Select the option that is related to the third number in the same way as the second number is related to the first number. 5 : 150 :: 8 : ?
576
This question is a number analogy problem where we need to find the relationship between the first pair of numbers and apply the same relationship to the third number to find the fourth number.
Understanding Number Analogies
Number analogies are a common type of question in reasoning tests. They require you to identify a specific mathematical or logical rule that connects two numbers and then use that rule to connect a third number to a fourth, unknown number. The key is to find the exact pattern.
Analyzing the First Pair: 5 : 150
We are given the first pair as 5 and 150. We need to figure out how 5 is related to 150. Let's explore a few possible relationships:
- Is it multiplication? \(5 \times \text{something} = 150\). \(150 / 5 = 30\). So, \(5 \times 30 = 150\).
- Is it related to squares or cubes? \(5^2 = 25\), \(5^3 = 125\).
Discovering the Pattern
Let's look closer at the numbers 5 and 150. We know \(5^3 = 125\). The difference between 150 and 125 is \(150 - 125 = 25\). Notice that 25 is the square of 5 (\(5^2 = 25\)).
This suggests a potential pattern: the second number might be the cube of the first number plus the square of the first number. Let's express this relationship as \(n : n^3 + n^2\), where 'n' is the first number in the pair.
Let's verify this pattern for the first pair (n=5):
\(5^3 + 5^2 = 125 + 25 = 150\)
The pattern \(n^3 + n^2\) holds true for the first pair (5 : 150).
Applying the Pattern to Find the Missing Number
Now, we apply the same relationship to the third number, which is 8. Here, n = 8.
According to the pattern \(n : n^3 + n^2\), the related number for 8 will be \(8^3 + 8^2\).
Calculating the values:
- Calculate the cube of 8: \(8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512\).
- Calculate the square of 8: \(8^2 = 8 \times 8 = 64\).
Now, add these two values together:
\(512 + 64 = 576\)
So, the number related to 8, based on the pattern \(n^3 + n^2\), is 576.
Comparing with Options
Let's look at the given options to see if our calculated number, 576, is present:
- Option 1: 864
- Option 2: 576
- Option 3: 262
- Option 4: 186
Our calculated value, 576, matches Option 2.
Conclusion on the Number Analogy
The number analogy 5 : 150 :: 8 : ? is solved by identifying the pattern \(n : n^3 + n^2\). Applying this pattern to the number 8, we find the missing number is 576.
Number Analogy Revision Table
| Concept | Explanation | Example |
|---|---|---|
| Number Analogy | Identifying the relationship between numbers in a pair and applying it to another pair. | Find the pattern in 2:8 and apply to 3:? |
| Pattern Types | Can involve arithmetic (+, -, ×, ÷), squares (\(n^2\)), cubes (\(n^3\)), or combinations. | \(n \times k\), \(n^2\), \(n^3+n\), etc. |
| Solving Steps | 1. Analyze the first pair. 2. Identify the rule/pattern. 3. Apply the pattern to the third number. 4. Calculate the result. 5. Check options. | Steps used for 5:150 :: 8:? |
Additional Information on Analogies
Analogies are common in many competitive exams as they test logical thinking and pattern recognition skills. While number analogies focus on mathematical relationships, other types of analogies involve word meanings, synonyms, antonyms, part-whole relationships, etc.
For number analogies, it's helpful to be familiar with common mathematical operations, squares up to at least 20, and cubes up to at least 10. Sometimes, prime numbers or sequences like Fibonacci series might also be the underlying pattern.
Practicing different types of number analogy problems helps improve speed and accuracy in identifying the pattern quickly. If one pattern doesn't fit the options, try another one.
Similar Questions
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Select the option that is related to the third number in the same way as the second number is related to the first number.
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