When a certain number is increased by 84, it becomes 107 percent of its original value. What is the original number?

Introduction / Context:
This question asks you to reverse a percentage increase. Instead of being given the original number and asked for the increased value, we are given the increased value as a relationship and the absolute increase. The task is to recover the original number using algebra and percentage concepts.

Given Data / Assumptions:
Let the original number be N. The number is increased by 84. After the increase, the result equals 107 percent of N. We need to find N.

Concept / Approach:
The phrase becomes 107 percent of itself means new value equals 1.07 times the original number. We are also told that new value equals N plus 84. Equating these two expressions for the new value produces a linear equation in N. Solving that equation yields the original number.

Step-by-Step Solution:
Let the original number be N. After an increase of 84, the new value is N + 84. We are told that this new value equals 107 percent of N. So N + 84 = 107/100 * N = 1.07N. Rearrange the equation: N + 84 = 1.07N. Bring N terms together: 84 = 1.07N - N = 0.07N. Thus N = 84 / 0.07. Compute 84 / 0.07 = 1200. So the original number is 1200.

Verification / Alternative check:
Check by forward calculation. Start with 1200. Increase by 84 to get 1284. Now find 107 percent of 1200: 107/100 * 1200 = 1.07 * 1200 = 1284. Since the new value computed directly from 107 percent matches the value obtained by adding 84, the solution is verified.

Why Other Options Are Wrong:
For 600, an increase of 84 gives 684, but 107 percent of 600 is 642. For 900, 900 plus 84 is 984 while 107 percent of 900 is 963. For 1500, 1500 plus 84 is 1584 but 107 percent of 1500 is 1605. For 1050, adding 84 yields 1134, while 107 percent of 1050 is 1123.5. None of these pairs agree, so the only correct original number is 1200.

Common Pitfalls:
Some learners wrongly treat 84 as 7 percent of the original number and then guess the original, but here 84 is the absolute increase, not a direct percentage. Others mistakenly compute 107 percent of 84. Writing the algebraic equation carefully prevents confusion and leads to a clear solution.

Final Answer:
The original number before the increase was 1200.