Finding the Number Relationship in Sets

The question asks us to find a set of numbers that shares the same relationship as the given sets: (15, 135, 6) and (12, 84, 5).

To solve this, we first need to carefully examine the numbers in the provided sets and identify the underlying pattern or rule connecting them.

Analyzing the Given Number Sets

Let's look at the first set: (15, 135, 6). The numbers are 15, 135, and 6. The middle number, 135, is significantly larger than the other two.

Let's look at the second set: (12, 84, 5). The numbers are 12, 84, and 5. Similarly, the middle number, 84, is larger than the other two.

This suggests that the middle number might be a result of some operation involving the first and third numbers.

Identifying the Pattern or Rule

Let's try to find a relationship between the first number, the third number, and the middle number.

Consider the ratio of the middle number to the first number:

• For (15, 135, 6): $\frac{135}{15} = 9$
• For (12, 84, 5): $\frac{84}{12} = 7$

The resulting numbers are 9 and 7. Now let's see how these relate to the third numbers in their respective sets, which are 6 and 5.

• For set 1: The ratio is 9, and the third number is 6. The difference is $9 - 6 = 3$. The sum is $9 + 6 = 15$. Notice that 15 is the first number.
• For set 2: The ratio is 7, and the third number is 5. The difference is $7 - 5 = 2$. The sum is $7 + 5 = 12$. Notice that 12 is the first number.

This observation gives us a potential relationship:

The ratio of the Middle Number to the First Number is equal to the First Number minus the Third Number.

Let's write this as an equation:

$\frac{\text{Middle Number}}{\text{First Number}} = \text{First Number} - \text{Third Number}$

Multiplying both sides by the First Number, we get:

Middle Number = First Number $\times$ (First Number - Third Number)

Verifying the Relationship with Given Sets

Let's check if this rule holds true for both initial sets:

Set 1: (15, 135, 6)

First Number = 15, Third Number = 6

Middle Number = $15 \times (15 - 6) = 15 \times 9 = 135$.

This matches the middle number in the set (15, 135, 6).

Set 2: (12, 84, 5)

First Number = 12, Third Number = 5

Middle Number = $12 \times (12 - 5) = 12 \times 7 = 84$.

This matches the middle number in the set (12, 84, 5).

The relationship holds for both given sets.

Checking the Options Based on the Relationship

Now we will apply the identified relationship (Middle Number = First Number $\times$ (First Number - Third Number)) to each of the given options to find the matching set.

Option 1: (21, 315, 6)

• First Number = 21
• Third Number = 6

Calculated Middle Number = $21 \times (21 - 6) = 21 \times 15 = 315$.

The calculated middle number (315) matches the middle number in the option (21, 315, 6).

This option follows the same relationship.

Option 2: (9, 89, 3)

• First Number = 9
• Third Number = 3

Calculated Middle Number = $9 \times (9 - 3) = 9 \times 6 = 54$.

The calculated middle number (54) does not match the middle number in the option (9, 89, 3).

This option does not follow the same relationship.

Option 3: (24, 145, 18)

• First Number = 24
• Third Number = 18

Calculated Middle Number = $24 \times (24 - 18) = 24 \times 6 = 144$.

The calculated middle number (144) does not match the middle number in the option (24, 145, 18).

This option does not follow the same relationship.

Option 4: (17, 165, 7)

• First Number = 17
• Third Number = 7

Calculated Middle Number = $17 \times (17 - 7) = 17 \times 10 = 170$.

The calculated middle number (170) does not match the middle number in the option (17, 165, 7).

This option does not follow the same relationship.

Conclusion

Based on the analysis, only Option 1 (21, 315, 6) follows the same mathematical relationship as the given sets (15, 135, 6) and (12, 84, 5), which is: Middle Number = First Number $\times$ (First Number - Third Number).

Revision Table: Number Analogy Practice

Additional Information on Number Analogy

Number analogy questions test your ability to identify mathematical relationships between numbers within a set and apply that relationship to other sets. These questions are common in logical reasoning and quantitative aptitude sections of various exams.

Common types of relationships in number analogy include:

• Arithmetic operations (addition, subtraction, multiplication, division).
• Operations involving squares, cubes, square roots, cube roots.
• Relationships based on prime numbers, composite numbers, odd/even numbers.
• Relationships involving digits of the numbers.
• Combinations of multiple operations.

Tips for solving number analogy problems:

• Look for simple relationships first (addition, subtraction, multiplication, division).
• Check for ratios or differences between numbers.
• Consider operations involving pairs of numbers to get the third number.
• Test your identified pattern on all given examples before applying it to the options.
• Don't assume the first pattern you see is the only one; sometimes complex patterns exist.

Practicing different types of number analogy problems helps in recognizing patterns more quickly during exams.