Analyzing the Number Pattern Question

The question asks us to identify a pattern connecting the three numbers within each set provided in the format (First Number, Middle Number, Third Number). We are given three sets:

• (5, 14, 3)
• (3, 24, 7)
• (9, 30, ?)

We need to find the value of the question mark (?) in the third set by discovering the rule that relates the first, middle, and third numbers in the first two sets. The rule must use the whole numbers as they are, without breaking them down into individual digits.

Discovering the Pattern Rule

Let's examine the relationship between the numbers in the first two sets to find a consistent pattern.

Examining the First Set: (5, 14, 3)

Here, the first number is 5, the middle number is 14, and the third number is 3.

Let's try different combinations of operations involving the first and third numbers to get the middle number 14.

• Sum: \(5 + 3 = 8\) (Not 14)
• Difference: \(5 - 3 = 2\) (Not 14)
• Product: \(5 \times 3 = 15\) (Close to 14)

If we use the product, \(5 \times 3 = 15\), we can get 14 by subtracting 1 (\(15 - 1 = 14\)). Let's see if this rule works for the second set.

Rule candidate 1: Middle Number = (First Number \(\times\) Third Number) - 1

Examining the Second Set: (3, 24, 7)

Here, the first number is 3, the middle number is 24, and the third number is 7.

Let's test Rule candidate 1:

• (First Number \(\times\) Third Number) - 1 = \((3 \times 7) - 1 = 21 - 1 = 20\) (This is not 24).

So, Rule candidate 1 is incorrect.

Let's try another approach. How else can we combine 3 and 7 to get 24? What if we multiply one of the numbers by a constant and add the other?

• Try multiplying the first number: \((3 \times \text{constant}) + 7 = 24\). If constant is 3, \(3 \times 3 + 7 = 9 + 7 = 16\) (Not 24).
• Try multiplying the third number: \((7 \times \text{constant}) + 3 = 24\). \(7 \times \text{constant} = 24 - 3 = 21\). Constant = \(21 / 7 = 3\).

This gives us Rule candidate 2: Middle Number = (Third Number \(\times\) 3) + First Number.

Verifying Rule Candidate 2 on Both Sets

Let's test Rule candidate 2:

• Set 1: (5, 14, 3)
  Middle Number = (Third Number \(\times\) 3) + First Number
  \(14 = (3 \times 3) + 5\)
  \(14 = 9 + 5\)
  \(14 = 14\)
  This rule works for the first set.
• Set 2: (3, 24, 7)
  Middle Number = (Third Number \(\times\) 3) + First Number
  \(24 = (7 \times 3) + 3\)
  \(24 = 21 + 3\)
  \(24 = 24\)
  This rule works for the second set.

Since Rule candidate 2 works for both given sets, this is the correct pattern rule.

Finding the Missing Number

Now we apply the discovered pattern rule to the third set: (9, 30, ?)

The rule is: Middle Number = (Third Number \(\times\) 3) + First Number.

In the third set:

• First Number = 9
• Middle Number = 30
• Third Number = ?

Substitute these values into the rule:

\(30 = (? \times 3) + 9\)

Now, we solve for ?:

\(30 - 9 = ? \times 3\)

\(21 = ? \times 3\)

\(? = \frac{21}{3}\)

\(? = 7\)

The missing number that replaces the question mark is 7.

Conclusion

The pattern followed in the given sets is Middle Number = (Third Number \(\times\) 3) + First Number. Applying this rule to the third set (9, 30, ?) reveals that the missing number is 7.

The final answer is 7.

Pattern Analysis Revision Table

Additional Information on Number Patterns

Number pattern questions are common in logical reasoning and quantitative aptitude tests. They require you to find the relationship or rule that connects a series of numbers. This rule can involve various mathematical operations like addition, subtraction, multiplication, division, squaring, cubing, or combinations of these.

Key strategies for solving number pattern problems:

• Look for differences between consecutive numbers.
• Look for ratios between consecutive numbers.
• Check for squares, cubes, or their roots.
• Consider alternating patterns.
• Try combining different elements if the pattern involves multiple numbers in a set, as seen in this problem.
• Always test your discovered rule on all the provided examples to ensure consistency.

Practice with different types of number patterns helps improve problem-solving skills and pattern recognition abilities.