Solving the Number Analogy: 2 : 10 : : 8 : ?

This question asks us to find a number that shares the same relationship with 8 as 10 shares with 2. These types of questions test our ability to identify patterns and logical rules applied to numbers.

Identifying the Pattern in the First Pair (2 : 10)

Let's look closely at the numbers 2 and 10 and try to find a mathematical operation or rule connecting them. Some common patterns in number analogies involve squaring, cubing, adding, subtracting, or multiplying the number itself or numbers related to it.

• Could it be simple multiplication? $2 \times 5 = 10$. If we apply this to 8, $8 \times 5 = 40$. 40 is not among the options.
• Could it involve squaring the number? $2^2 = 4$. To get 10 from 4, we need to add 6. $2^2 + 6 = 10$. If we apply this rule (number squared plus 6) to 8, we get $8^2 + 6 = 64 + 6 = 70$. 70 is not an option.
• Could it involve cubing the number? $2^3 = 8$. To get 10 from 8, we need to add 2. $2^3 + 2 = 10$. If we apply this rule (number cubed plus 2) to 8, we get $8^3 + 2 = 512 + 2 = 514$. 514 is not an option.

Let's consider patterns involving the number itself and a number related to it, like the next consecutive integer.

• Consider the number 2. The next consecutive integer is 3. What if we square the next integer and add or subtract something? $3^2 = 9$. To get 10 from 9, we add 1. So, the pattern could be $(n+1)^2 + 1$, where $n$ is the first number in the pair.

Verifying the Identified Pattern

Let's test the pattern $n : (n+1)^2 + 1$ with the first pair (2 : 10), where $n=2$.

Applying the rule: $(2+1)^2 + 1 = 3^2 + 1 = 9 + 1 = 10$.

This matches the given second number, 10. So, the rule seems correct.

Applying the Pattern to the Second Pair (8 : ?)

Now we apply the same rule $n : (n+1)^2 + 1$ to the number 8. Here, $n=8$.

Applying the rule: $(8+1)^2 + 1 = 9^2 + 1 = 81 + 1 = 82$.

Comparing with the Options

The number we found is 82. Let's check the given options:

1. 82
2. 648
3. 132
4. 76

Our calculated number, 82, is present as option 1.

Conclusion

The relationship between the numbers in the analogy 2 : 10 : : 8 : ? follows the pattern $n : (n+1)^2 + 1$. When this pattern is applied to 8, the related number is 82.

Revision Table: Key Concepts in Number Analogies

Number analogies often involve identifying a rule based on mathematical operations between the numbers in the first pair and applying that rule to the second pair.

• Common operations include squaring, cubing, addition, subtraction, multiplication, division.
• Patterns might relate a number $n$ to $n^2$, $n^3$, $n+k$, $n-k$, $nk$, $n/k$, $n^2 \pm k$, $n^3 \pm k$, or operations involving $n+1$, $n-1$, etc.
• Always verify the rule found on the first pair before applying it to the second pair.

Additional Information: Strategies for Solving Number Analogies

Here are some strategies that can help you solve number analogy questions:

• Look for simple relationships: Check for addition, subtraction, multiplication, or division first.
• Consider squares and cubes: See if the numbers are squares or cubes, or close to squares or cubes.
• Check differences and ratios: Sometimes the relationship is in the difference or ratio between the numbers.
• Look for consecutive numbers: The pattern might involve the number and the next or previous integer.
• Prime numbers: Sometimes the pattern relates to prime numbers.
• Combined operations: The rule might involve a combination of operations, like squaring and adding.
• Test options: If you are stuck, sometimes looking at the options can give you clues about the type of relationship (e.g., large numbers might suggest squares or cubes).

Practicing different types of number analogies helps you become familiar with common patterns and improve your logical reasoning skills.