Select the option in which the numbers are related in the same way as are the numbers of the following sets. NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 - Operations on 13 such as adding/deleting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed) (4, 16, 64) (7, 49, 343)
(9, 81, 729)
Understanding Number Relations in Sets
The question asks us to find the relationship between the numbers in the given sets and then apply the same relationship to the options to find the matching set. We are given two sets of numbers:
- (4, 16, 64)
- (7, 49, 343)
We need to figure out how the second and third numbers are related to the first number in each set.
Analyzing the Given Number Sets
Let's look at the first set (4, 16, 64):
- The first number is 4.
- The second number is 16. We can see that $4 \times 4 = 16$. This can also be written as $4^2$.
- The third number is 64. We can see that $16 \times 4 = 64$. Since $16 = 4^2$, this means the third number is $4^2 \times 4 = 4^{(2+1)} = 4^3$.
So, the relationship in the first set appears to be: if the first number is 'a', the second number is $a^2$ and the third number is $a^3$. Let's verify this with the second set.
Now let's look at the second set (7, 49, 343):
- The first number is 7.
- The second number is 49. $7 \times 7 = 49$, which is $7^2$. This matches the pattern.
- The third number is 343. $49 \times 7 = 343$. Since $49 = 7^2$, this means the third number is $7^2 \times 7 = 7^{(2+1)} = 7^3$. This also matches the pattern.
The established relationship for the given sets is that if the first number is 'a', the set is $(a, a^2, a^3)$.
Testing the Options
Now, we will check each option to see which one follows the same $(a, a^2, a^3)$ relationship.
| Option | Numbers | First number (a) | Second number (a<sup>2</sup>) | Third number (a<sup>3</sup>) | Does it match? |
|---|---|---|---|---|---|
| 1 | (9, 81, 729) | 9 | $9^2 = 81$ | $9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729$ | Yes |
| 2 | (11, 121, 1231) | 11 | $11^2 = 121$ | $11^3 = 11 \times 11 \times 11 = 121 \times 11 = 1331$ | No (Third number is 1231, not 1331) |
| 3 | (5, 25, 115) | 5 | $5^2 = 25$ | $5^3 = 5 \times 5 \times 5 = 25 \times 5 = 125$ | No (Third number is 115, not 125) |
| 4 | (8, 64, 412) | 8 | $8^2 = 64$ | $8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512$ | No (Third number is 412, not 512) |
Based on the analysis, only Option 1, (9, 81, 729), follows the pattern where the second number is the square of the first number and the third number is the cube of the first number.
Conclusion
The set of numbers (9, 81, 729) exhibits the same relationship as the given sets (4, 16, 64) and (7, 49, 343), which is $(a, a^2, a^3)$.
Revision Table: Number Relations
Let's quickly summarize the pattern and verification:
| Set Type | Relationship | Example 1 | Example 2 | Matching Option |
|---|---|---|---|---|
| Given | $(a, a^2, a^3)$ | (4, $4^2$, $4^3$) = (4, 16, 64) | (7, $7^2$, $7^3$) = (7, 49, 343) | - |
| Option 1 | $(9, 9^2, 9^3)$ | (9, 81, 729) | - | Yes |
Additional Information: Understanding Powers and Cubes
The relationship found in this number set problem involves powers, specifically squares and cubes of a number. Let's clarify these terms:
- Power: A power of a number indicates how many times to multiply the base number by itself. It is written as base<sup>exponent</sup>.
- Square: The square of a number 'a' is 'a' raised to the power of 2, written as $a^2$. It means $a \times a$. For example, $9^2 = 9 \times 9 = 81$.
- Cube: The cube of a number 'a' is 'a' raised to the power of 3, written as $a^3$. It means $a \times a \times a$. For example, $9^3 = 9 \times 9 \times 9 = 729$.
Number series and pattern recognition questions like this one often test the ability to identify simple mathematical relationships such as arithmetic progression, geometric progression, squares, cubes, or combinations of operations.
Important Questions from Similarity
'Alone' is related to 'Unaccompanied' in the same way as 'Amount' is related to ______.
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.
97 : 2 :: 63 : ? :: 74 : 3
Select the set in which the numbers are related in the same way as are the numbers of the following sets.
(15, 135, 6)(12, 84, 5)
Select the option in which the numbers are related in the same way as are the numbers in the given set.
(146, 102, 58)
In the given word pairs, the first word is related to the second word following a certain logic. Study the given pairs carefully, and from the given options, select the pair that follows the same logic.
Dry ∶ Wet
