Analogy Pattern: 1000:121 :: ?

The question asks us to find a pair of numbers that has the same relationship as the given pair 1000 : 121.

Identifying the Relationship

Let's analyze the given pair 1000 : 121.

• The first number, 1000, can be expressed as a cube: $1000 = 10^3$.
• The second number, 121, can be expressed as a square: $121 = 11^2$.

Observe that the base of the square (11) is one greater than the base of the cube (10). Thus, the pattern is $n^3 : (n+1)^2$.

For the given pair, $n=10$. We have $10^3 = 1000$ and $(10+1)^2 = 11^2 = 121$. This confirms the pattern.

Applying the Pattern to Options

Now, we apply the identified pattern $n^3 : (n+1)^2$ to the given options:

• Option 1: 64 : 8
  If $n^3 = 64$, then $n = 4$.
  The second term should be $(4+1)^2 = 5^2 = 25$.
  The pair 64 : 8 does not fit the pattern.
• Option 2: 125 : 36
  If $n^3 = 125$, then $n = 5$.
  The second term should be $(5+1)^2 = 6^2 = 36$.
  The pair 125 : 36 perfectly fits the pattern.
• Option 3: 27 : 64
  If $n^3 = 27$, then $n = 3$.
  The second term should be $(3+1)^2 = 4^2 = 16$.
  The pair 27 : 64 does not fit the pattern.
• Option 4: 343 : 81
  If $n^3 = 343$, then $n = 7$.
  The second term should be $(7+1)^2 = 8^2 = 64$.
  The pair 343 : 81 does not fit the pattern.

Conclusion

The only pair that follows the established relationship $n^3 : (n+1)^2$ is 125 : 36.