Understanding the Analogy Question: 27:18 :: 102:?

This question is a numerical analogy. In an analogy, two pairs of numbers are related to each other in the same way. We are given the first pair, 27 and 18, and the first number of the second pair, 102. We need to find the missing term in the second pair that has the same relationship with 102 as 18 has with 27.

Finding the Relationship Between 27 and 18

Let's analyze how the number 27 is related to 18. We can look for simple arithmetic operations or ratios that connect them.

• Is it subtraction? $27 - 18 = 9$. If we subtract 9 from 102, we get $102 - 9 = 93$. 93 is not among the options.
• Is it a multiplication or division relationship? Let's see what we need to multiply 27 by to get 18. Let the relationship be $27 \times k = 18$. To find $k$, we can divide 18 by 27: $$k = \frac{18}{27}$$ We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 9. $$k = \frac{18 \div 9}{27 \div 9} = \frac{2}{3}$$ So, the relationship is that the second number is $\frac{2}{3}$ times the first number. Let's check this: $27 \times \frac{2}{3} = \frac{27 \times 2}{3} = \frac{54}{3} = 18$. This relationship holds true for the first pair.

Applying the Relationship to 102

Now that we have found the relationship (multiplying by $\frac{2}{3}$), we can apply it to the first number of the second pair, which is 102, to find the missing term.

Let the missing term be $x$. According to the analogy, the relationship between 102 and $x$ should be the same as between 27 and 18. So, we have: $$102 \times \frac{2}{3} = x$$

Calculating the Missing Term

Let's perform the calculation:

We can simplify this by dividing 102 by 3 first:

Now, multiply the result by 2:

The missing term in the analogy is 68.

Checking the Options

Let's look at the given options:

1. 54
2. 68
3. 76
4. 84

Our calculated missing term, 68, is present as Option 2.

Conclusion

The relationship in the analogy 27:18 :: 102:? is that the second number is obtained by multiplying the first number by $\frac{2}{3}$. Applying this rule to 102 gives us 68.

Therefore, the correct missing term is 68.

Revision Table: Analogy Patterns

Additional Information: Solving Analogy Questions

Solving analogy questions, especially numerical ones, requires identifying the hidden rule or pattern that connects the first pair of terms. Here are some tips:

• Look for simple arithmetic operations first (addition, subtraction, multiplication, division).
• Consider ratios or fractions connecting the terms.
• Check for squares, cubes, or their roots.
• Examine the digits of the numbers (sum of digits, product of digits, etc.).
• Look for patterns involving prime numbers or factors.
• Sometimes, the pattern might be a combination of operations.
• Test your proposed pattern with the first pair before applying it to the second pair.
• Always check your result against the given options.

Practice with different types of numerical analogies helps in quickly recognizing potential patterns.