Understanding Percentage Changes: Increase and Decrease

This problem involves successive percentage changes applied to a number. First, the number is increased by a certain percentage, and then the new number is decreased by the same percentage. We need to find the overall or net percentage change.

Let's explore two common methods to solve this type of problem: using a base number and using a formula.

Method 1: Using a Base Number

It's often helpful to assume an initial number, especially when dealing with percentage changes. A base of 100 is convenient for calculations.

• Let the original number be 100.
• The number is increased by 30%. The increase amount is 30% of 100.
  • Increase amount $$= \frac{30}{100} \times 100 = 30$$
• The number after the increase is the original number plus the increase amount.
  • Number after increase $$= 100 + 30 = 130$$
• Now, this new number (130) is decreased by 30%. The decrease amount is 30% of 130.
  • Decrease amount $$= \frac{30}{100} \times 130$$
  • $$= \frac{3}{10} \times 130$$
  • $$= 3 \times 13 = 39$$
• The final number after the decrease is the number after increase minus the decrease amount.
  • Final number $$= 130 - 39 = 91$$
• To find the net change, compare the final number to the original number.
  • Original number $$= 100$$
  • Final number $$= 91$$
  • Change $$= \text{Final number} - \text{Original number} = 91 - 100 = -9$$
• Since the change is negative (-9), it represents a decrease. To find the percentage change, we compare the change to the original number.
  • Net percentage change $$= \frac{\text{Change}}{\text{Original number}} \times 100\%$$
  • $$= \frac{-9}{100} \times 100\%$$
  • $$= -9\%$$

A net change of -9% means there is a 9% decrease.

Method 2: Using the Successive Percentage Change Formula

For two successive percentage changes, say $x\%$ and $y\%$, the net percentage change can be calculated using the formula:

Net change $$= \left(x + y + \frac{xy}{100}\right)\%$$

In this problem, we have an increase of 30% followed by a decrease of 30%. So, we can represent the changes as:

• First change, $x = +30\%$ (positive for increase)
• Second change, $y = -30\%$ (negative for decrease)

Now, substitute these values into the formula:

• Net change $$= \left(+30 + (-30) + \frac{(+30)(-30)}{100}\right)\%$$
• $$= \left(30 - 30 + \frac{-900}{100}\right)\%$$
• $$= \left(0 - 9\right)\%$$
• $$= -9\%$$

The result is -9%. A negative result indicates a net decrease. Therefore, the net effect is a 9% decrease.

Conclusion on Net Change

Both methods show that when a number is increased by 30% and then decreased by 30%, the net effect is a decrease. The magnitude of the decrease is 9%.

This confirms the net decrease of 9%.

Revision Table: Key Concepts

Additional Information on Successive Percentage Changes

When a percentage increase and an equal percentage decrease are applied successively, the net change is always a decrease. The percentage decrease is equal to the square of the percentage change divided by 100.

In this problem, the percentage change is 30%. Using this concept:

• Percentage decrease $$= \frac{(\text{Percentage Change})^2}{100}\%$$
• $$= \frac{(30)^2}{100}\%$$
• $$= \frac{900}{100}\%$$
• $$= 9\%$$

This simple calculation also confirms the 9% decrease. This particular scenario (equal increase and decrease) is a common trick question, and understanding this shortcut can save time during exams.

It's important to remember that the second percentage change (decrease) is applied to the new value obtained after the first change (increase), not to the original value. This is why the net change is not zero.