Solving Real-Life Percent Change Scenarios

Introduction

Key Concepts

1. Understanding Percent Change

Percent change measures the extent to which a value increases or decreases in relation to its original amount. It's a critical concept in mathematics, particularly in areas like finance, economics, and data analysis. The basic formula to calculate percent change is:

$$ \text{Percent Change} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100\% $$

This formula helps in determining whether there has been an increase or decrease and by what percentage.

2. Percentage Increase

Percentage increase occurs when a value grows compared to its original amount. It's calculated using the same formula for percent change, where the new value is greater than the original.

Example: If a shirt originally costs $50 and its price increases to $65, the percentage increase is:

$$ \text{Percentage Increase} = \left( \frac{65 - 50}{50} \right) \times 100\% = 30\% $$

This means the shirt's price has increased by 30%.

3. Percentage Decrease

Percentage decrease happens when a value diminishes in comparison to its original amount. The calculation is identical to percentage increase, but the new value is less than the original.

Example: If a laptop originally costs $1200 and its price drops to $900, the percentage decrease is:

$$ \text{Percentage Decrease} = \left( \frac{1200 - 900}{1200} \right) \times 100\% = 25\% $$

This indicates a 25% decrease in the laptop's price.

4. Applications of Percent Change

Percent change is widely applicable in various fields:

• Finance: Calculating interest rates, investment growth, and depreciation.
• Economics: Analyzing inflation rates, unemployment rates, and economic growth.
• Business: Assessing sales growth, profit margins, and market share changes.
• Everyday Life: Comparing discounts, price hikes, and budgeting.

5. Compound Percent Change

Sometimes, percent changes occur sequentially, requiring compound calculations. This is common in finance when dealing with compound interest or multiple consecutive discounts.

Example: If a product's price increases by 10% one year and then decreases by 5% the next year, the compound percent change is:

$$ \text{First Increase: } 100 \times 1.10 = 110 $$ $$ \text{Second Decrease: } 110 \times 0.95 = 104.5 $$ $$ \text{Overall Change: } \frac{104.5 - 100}{100} \times 100\% = 4.5\% \text{ increase} $$

Despite an increase and a decrease, the overall effect is a net 4.5% increase.

6. Solving Real-Life Scenarios

Applying percent change concepts to real-life situations involves interpreting data, setting up equations, and solving for unknowns. Let's explore several practical scenarios.

Scenario 1: Price Discounts

A store offers a 20% discount on a jacket originally priced at $80. To find the sale price:

$$ \text{Discount Amount} = 80 \times 0.20 = 16 $$ $$ \text{Sale Price} = 80 - 16 = 64 \text{ dollars} $$

Thus, the jacket costs $64 after the discount.

Scenario 2: Salary Increase

An employee receives a 15% salary increase, raising their annual salary from $40,000 to:

$$ \text{Increase Amount} = 40,000 \times 0.15 = 6,000 $$ $$ \text{New Salary} = 40,000 + 6,000 = 46,000 \text{ dollars} $$

The employee's new salary is $46,000.

Scenario 3: Population Decline

A town's population decreases by 5% over a year. If the initial population was 20,000, the new population is:

$$ \text{Decrease Amount} = 20,000 \times 0.05 = 1,000 $$ $$ \text{New Population} = 20,000 - 1,000 = 19,000 $$

The population after the decline is 19,000.

7. Percentage Change in Multiple Steps

In some scenarios, multiple percent changes occur in sequence. It's essential to apply each change step-by-step.

Example: A product undergoes a 10% increase followed by a 20% decrease. Starting with $50:

$$ \text{After 10% Increase: } 50 \times 1.10 = 55 $$ $$ \text{After 20% Decrease: } 55 \times 0.80 = 44 $$ $$ \text{Overall Change: } \frac{44 - 50}{50} \times 100\% = -12\% $$

Despite an initial increase, the final price is 12% lower than the original.

8. Working Backwards with Percent Change

Sometimes, you may need to find the original value given the new value and the percent change.

Example: After a 25% decrease, the price of a bike is $150. Find the original price.

Let the original price be \( x \):

$$ x - 0.25x = 150 $$ $$ 0.75x = 150 $$ $$ x = \frac{150}{0.75} = 200 $$

The original price was $200.

9. Percentage Change in Investments

Investors frequently use percent change to assess the performance of their investments.

Example: An investment grows from $5,000 to $6,500 in a year. The percent increase is:

$$ \text{Percent Increase} = \left( \frac{6,500 - 5,000}{5,000} \right) \times 100\% = 30\% $$

The investment grew by 30%.

10. Analyzing Multiple Percent Changes

When multiple percent changes are involved, it's crucial to consider their cumulative effect.

Example: A population increases by 10% one year and decreases by 10% the next year.

$$ \text{Initial Population} = 1000 $$ $$ \text{After 10% Increase: } 1000 \times 1.10 = 1100 $$ $$ \text{After 10% Decrease: } 1100 \times 0.90 = 990 $$ $$ \text{Overall Change: } \frac{990 - 1000}{1000} \times 100\% = -1\% $$

Despite equal percentage increases and decreases, the population decreased by 1%.

11. Solving Percentage Change Problems Using Algebra

Algebra can be used to solve more complex percent change problems, especially when dealing with unknowns.

Example: A price increases by \( p\% \) to become $180. The original price was $150. Find \( p \).

Using the percent change formula:

$$ 150 \times \left(1 + \frac{p}{100}\right) = 180 $$ $$ 1 + \frac{p}{100} = \frac{180}{150} = 1.2 $$ $$ \frac{p}{100} = 0.2 $$ $$ p = 20\% $$

The price increased by 20%.

12. Real-Life Applications and Practice Problems

Practicing real-life scenarios enhances understanding and application of percent change concepts.

Practice Problem 1: A laptop’s price decreases from $1200 to $960. What is the percentage decrease?

Solution:

$$ \text{Percentage Decrease} = \left( \frac{1200 - 960}{1200} \right) \times 100\% = 20\% $$

Practice Problem 2: A student’s grade improves from 75% to 90%. What is the percentage increase?

Solution:

$$ \text{Percentage Increase} = \left( \frac{90 - 75}{75} \right) \times 100\% = 20\% $$

Practice Problem 3: An item is first discounted by 15% and then by an additional 10%. If the final price is $85.50, what was the original price?

Solution:

$$ \text{Let Original Price} = x $$ $$ x \times 0.85 \times 0.90 = 85.50 $$ $$ x \times 0.765 = 85.50 $$ $$ x = \frac{85.50}{0.765} = 111.76 $$

The original price was approximately $111.76.

Comparison Table

Aspect	Percentage Increase	Percentage Decrease
Definition	Value grows compared to the original amount.	Value diminishes compared to the original amount.
Formula	$\left( \frac{\text{New} - \text{Original}}{\text{Original}} \right) \times 100\%$	$\left( \frac{\text{Original} - \text{New}}{\text{Original}} \right) \times 100\%$
Example	Price increases from $50 to $65 (30% increase).	Price decreases from $1200 to $900 (25% decrease).
Applications	Salary hikes, investment growth, sales increases.	Discounts, population decline, depreciation.
Impact of Multiple Changes	Sequential increases compound positively.	Sequential decreases compound negatively.

Summary and Key Takeaways