Percent Change Word Problems Worksheet

Mastering Percent Change: A Comprehensive Guide with Worked Examples and Practice Problems

Understanding percent change is a crucial skill in mathematics, with applications spanning various fields like finance, science, and everyday life. This comprehensive guide will equip you with the knowledge and tools to confidently tackle percent change word problems. We'll cover the fundamentals, delve into different types of problems (including percent increase and percent decrease), provide step-by-step solutions to example problems, and offer a worksheet with practice problems to solidify your understanding. This guide will address percent change problems frequently encountered in classrooms and real-world scenarios.

Understanding Percent Change

Percent change represents the relative change between an old value and a new value. It shows how much a value has increased or decreased in percentage terms. The formula for calculating percent change is:

Percent Change = [(New Value - Old Value) / Old Value] x 100%

This formula tells us the percentage change, whether an increase or a decrease, relative to the original value.

Let's break down the formula:

• New Value: This is the value after the change has occurred.
• Old Value: This is the initial or original value before the change.
• (New Value - Old Value): This calculates the difference between the new and old values. A positive result indicates an increase, while a negative result indicates a decrease.
• / Old Value: This division normalizes the difference against the original value.
• x 100%: Multiplying by 100% converts the result into a percentage.

Percent Increase Problems

Percent increase problems involve calculating the percentage by which a value has grown. The process remains the same as the general percent change formula, but the result will always be a positive percentage.

Example 1: Percent Increase

The price of a bicycle increased from $200 to $250. What is the percent increase?

Solution:

1. Identify the Old Value and New Value: Old Value = $200, New Value = $250

2. Apply the Formula: Percent Increase = [(250 - 200) / 200] x 100%

3. Calculate: Percent Increase = (50 / 200) x 100% = 0.25 x 100% = 25%

Therefore, the price of the bicycle increased by 25%.

Example 2: Percent Increase with More Complex Values

A company's profit increased from $15,000 to $21,750. Calculate the percent increase in profit.

Solution:

1. Identify the Old Value and New Value: Old Value = $15,000, New Value = $21,750

2. Apply the Formula: Percent Increase = [(21,750 - 15,000) / 15,000] x 100%

3. Calculate: Percent Increase = (6750 / 15,000) x 100% = 0.45 x 100% = 45%

The company's profit increased by 45%.

Percent Decrease Problems

Percent decrease problems are similar, but the difference (New Value - Old Value) will be negative, resulting in a negative percentage change. However, we usually report percent decrease as a positive number, representing the magnitude of the decrease.

Example 3: Percent Decrease

The population of a town decreased from 5000 to 4500. What is the percent decrease?

Solution:

1. Identify the Old Value and New Value: Old Value = 5000, New Value = 4500

2. Apply the Formula: Percent Decrease = [(4500 - 5000) / 5000] x 100%

3. Calculate: Percent Decrease = (-500 / 5000) x 100% = -0.1 x 100% = -10%

We report the percent decrease as 10%.

Example 4: Percent Decrease in Sales

A store's sales dropped from $100,000 to $85,000. Calculate the percent decrease in sales.

Solution:

1. Identify the Old Value and New Value: Old Value = $100,000, New Value = $85,000

2. Apply the Formula: Percent Decrease = [(85,000 - 100,000) / 100,000] x 100%

3. Calculate: Percent Decrease = (-15,000 / 100,000) x 100% = -0.15 x 100% = -15%

The percent decrease in sales is 15%.

Finding the New Value After a Percent Change

Sometimes, you'll know the original value and the percent change, and you need to find the new value. Here's how:

Example 5: Finding the New Value After an Increase

The price of a shirt is $30, and it's increased by 15%. What is the new price?

Solution:

1. Calculate the increase amount: 15% of $30 = (15/100) x $30 = $4.50

2. Add the increase to the original price: $30 + $4.50 = $34.50

The new price of the shirt is $34.50.

Example 6: Finding the New Value After a Decrease

A car's value is $20,000, and it depreciates by 10%. What is the new value?

Solution:

1. Calculate the decrease amount: 10% of $20,000 = (10/100) x $20,000 = $2,000

2. Subtract the decrease from the original value: $20,000 - $2,000 = $18,000

The new value of the car is $18,000.

Finding the Original Value After a Percent Change

This is the inverse of the previous problem. You know the new value and the percent change, and need to find the original value. This often requires working backwards with the percent change formula.

Example 7: Finding the Original Value After an Increase

The price of a house is now $315,000 after a 5% increase. What was the original price?

Solution:

Let x be the original price. After a 5% increase, the new price is 1.05x (because it’s 100% + 5%). We know the new price is $315,000, so:

1.05x = $315,000

x = $315,000 / 1.05 = $300,000

The original price of the house was $300,000.

Example 8: Finding the Original Value After a Decrease

After a 20% discount, a laptop costs $800. What was the original price?

Solution:

Let x be the original price. After a 20% discount, the new price is 0.8x (because it’s 100% - 20%). We know the new price is $800, so:

0.8x = $800

x = $800 / 0.8 = $1000

The original price of the laptop was $1000.

Percent Change Word Problems Worksheet

Now, let's put your skills to the test! Try solving the following problems. Remember to show your work.

Problem 1: A store's revenue increased from $50,000 to $65,000. What is the percent increase in revenue?

Problem 2: The number of students in a school decreased from 1200 to 1080. What is the percent decrease in the number of students?

Problem 3: A painting's value appreciated by 25% and is now worth $1250. What was its original value?

Problem 4: After a 15% discount, a TV costs $765. What was the original price of the TV?

Problem 5: A company's stock price increased from $25 per share to $37.50 per share. What is the percent increase in the stock price?

Problem 6: A farmer's crop yield decreased by 10% this year and he harvested only 810 bushels. How many bushels did he harvest last year?

Problem 7: The price of a gallon of gas increased by 8% from last week's price of $3.50. What is the new price of a gallon of gas?

Problem 8: A dress is on sale for $45 after a 20% discount. What was the original price of the dress?

Problem 9: The population of a city grew from 150,000 to 165,000. What is the percent increase in population?

Problem 10: A car depreciated in value by 12% and is now worth $17,600. What was its original value?

Solutions to Worksheet Problems

Problem 1: 30% increase

Problem 2: 10% decrease

Problem 3: $1000

Problem 4: $900

Problem 5: 50% increase

Problem 6: 900 bushels

Problem 7: $3.78

Problem 8: $56.25

Problem 9: 10% increase

Problem 10: $20,000

Frequently Asked Questions (FAQ)

Q: What if the new value is less than the old value?

A: If the new value is less than the old value, your answer will be a negative percentage. This indicates a percentage decrease, which is typically reported as a positive value representing the magnitude of the decrease.

Q: Can I use the percent change formula for more than just price changes?

A: Absolutely! The percent change formula applies to any scenario involving a change in value. Examples include population changes, growth rates, changes in weight, and many more.

Q: What if I'm given the percent change and the new value, and I need to find the old value? How do I solve for the old value?

A: You need to work backward using the percent change formula. This usually involves setting up an algebraic equation and solving for the unknown (old value). The examples above demonstrate this process.

Q: Are there different formulas for percent increase and percent decrease?

A: While you could technically derive separate formulas, it's most efficient and accurate to use the general percent change formula. The sign of the result will indicate whether it's an increase or decrease.

Conclusion

Mastering percent change problems requires understanding the underlying concept and the formula. Practice is key to developing fluency and confidence. By working through the examples and completing the worksheet, you've significantly improved your ability to tackle a wide range of percent change word problems. Remember to always clearly identify the old and new values before applying the formula. With consistent practice, you'll become proficient in solving these problems in any context.