Finding the Odd Number Pair Based on Logic and Relation

This question asks us to identify the number pair that does not follow the same logic or rule as the other pairs. We are given four pairs of numbers, and we need to find the mathematical relationship between the first number and the second number in each pair.

The note specifies that operations should be performed on the whole numbers, not on their constituent digits. This means we should look for relationships like multiplication, division, addition, or subtraction involving the entire number.

Analyzing Each Number Pair

Let's examine each pair and try to find the relationship between the number on the left side and the number on the right side of the dash (–).

1. 3375 – 13500
2. Let's see how 13500 is related to 3375. A common relationship could be multiplication. Let's divide 13500 by 3375.
3. \[ \frac{13500}{3375} = 4 \]
4. So, the relationship here is that the second number is 4 times the first number: \( 3375 \times 4 = 13500 \).
5. 2125 – 8600
6. Let's apply the same logic. Let's divide 8600 by 2125.
7. \[ \frac{8600}{2125} = 4.047... \]
8. The result is not a whole number like 4. Let's see what happens if we multiply the first number by 4: \( 2125 \times 4 = 8500 \).
9. The second number is 8600. The difference is \( 8600 - 8500 = 100 \).
10. So, the relationship here seems to be \( 2125 \times 4 + 100 = 8600 \).
11. 2685 – 10740
12. Let's divide 10740 by 2685.
13. \[ \frac{10740}{2685} = 4 \]
14. So, the relationship here is that the second number is 4 times the first number: \( 2685 \times 4 = 10740 \).
15. 1595 – 6380
16. Let's divide 6380 by 1595.
17. \[ \frac{6380}{1595} = 4 \]
18. So, the relationship here is that the second number is 4 times the first number: \( 1595 \times 4 = 6380 \).

Identifying the Odd One Out

Let's summarize the relationships found for each pair:

• Pair 1 (3375 – 13500): Second number = First number \( \times 4 \)
• Pair 2 (2125 – 8600): Second number = First number \( \times 4 + 100 \)
• Pair 3 (2685 – 10740): Second number = First number \( \times 4 \)
• Pair 4 (1595 – 6380): Second number = First number \( \times 4 \)

We can see that pairs 1, 3, and 4 all follow the same logic: the second number is obtained by multiplying the first number by 4. However, pair 2 follows a different logic: multiplying the first number by 4 and then adding 100.

Therefore, the odd one out from the given alternatives is the pair that does not follow the \( \times 4 \) rule, which is 2125 – 8600.

Let's present this in a table for clarity.

As the table clearly shows, the logic for pairs 1, 3, and 4 is multiplication by 4, while the logic for pair 2 is multiplication by 4 followed by the addition of 100. Hence, pair 2 is the odd one out.

Revision Table: Key Logic and Relation Concepts

Additional Information: Strategies for Odd One Out Questions

When tackling 'Odd One Out' questions involving numbers or number pairs based on logic, here are some useful strategies:

• Look for simple arithmetic relations: Check for addition, subtraction, multiplication, or division between the numbers in each pair.
• Consider squares, cubes, and roots: Sometimes the relation involves squaring, cubing, or taking the root of a number.
• Check for sequences or series logic: While less common for isolated pairs, be aware of potential patterns if multiple pairs suggest a progression.
• Analyze the structure of the numbers: Although the question explicitly says not to break digits, sometimes properties of the whole number (like being even/odd, prime/composite, sum/product of digits - *when allowed*) can be relevant. In this specific problem, digit-based logic was disallowed.
• Test a hypothesis: If you find a potential rule in one pair, test it on the others to see if it holds true.
• Calculate ratios or differences: Calculating the ratio (second/first) or the difference (second - first) for all pairs can quickly reveal a consistent pattern or the outlier.

In this problem, calculating the ratio \( \frac{\text{Second Number}}{\text{First Number}} \) was a very efficient way to identify the underlying rule and the pair that deviated from it.