Analysing Number Set Patterns

This question asks us to find a set of numbers among the options that follows the same relationship or pattern as the given sets of numbers. We need to identify the rule connecting the numbers in the original sets and then apply that rule to the options.

Understanding the Given Number Sets and Finding the Pattern

We are given two sets:

• (9, 3, 90)
• (5, 4, 41)

Let's look at the first set (9, 3, 90). We need to find a relationship between 9, 3, and 90 using whole number operations.

Let's consider common mathematical operations between the first two numbers (9 and 3) and see how they might relate to the third number (90).

• Sum: $9 + 3 = 12$ (Not 90)
• Difference: $9 - 3 = 6$ (Not 90)
• Product: $9 \times 3 = 27$ (Not 90)
• Squares: Let's try squaring the numbers.

Square of the first number: $9^2 = 9 \times 9 = 81$

Square of the second number: $3^2 = 3 \times 3 = 9$

Let's see if the sum of their squares equals the third number:

\(81 + 9 = 90\)

Yes, this matches the third number in the first set!

Now, let's check if this pattern holds for the second set (5, 4, 41).

Square of the first number: $5^2 = 5 \times 5 = 25$

Square of the second number: $4^2 = 4 \times 4 = 16$

Sum of their squares:

\(25 + 16 = 41\)

This also matches the third number in the second set.

So, the pattern is confirmed: If a set is represented as (a, b, c), then the relationship is $a^2 + b^2 = c$. The third number is the sum of the squares of the first two numbers.

Applying the Pattern to the Options

We will now test each option to see which one follows the pattern $a^2 + b^2 = c$.

Option 1: (13, 7, 218)

Here, $a = 13$ and $b = 7$. Let's calculate $a^2 + b^2$:

\(13^2 + 7^2 = (13 \times 13) + (7 \times 7) = 169 + 49 = 218\)

The calculated value is 218, which matches the third number in the set (218). This option follows the pattern.

Option 2: (7, 8, 117)

Here, $a = 7$ and $b = 8$. Let's calculate $a^2 + b^2$:

\(7^2 + 8^2 = (7 \times 7) + (8 \times 8) = 49 + 64 = 113\)

The calculated value is 113, which does not match the third number in the set (117). This option does not follow the pattern.

Option 3: (11, 8, 182)

Here, $a = 11$ and $b = 8$. Let's calculate $a^2 + b^2$:

\(11^2 + 8^2 = (11 \times 11) + (8 \times 8) = 121 + 64 = 185\)

The calculated value is 185, which does not match the third number in the set (182). This option does not follow the pattern.

Option 4: (4, 12, 162)

Here, $a = 4$ and $b = 12$. Let's calculate $a^2 + b^2$:

\(4^2 + 12^2 = (4 \times 4) + (12 \times 12) = 16 + 144 = 160\)

The calculated value is 160, which does not match the third number in the set (162). This option does not follow the pattern.

Identifying the Matching Number Set

Based on our analysis, only Option 1, the set (13, 7, 218), follows the same pattern ($a^2 + b^2 = c$) as the given sets (9, 3, 90) and (5, 4, 41).

Therefore, the set in which the numbers are related in the same way is (13, 7, 218).

Revision Table: Number Set Patterns

Understanding common patterns helps in solving number analogy questions quickly.

Additional Information: Common Number Patterns

Number analogy questions often test your ability to spot mathematical relationships. Here are some common patterns to look out for:

• Basic Operations: Addition, subtraction, multiplication, division.
• Squares and Cubes: Numbers might be squares or cubes of other numbers in the set, or the relationship might involve sums or differences of squares/cubes.
• Multiples and Factors: One number might be a multiple or factor of another.
• Arithmetic Progression (AP): The difference between consecutive numbers is constant.
• Geometric Progression (GP): The ratio between consecutive numbers is constant.
• Combinations of Operations: The pattern might involve more than one operation, such as $a \times b + c = d$ or $(a+b) \times c = d$.

Practicing with different types of number sets will help you identify patterns faster.