Understanding Percentage Changes and Differences

This problem involves a sequence of operations on a number using percentages and then comparing the differences between the resulting numbers. Let's break it down step by step to find the solution.

Step 1: Define the Original Number

Let the original number be represented by a variable. A common way to handle percentage problems is to assume a base value, like 100, or use a variable. Using a variable makes the solution general. Let the original number be \(x\).

Step 2: Calculate the Second Number

The question states that the original number is decreased by 20% to get the second number. A decrease of 20% means the number becomes \(100\% - 20\% = 80\%\) of the original number.

Second number = Original number - 20% of Original number

Second number = \(x - 0.20x = 0.80x\)

So, the second number is \(0.80x\).

Step 3: Calculate the Third Number

The number so obtained (the second number) is increased by 200% to get the third number. An increase of 200% means adding 200% of the second number to the second number. \(200\%\) of a number is \(2\) times the number.

Third number = Second number + 200% of Second number

Third number = \(0.80x + 200\% \text{ of } 0.80x\)

Third number = \(0.80x + 2.00 \times 0.80x\)

Third number = \(0.80x + 1.60x\)

Third number = \(2.40x\)

So, the third number is \(2.40x\).

Step 4: Calculate the First Difference

The first difference is the difference between the third number and the original number.

Difference 1 = Third number - Original number

Difference 1 = \(2.40x - x = 1.40x\)

Step 5: Calculate the Second Difference

The second difference is the difference between the second and the third number.

Difference 2 = Second number - Third number

Difference 2 = \(0.80x - 2.40x = -1.60x\)

When comparing differences using percentages in this context, we usually compare the magnitudes. The magnitude of the second difference is \(| -1.60x | = 1.60x\).

Step 6: Compare the Differences as a Percentage

We need to determine if the first difference (\(1.40x\)) is what percentage more or less than the second difference (\(1.60x\)).

• First, compare the values: \(1.40x\) is smaller than \(1.60x\). So, the first difference is less than the second difference.
• Next, calculate the amount by which the first difference is less than the second difference: Amount less = Second difference - First difference = \(1.60x - 1.40x = 0.20x\).
• Finally, calculate this amount as a percentage of the second difference:

Percentage difference = \(\frac{\text{Amount less}}{\text{Second difference}} \times 100\%\)

Percentage difference = \(\frac{0.20x}{1.60x} \times 100\%\)

Percentage difference = \(\frac{0.20}{1.60} \times 100\%\)

Percentage difference = \(\frac{20}{160} \times 100\%\)

Percentage difference = \(\frac{1}{8} \times 100\%\)

Percentage difference = \(12.5\%\)

So, the difference of the third number and the original number (\(1.40x\)) is 12.5% less than the difference of the second and the third number (\(1.60x\)).

Conclusion on Percentage Difference

The difference of the third number and the original number is 12.5% less than the difference of the second and the third number (magnitude). This matches one of the given options.

Revision Table: Key Calculations

Additional Information on Percentage Concepts

Percentage is a way of expressing a part of a whole in terms of 100. It is denoted by the symbol '%'.

• Percentage Increase: When a quantity increases, the percentage increase is calculated based on the original quantity. Formula: \(\frac{\text{Increase}}{\text{Original Quantity}} \times 100\%\).
• Percentage Decrease: When a quantity decreases, the percentage decrease is calculated based on the original quantity. Formula: \(\frac{\text{Decrease}}{\text{Original Quantity}} \times 100\%\).
• Percentage Point Difference: This is the simple arithmetic difference between two percentages. For example, the difference between 20% and 30% is 10 percentage points.
• Percentage Change vs. Percentage Point Change: It's crucial not to confuse percentage change (which is relative to a base) with percentage point change (an absolute difference).

In this problem, we used percentage changes to find new numbers and then calculated a percentage difference between two derived quantities (differences between numbers). The base for the final percentage comparison was the magnitude of the difference between the second and third numbers.