Select the option that is related to the third number in the same way as the second number is related to the first number and the sixth number is related to the fifth number.
6 : 21 :: 4 : ? :: 8 : 36

Question

Select the option that is related to the third number in the same way as the second number is related to the first number and the sixth number is related to the fifth number.6 : 21 :: 4 : ? :: 8 : 36

Prepp Team · Accepted Answer

Solving Number Analogy Problems

This question asks us to find a relationship between numbers that holds true across different pairs. We are given the analogy 6 : 21 :: 4 : ? :: 8 : 36. We need to identify the pattern connecting the first number to the second number in the pairs (6 and 21, 8 and 36) and apply it to the third number (4) to find the missing fourth number.

Identifying the Pattern in the Number Analogy

Let's look closely at the given pairs:

First pair: 6 and 21
Second pair: 8 and 36

We need to find a mathematical operation or relationship that connects 6 to 21 and the same relationship that connects 8 to 36.

Let's try to find a pattern based on simple operations:

Addition/Subtraction: $21 - 6 = 15$. $36 - 8 = 28$. The difference is not constant.
Multiplication: $6 \times 3.5 = 21$. $8 \times 4.5 = 36$. The multiplier changes.

Let's consider operations involving the number itself, maybe its half or square.

Consider the first number, 6. Half of 6 is $6/2 = 3$. The next integer after 6 is $6+1 = 7$. Let's multiply these two results: $3 \times 7 = 21$. This matches the second number in the first pair.

Let's check if this pattern holds for the second pair, 8 and 36.

Consider the first number, 8. Half of 8 is $8/2 = 4$. The next integer after 8 is $8+1 = 9$. Let's multiply these two results: $4 \times 9 = 36$. This matches the second number in the second pair.

The pattern appears to be: (First Number / 2) × (First Number + 1) = Second Number.

Applying the Pattern to Find the Missing Number

Now we apply the identified pattern to the third number, 4, to find the missing fourth number.

The third number is 4.

Half of 4 is $4/2 = 2$.
The next integer after 4 is $4+1 = 5$.

According to the pattern, the missing number should be the product of these two results:

$2 \times 5 = 10$.

Therefore, the missing number in the analogy 6 : 21 :: 4 : ? :: 8 : 36 is 10.

Summary of the Pattern and Solution

Pair	First Number	Second Number	Pattern Check
1st	6	21	$(6/2) \times (6+1) = 3 \times 7 = 21$ (Correct)
2nd	8	36	$(8/2) \times (8+1) = 4 \times 9 = 36$ (Correct)
3rd (Missing)	4	?	$(4/2) \times (4+1) = 2 \times 5 = 10$

The missing number is 10.

Revision Table: Key Concepts in Number Analogy

Concept	Description	Example (from this problem)
Number Analogy	Finding a relationship between two numbers and applying the same relationship to another number or pair.	Finding the relationship between 6 and 21, and between 8 and 36, to apply it to 4.
Pattern Recognition	Identifying the rule, sequence, or mathematical operation connecting the numbers.	Discovering the pattern: $(n/2) \times (n+1)$.
Applying the Pattern	Using the identified rule to find the missing element.	Using $(4/2) \times (4+1)$ to find the missing number.

Additional Information: Common Number Analogy Patterns

Number analogy problems can involve various patterns. Some common types include:

Arithmetic Operations: Addition, subtraction, multiplication, division.
Square and Cube Relations: Numbers might be squares or cubes, or related to squares/cubes (e.g., $n^2 \pm c$, $n^3 \pm c$).
Digit Operations: Sum of digits, product of digits.
Consecutive Numbers: Relationships involving the next or previous numbers.
Combinations of Operations: A mix of arithmetic, squares/cubes, etc.
Specific Series: Prime numbers, odd/even numbers, Fibonacci sequence.

To solve number analogy problems effectively, it's helpful to practice identifying different types of relationships and testing various possibilities systematically.