When a number is increased by 120, it becomes 130% of its original value. What is the original number?

Introduction / Context:
This question deals with converting a word statement involving percentage increase into an algebraic equation. When a number is said to become 130% of itself, it means the new value equals 1.3 times the original number. Such problems are common in percentage and profit and loss chapters, where we often move between absolute changes and percentage descriptions. Proper translation into equations makes these questions straightforward to solve.

Given Data / Assumptions:

• Let the original number be N.
• When increased by 120, the new number is N + 120.
• This new value N + 120 equals 130% of the original number.
• 130% of N means (130/100) * N = 1.3N.
• We are required to find the numerical value of N.

Concept / Approach:
The key idea is to set up an equation based on the statement that the new value equals 130% of the original. Once the equation is formed, solving for N reduces to simple algebra. Specifically, we equate N + 120 to 1.3N, then isolate N by moving all terms with N to one side and constants to the other. This pattern repeats in many similar percentage increase or salary raise problems.

Step-by-Step Solution:
Step 1: Let the original number be N. Step 2: After an increase of 120, the new value is N + 120. Step 3: According to the question, N + 120 = 130% of N = 1.3N. Step 4: Write the equation: N + 120 = 1.3N. Step 5: Rearrange terms: 120 = 1.3N − N = 0.3N. Step 6: Solve for N: N = 120 / 0.3 = 400.

Verification / Alternative check:
Substitute N = 400 back into the condition. Increasing 400 by 120 gives 400 + 120 = 520. Now calculate 130% of 400: (130/100) * 400 = 1.3 * 400 = 520. Since both methods give the same new value, 520, the solution N = 400 satisfies the condition perfectly. This confirms that our algebraic manipulation is correct and that the original number is indeed 400.

Why Other Options Are Wrong:

• 520 is the new value after increase, not the original number.
• 460 would give 460 + 120 = 580, but 130% of 460 is 598, which does not match.
• 580 would produce 580 + 120 = 700, while 130% of 580 is 754, so it clearly fails the given condition.

Common Pitfalls:
Learners sometimes treat 130% as 130N instead of 1.3N or (130/100) * N. Another frequent error is to set the equation incorrectly as N + 120 = 0.3N, mixing up the meaning of percentage. Care must also be taken with decimal division, since 120 / 0.3 equals 400, not 40. Writing all steps clearly and checking the final answer by substitution is a good habit to avoid such mistakes.

Final Answer:
The original number is 400.