A number is decreased by 10% and the result is 54. To make this value equal to 72, by what percentage should 54 be increased?

Introduction / Context:
This problem combines successive percentage changes and asks for the percentage increase needed to move from one specific value to another. The initial information that the number was decreased by 10% to become 54 is not directly required to find the percentage increase from 54 to 72, but it provides context about how the number changed earlier. Being able to compute percentage increase between two given numbers is a very common skill tested in aptitude examinations.

Given Data / Assumptions:

• After a 10% decrease from some original number, the resulting value is 54.
• We now want to increase 54 to 72.
• We are asked for the percentage increase required on 54 to reach 72.
• The base for the percentage increase calculation is 54.

Concept / Approach:
Percentage increase is calculated as (increase / original value) * 100. Here the original value for the second change is 54 and the final value is 72. So the increase is 72 − 54 and the percentage increase is (72 − 54) / 54 * 100. The earlier information about a 10% decrease can be used to find the original number if needed, but it is not essential for solving the specific question about the increase from 54 to 72.

Step-by-Step Solution:
Step 1: Current value = 54. Step 2: Target value = 72. Step 3: Absolute increase required = 72 − 54 = 18. Step 4: Base for percentage calculation = 54. Step 5: Percentage increase = (increase / base) * 100 = (18 / 54) * 100. Step 6: Simplify the fraction 18 / 54 = 1/3. Step 7: Therefore, percentage increase = (1/3) * 100% = 33.33%.

Verification / Alternative check:
We can also view 72 as a multiple of 54. Compute 72 / 54 = 4/3 ≈ 1.3333. This means 72 is 1.3333 times 54, or 133.33% of 54. The extra above 100% is 33.33%. Thus the required percentage increase is 33.33%, consistent with the previous method. The earlier information about a 10% decrease could be checked: if after a 10% decrease the value is 54, then original number is 54 / 0.9 = 60. But this does not affect the needed increase from 54 to 72.

Why Other Options Are Wrong:

• 25% would give a new value of 54 * 1.25 = 67.5, which is less than 72.
• 30% would give 54 * 1.3 = 70.2, still short of 72.
• 20% would produce 54 * 1.2 = 64.8, again too small.
• 40% would give 54 * 1.4 = 75.6, which exceeds 72.

Common Pitfalls:
Some students unnecessarily involve the original number before the 10% decrease and attempt a more complex approach. Others mistakenly compute (72 − 54) / 72 instead of dividing by 54, which answers how much smaller 54 is than 72 rather than how much 54 must grow. Another frequent error is rounding 1/3 to 30% or 35% instead of 33.33%. Always carefully identify the correct base for percentage calculations and keep sufficient precision in fractions like 1/3.

Final Answer:
The value 54 must be increased by 33.33% to become 72.