When a number is increased by 105, it becomes 135% of its original value.\nWhat is the original number?

Introduction / Context:
This question again tests your ability to interpret a verbal description of a percentage increase and convert it into an algebraic equation. The information links the increase in absolute terms to a statement about the final value being a fixed percentage of the original. Solving such problems is a basic yet important part of percentage based aptitude questions.

Given Data / Assumptions:

• Let the original number be N.
• When the number is increased by 105, the result becomes 135% of the original number.
• We have to determine the value of N.

Concept / Approach:
If the original number is N, then a 35% increase leads to a final value of 1.35 * N, which represents 135% of N. The statement in the question can be written as N + 105 = 1.35 * N. This is a simple linear equation in N. Once we solve for N, we can check the result by substituting it back into the condition given.

Step-by-Step Solution:
Step 1: Let the original number be N.Step 2: After increasing N by 105, the new value is N + 105.Step 3: According to the question, this new value equals 135% of N, that is 1.35 * N.Step 4: Set up the equation: N + 105 = 1.35 * N.Step 5: Rearrange by subtracting N from both sides: 105 = 1.35 * N - N = 0.35 * N.Step 6: Solve for N: N = 105 / 0.35.Step 7: Compute the division: 105 / 0.35 = 300.

Verification / Alternative check:
To check the result, take N = 300 and calculate 135% of it. 135% of 300 is 300 * 135 / 100 = 300 * 1.35 = 405. Now apply the original condition: if we increase 300 by 105, we get 300 + 105 = 405, which matches 135% of 300. This confirms that the original number must be 300.

Why Other Options Are Wrong:
If N = 450, then 135% of 450 is 607.5, but 450 + 105 = 555, so the condition fails.
If N = 600, then 135% of 600 is 810, but 600 + 105 = 705, so this does not match.
If N = 400, then 135% of 400 is 540, but 400 + 105 = 505, which again does not satisfy the equation.
210 as a value gives 135% equal to 283.5, while 210 + 105 = 315, which is not the same.

Common Pitfalls:
Students sometimes misinterpret the phrase "becomes 135% of itself" and treat 135 as the increase instead of 35. Others incorrectly set up the equation as 105 = 1.35 * N, ignoring the original N that must be added. Always express the final amount as original + change and also as original multiplied by the given percentage factor. Equating these two expressions leads directly to the correct equation for N.

Final Answer:
The original number is 300.