Problem 24 - Prealgebra

Question

CHALLENGE Explain why a \(10 \%\) increase followed by a \(10 \%\) decrease is less than the original amount if the original amount was positive.

Step-by-Step Solution

Answer

A 10% increase followed by a 10% decrease results in 99% of the original amount, less than the original.

Step 1: Understanding the Original Amount

Let's call the original amount \( A \). This is the starting point, from which we will calculate the effects of a 10% increase and a 10% decrease.

Step 2: Calculating the 10% Increase

When the amount increases by 10%, it becomes \( A + 0.1A = 1.1A \). This represents a 10% increase from the original value \( A \).

Step 3: Applying the 10% Decrease

Now, we apply a 10% decrease to the new amount \( 1.1A \). That is, we subtract 10% of \( 1.1A \): \( 1.1A - 0.1(1.1A) = 1.1A - 0.11A = 0.99A \).

Step 4: Comparing to the Original Amount

After the increase and subsequent decrease, the amount is \( 0.99A \). Since \( 0.99A < A \), this shows that the final amount is less than the original amount.

Key Concepts

Increase and Decrease in Percentages

When we talk about increases and decreases in percentages, we're discussing how the original value changes by a certain proportion.
For instance, if you increase a quantity by 10%, you are essentially adding 10% of that original quantity to itself. This means multiplying the original number by 1.1.
On the other hand, decreasing by a percentage means subtracting that percentage of the current value from itself. So, a 10% decrease means you multiply the current value by 0.9. Important points to remember:

• An increase is multiplying by (1 + ext{percentage as a decimal})
• A decrease is multiplying by (1 - ext{percentage as a decimal})

This method helps in calculating new values after a percentage adjustment.

Effect of Consecutive Percentage Changes

Consecutive percentage changes can be a bit tricky. You might think that if you increase and then decrease by the same percentage, you would end up where you started, right? Incorrect! This isn't the case.
Consider an original amount, say 100. First, you increase it by 10%, bringing it to 110.
Then, you decrease this new amount by 10%. Instead of returning to 100, you arrive at 99. This happens because the 10% decrease is applied to 110, not the original 100.
Here's why consecutive changes behave this way:

• Each percentage change is based on the current amount, not the original.
• Increase: Original amount is scaled up.
• Decrease: The higher new amount is scaled down, reducing more than if calculated from the original.

Understanding this subverts the intuitive thought that equal increases and decreases would cancel each other out.

Mathematical Reasoning for Percentage Change

The math of percentage change is simple yet powerful. When you calculate a percentage increase, you multiply the original amount by a factor greater than 1 (like 1.1 for a 10% increase).
When calculating a decrease, you multiply the amount by a factor less than 1 (like 0.9 for a 10% decrease). Why is this important?

• Each operation is relative to the value it's applied to.
• Increases multiply the current amount, boosts value.
• Decreases cut down an already elevated value.

So, a 10% increase followed by a 10% decrease is not neutral. Instead, it cuts more from the bigger, increased value than it added to the original. Hence, the compound result can be less than the starting value, defying simple assumptions with clear math logic.