Select the set in which the numbers are related in the same way as are the numbers of the following set. (NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits.) (60, 5, 6) (48, 8, 3)
(24, 4, 3)
Understanding Number Analogies and Set Relations
This question asks us to identify a numerical set that shares the same relationship between its numbers as the two given sets. The given sets are (60, 5, 6) and (48, 8, 3). We are instructed to perform operations on the whole numbers without breaking them down into constituent digits.
Analyzing the Relationship in the Given Sets
Let's examine the first set (60, 5, 6). We need to find how 60 relates to 5 and 6.
Let's examine the second set (48, 8, 3). We need to find how 48 relates to 8 and 3.
We look for a consistent mathematical operation or combination of operations that connects the three numbers in both sets.
Step-by-Step Pattern Discovery
Consider the first set (60, 5, 6):
- Could the first number be related to the product of the other two? $5 \times 6 = 30$. How does 30 relate to 60? $30 \times 2 = 60$.
Let's test this potential pattern on the second set (48, 8, 3):
- Consider the product of the second and third numbers: $8 \times 3 = 24$.
- Apply the 'multiply by 2' step: $24 \times 2 = 48$.
This matches the first number in the second set (48). The pattern seems to be consistent across both given sets.
Derived Relationship Pattern
The relationship found is: The first number is equal to the product of the second and third numbers, multiplied by 2.
In mathematical terms, if the set is (A, B, C), the pattern is $A = (B \times C) \times 2$.
Checking Options Based on the Pattern
Now, we will apply this relationship pattern to each of the given options to find the set that follows the same rule.
| Option Set | Second Number (B) | Third Number (C) | Calculation: $(B \times C) \times 2$ | First Number (A) | Does it Match? |
|---|---|---|---|---|---|
| (17, 3, 4) | 3 | 4 | $(3 \times 4) \times 2 = 12 \times 2 = 24$ | 17 | No ($24 \neq 17$) |
| (25, 4, 6) | 4 | 6 | $(4 \times 6) \times 2 = 24 \times 2 = 48$ | 25 | No ($48 \neq 25$) |
| (24, 4, 3) | 4 | 3 | $(4 \times 3) \times 2 = 12 \times 2 = 24$ | 24 | Yes ($24 = 24$) |
| (25, 4, 3) | 4 | 3 | $(4 \times 3) \times 2 = 12 \times 2 = 24$ | 25 | No ($24 \neq 25$) |
Based on the analysis, the set (24, 4, 3) is the only option that satisfies the relationship pattern found in the given sets: The first number (24) is equal to the product of the second (4) and third (3) numbers, multiplied by 2 ($ (4 \times 3) \times 2 = 12 \times 2 = 24 $).
Conclusion
The set (24, 4, 3) is related in the same way as the numbers of the given sets (60, 5, 6) and (48, 8, 3).
Revision Table: Key Concepts in Number Sets
| Concept | Description | Application in this Problem |
|---|---|---|
| Number Analogy | Finding a relationship or pattern between numbers in a set or across different sets. | Identifying the mathematical rule connecting the three numbers in the given sets. |
| Set Relation | The specific rule or formula that describes how the elements within a set are connected. | The pattern $A = (B \times C) \times 2$ found for sets (A, B, C). |
| Pattern Recognition | The ability to identify recurring sequences or relationships. | Crucial for solving number analogy and set relation problems. |
Additional Information: Tips for Solving Number Analogy Problems
Solving number analogy problems involves logical reasoning and mathematical skills. Here are some tips:
- Analyze the Given Sets Carefully: Look for relationships involving addition, subtraction, multiplication, division, squares, cubes, etc.
- Test Simple Operations First: Start with basic arithmetic operations between the numbers.
- Look for Relationships Between Positions: How does the first number relate to the second and third? How does the second relate to the first and third?
- Consider Two-Step Operations: Sometimes the pattern involves a combination of operations (e.g., multiply and then add, divide and then square).
- Check for Consistency: The identified pattern must work for all the example sets provided in the question.
- Systematically Check Options: Once a pattern is found, apply it rigorously to each option to see which one fits.
Practicing different types of number analogy questions helps improve pattern recognition skills.
Important Questions from Miscellaneous
A stone is thrown horizontally from the top of a 20 m high building with a speed of 12 m/s. It hits the ground at a distance R from the building. Taking g = 10 m/s2 and neglecting air resistance will give :
A sphere of volume V is made of a material with lower density than water. While on Earth, it floats on water with its volume f1V (f1 < 1) submerged. On the other hand, on a spaceship accelerating with acceleration a < g (g is the acceleration due to gravity on Earth) in outer space, its submerged volume in water is f2V. Then:
A railway wagon (open at the top) of mass M1 is moving with speed v1 along a straight track. As a result of rain, after some time it gets partially filled with water so that the mass of the wagon becomes M2 and speed becomes v2. Taking the rain to be falling vertically and the water stationery inside the wagon, the relation between the two speeds v1 and v2 is :
Consider the following statements:
1. Distance between the longitudes becomes zero on North Pole and South Pole.
2. Distance between the longitudes is maximum on the Equator.
3. Number of longitudes is more than number of latitudes.
Which of the statements given above is/are correct?
One block of 2⋅0 kg mass is placed on top of another block of 3⋅0 kg mass. The coefficient of static friction between the two blocks is 0⋅2. The bottom block is pulled with a horizontal force F such that both the blocks move together without slipping. Taking acceleration due to gravity as 10 m/s2, the maximum value of the frictional force is :
