Select the option that is related to the fifth number in the same way as the second number is related to the first number and the fourth number is related to the third number. 864 ∶ 12 ∶ 1372 ∶ 14 ∶ 2048 ∶ ?
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Understanding Numerical Analogies and Patterns
This question asks us to find the missing number in a numerical analogy. A numerical analogy establishes a relationship between two numbers, and we need to apply the same relationship to find a corresponding number for another given number. The given analogy is 864 ∶ 12 ∶ 1372 ∶ 14 ∶ 2048 ∶ ?
We are given three pairs (the third pair is incomplete). We need to identify the pattern or rule connecting the numbers in the first two complete pairs and then use that rule to find the missing number in the third pair.
Analyzing the First Numerical Pair: 864 and 12
Let's examine the relationship between 864 and 12. We can try simple arithmetic operations. 864 is much larger than 12.
- Is 864 a multiple of 12? $864 \div 12 = 72$. The multiple is 72. This might be part of the pattern, but we need to see if 72 is related to 12 in some way.
- Let's consider powers of 12 or numbers related to 12. $12^2 = 144$, $12^3 = 1728$. Neither 144 nor 1728 directly relates to 864 or 72 in an obvious way.
- What if we look at half of 12, which is 6? $6^2 = 36$, $6^3 = 216$.
- Let's see if 864 is related to $6^3 = 216$. $864 \div 216 = 4$.
So, we found a potential relationship: $864 = 4 \times (12 \div 2)^3$. Let's verify this rule with the second pair.
Analyzing the Second Numerical Pair: 1372 and 14
Now we apply the hypothesized rule to the pair 1372 and 14. According to our rule, $1372$ should be equal to $4 \times (14 \div 2)^3$.
- Half of 14 is 7.
- Cube of 7 is $7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343$.
- Now multiply 343 by 4: $343 \times 4 = 1372$.
The rule holds true for the second pair: $1372 = 4 \times (14 \div 2)^3$.
Applying the Pattern to Find the Missing Number
We have identified the pattern: The first number in each pair is equal to 4 times the cube of half of the second number.
Let the missing number in the third pair be $x$. The pair is 2048 and $x$. According to the rule:
$$2048 = 4 \times \left(\frac{x}{2}\right)^3$$
Now, we need to solve this equation for $x$.
Divide both sides by 4:
$$\frac{2048}{4} = \left(\frac{x}{2}\right)^3$$
$$512 = \left(\frac{x}{2}\right)^3$$
We need to find the number whose cube is 512. We know that $8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512$.
So, we have:
$$8^3 = \left(\frac{x}{2}\right)^3$$
Taking the cube root of both sides:
$$8 = \frac{x}{2}$$
Multiply both sides by 2:
$$x = 8 \times 2$$
$$x = 16$$
The missing number is 16.
Conclusion
Based on the pattern observed in the first two pairs, where the first number is 4 times the cube of half the second number, the missing number related to 2048 is 16.
The analogy is: 864 ∶ 12 :: 1372 ∶ 14 :: 2048 ∶ 16
Revision Table: Numerical Analogy Pattern
| Pair | First Number | Second Number | Relationship Check: $4 \times (\text{Second Number} / 2)^3$ |
|---|---|---|---|
| 1 | 864 | 12 | $4 \times (12/2)^3 = 4 \times 6^3 = 4 \times 216 = 864$ (Holds) |
| 2 | 1372 | 14 | $4 \times (14/2)^3 = 4 \times 7^3 = 4 \times 343 = 1372$ (Holds) |
| 3 | 2048 | ? (Let it be $x$) | $4 \times (x/2)^3 = 2048 \implies (x/2)^3 = 512 \implies x/2 = 8 \implies x = 16$ |
Additional Information on Number Patterns and Reasoning
Numerical analogy and number pattern questions are common in reasoning and quantitative aptitude tests. They require you to identify the underlying rule that connects numbers in a sequence or pair. These rules can involve various mathematical operations:
- Basic arithmetic: addition, subtraction, multiplication, division.
- Powers and roots: squares, cubes, square roots, cube roots.
- Combinations of operations.
- Digit-based operations.
- Prime numbers, composite numbers, specific number series (like Fibonacci).
Solving these problems often involves trial and error, testing different common patterns until one fits all the given examples. Breaking down the numbers, looking for factors, and considering simple relationships first can be helpful strategies.
Practice with various types of number pattern questions improves your ability to quickly identify potential rules and apply them correctly.
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Important Questions from Analogy
Select the option that is related to the third term in the same way as the second term is related to the first term.
Day: Night :: Morning : ?Select the option that is related to the third term in the same way as the second term is related to the first term.
India : Country :: Delhi : ..........Select the option that is related to the third term in the same way as the second term is related to the first term.
MATE: NATE :: FATE: ?Select the option that is related to the third term in the same way as the second term is related to the first term.
13 : 196 :: 9 : .......
Select the related word pair from the given alternatives. Afghanistan : Kabul :: ....... : .........
