relationship holds true for both given sets.

Testing the Options Using the Relationship Rule

Now, we will apply the rule \(c = (b - a) + 10\) to each of the option sets to find the one that follows the same relationship.

Option 1: (301, 367, 78)

• First number \(a = 301\)
• Second number \(b = 367\)
• Third number \(c = 78\)

Applying the rule: \((367 - 301) + 10 = 66 + 10 = 76\).

The calculated value (76) is not equal to the third number in the set (78). So, Option 1 does not follow the relationship.

Option 2: (189, 213, 32)

• First number \(a = 189\)
• Second number \(b = 213\)
• Third number \(c = 32\)

Applying the rule: \((213 - 189) + 10 = 24 + 10 = 34\).

The calculated value (34) is not equal to the third number in the set (32). So, Option 2 does not follow the relationship.

Option 3: (145, 244, 89)

• First number \(a = 145\)
• Second number \(b = 244\)
• Third number \(c = 89\)

Applying the rule: \((244 - 145) + 10 = 99 + 10 = 109\).

The calculated value (109) is not equal to the third number in the set (89). So, Option 3 does not follow the relationship.

Option 4: (212, 258, 56)

• First number \(a = 212\)
• Second number \(b = 258\)
• Third number \(c = 56\)

Applying the rule: \((258 - 212) + 10 = 46 + 10 = 56\).

The calculated value (56) is equal to the third number in the set (56). So, Option 4 follows the relationship.

Conclusion

Only the set (212, 258, 56) from the options follows the same relationship where the third number is equal to the difference between the second and first numbers plus 10.

Thus, the set (212, 258, 56) shares the same number relationship as the given sets.

Number Set Relationship Revision Table

Additional Information on Number Relationships

Problems involving number sets or number analogies are common in reasoning and quantitative aptitude tests. They require you to quickly identify patterns and relationships using basic arithmetic operations. Here are some common types of relationships you might encounter:

• Arithmetic Operations: Sum, difference, product, division between numbers or combinations thereof.
• Squares and Cubes: Numbers might be squares, cubes, or related to squares/cubes (e.g., \(n^2+1\), \(n^3-1\)).
• Consecutive Numbers: Relationships based on consecutive integers.
• Digit Operations: Although excluded in this specific problem's note, sometimes relationships involve operations on the individual digits of the numbers.
• Prime Numbers: The numbers might be prime or follow a pattern based on prime numbers.

When approaching such problems, it's helpful to perform simple operations first (like finding sums, differences, products, or quotients) and see if they relate to the other numbers in the set. If simple operations don't work, consider squares, cubes, or combinations of operations. Always test your potential rule on all the given examples before applying it to the options.