Calculating Net Percentage Change After Sequential Modifications

This problem involves tracking the net effect of multiple percentage changes applied consecutively to an initial number.

Let's assume we start with an arbitrary number, say 100, to make the calculations easier to follow. We will apply the given percentage changes step-by-step.

Step 1: First Percentage Increase

The number is initially increased by 10%.

• Starting number = 100
• Increase amount = 10% of 100 = $\frac{10}{100} \times 100 = 10$
• Number after the first increase = $100 + 10 = 110$

Step 2: Percentage Decrease

The result from Step 1 (110) is then decreased by 10%.

• Number before decrease = 110
• Decrease amount = 10% of 110 = $\frac{10}{100} \times 110 = 11$
• Number after the decrease = $110 - 11 = 99$

Step 3: Second Percentage Increase

The result from Step 2 (99) is again increased by 10%.

• Number before increase = 99
• Increase amount = 10% of 99 = $\frac{10}{100} \times 99 = 9.9$
• Final number = $99 + 9.9 = 108.9$

Calculating the Net Change

Now, we compare the final number (108.9) with the original number (100) to find the net change.

• Original number = 100
• Final number = 108.9
• Net change = Final number - Original number = $108.9 - 100 = 8.9$

Since the final number (108.9) is greater than the original number (100), the net change is an increase.

Determining the Net Percentage Change

To find the net percentage change, we use the formula:

Net Percentage Change = $\frac{\text{Net Change}}{\text{Original Number}} \times 100\% $

Plugging in the values:

Net Percentage Change = $\frac{8.9}{100} \times 100\% = 8.9\%$

Therefore, there is a net increase of 8.9%.

Alternative Method Using Multipliers

We can also solve this using successive percentage multipliers.

• A 10% increase corresponds to a multiplier of $(1 + \frac{10}{100}) = 1.1$.
• A 10% decrease corresponds to a multiplier of $(1 - \frac{10}{100}) = 0.9$.

Applying these multipliers sequentially:

Final Value = Original Value $\times (1.1) \times (0.9) \times (1.1)$

Final Value = Original Value $\times (1.1 \times 0.9 \times 1.1)$

Final Value = Original Value $\times (0.99 \times 1.1)$

Final Value = Original Value $\times 1.089$

The net multiplier is 1.089.

To find the net percentage change:

Net Percentage Change = (Net Multiplier - 1) $\times 100\%$

Net Percentage Change = $(1.089 - 1) \times 100\% = 0.089 \times 100\% = 8.9\%$

Since the net multiplier (1.089) is greater than 1, this represents a net increase of 8.9%.

Conclusion on Net Change

Both methods confirm that after the sequence of a 10% increase, a 10% decrease, and another 10% increase, the net result is an 8.9% increase.