Percent Increase And Decrease Worksheet

Mastering Percent Increase and Decrease: A Comprehensive Worksheet Guide

Understanding percent increase and decrease is a fundamental skill in mathematics with wide-ranging applications in everyday life, from calculating sale prices and tax increases to analyzing financial data and understanding population growth. This comprehensive guide provides a step-by-step approach to mastering percent increase and decrease, accompanied by practical examples and a detailed worksheet to solidify your understanding. We'll cover everything from basic calculations to more complex scenarios, ensuring you develop a strong grasp of this crucial mathematical concept.

What are Percent Increase and Decrease?

Percent increase and decrease describe the relative change in a value over time or between two points. A percent increase represents a growth in value, while a percent decrease indicates a reduction. Both calculations utilize the same fundamental formula, but the interpretation of the result differs. The ability to calculate percent change is vital for interpreting data effectively and making informed decisions in various contexts.

Understanding the Formula: The Key to Solving Percent Change Problems

The core formula for calculating both percent increase and percent decrease is remarkably simple:

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

Let's break this down:

• New Value: This represents the value after the increase or decrease.
• Original Value: This is the starting value before the change.

The key difference lies in the interpretation of the result:

• Positive Result: Indicates a percent increase.
• Negative Result: Indicates a percent decrease.

The multiplication by 100% converts the decimal result into a percentage.

Step-by-Step Guide: Calculating Percent Increase and Decrease

Let's illustrate the process with detailed examples:

Example 1: Percent Increase

Suppose the price of a bicycle increased from $200 to $250. To calculate the percent increase:

1. Identify the Original Value: Original Value = $200
2. Identify the New Value: New Value = $250
3. Apply the Formula: Percent Change = [(250 - 200) / 200] x 100% = (50 / 200) x 100% = 0.25 x 100% = 25%

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

Example 2: Percent Decrease

A store is having a sale, reducing the price of a jacket from $150 to $120. Let's calculate the percent decrease:

1. Identify the Original Value: Original Value = $150
2. Identify the New Value: New Value = $120
3. Apply the Formula: Percent Change = [(120 - 150) / 150] x 100% = (-30 / 150) x 100% = -0.2 x 100% = -20%

The negative sign indicates a decrease. Therefore, the price of the jacket decreased by 20%.

Working Backwards: Finding the Original or New Value

The formula can also be manipulated to find the original or new value if one of them is unknown.

Finding the Original Value:

If you know the percent increase/decrease and the new value, you can rearrange the formula to find the original value:

Original Value = New Value / (1 + Percent Change as a decimal) (for increase) Original Value = New Value / (1 - Percent Change as a decimal) (for decrease)

Example 3: Finding the Original Value (Increase)

A house price increased by 15% to $345,000. What was the original price?

1. Percent Change as a decimal: 15% = 0.15
2. Apply the formula: Original Value = $345,000 / (1 + 0.15) = $345,000 / 1.15 = $300,000

Finding the New Value:

Similarly, if you know the original value and the percent increase/decrease, you can calculate the new value:

New Value = Original Value x (1 + Percent Change as a decimal) (for increase) New Value = Original Value x (1 - Percent Change as a decimal) (for decrease)

Example 4: Finding the New Value (Decrease)

A population of 5000 decreased by 8%. What is the new population?

1. Percent Change as a decimal: 8% = 0.08
2. Apply the formula: New Value = 5000 x (1 - 0.08) = 5000 x 0.92 = 4600

Percent Increase and Decrease Worksheet: Practice Problems

Now let's put your knowledge into practice with a series of problems. Try to solve these problems using the steps and formulas outlined above. Remember to show your work for each problem.

Section 1: Calculating Percent Increase/Decrease

1. A shirt originally priced at $25 is increased by 10%. What is the new price?
2. A car's value depreciated from $20,000 to $16,000. What is the percent decrease?
3. A savings account balance increased from $500 to $650. What is the percent increase?
4. The number of students in a school increased by 12% from 800 to what new number?
5. A company's profits decreased by 15% from $1,000,000 to what new amount?

Section 2: Finding the Original or New Value

1. A dress is now selling for $77 after a 10% discount. What was the original price?
2. The price of a laptop increased by 20% to $1200. What was the original price?
3. A town's population grew by 5% to reach 10,500. What was the original population?
4. After a 25% reduction, a phone costs $300. What was the original price?
5. A stock's value increased by 18% to reach $50. What was its original value?

Section 3: Challenging Problems

1. A store marks up its goods by 40%. If a customer purchases an item for $70, what was the original cost before markup?
2. A house increased in value by 25% in the first year and by 10% in the second year. What is the total percentage increase over the two years? (Hint: Consider the effect of the second increase on the already increased value.)
3. A company's sales decreased by 10% in one quarter and then increased by 15% in the next quarter. What is the net percentage change in sales over these two quarters?
4. John invested $1000. After one year, his investment grew by 10%, and in the second year, it decreased by 5%. What was the total percent change in his investment over the two years?
5. Sarah's salary increased by 8% this year. If her new salary is $64,800, what was her salary last year?

Scientific Explanation of Percent Change

Percent change is a concept grounded in the principles of ratios and proportions. The formula itself is derived from the definition of a ratio: the relationship between two quantities. In this case, we're comparing the change in value to the original value. The multiplication by 100% simply transforms the ratio into a more readily understandable percentage, making it easier to visualize and communicate the relative magnitude of the change. The consistent application of this ratio, whether calculating increase or decrease, ensures a standardized and easily comparable method across various applications. Understanding the core ratio underlying the formula helps solidify understanding and makes tackling more complex problems easier.

Frequently Asked Questions (FAQ)

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

A: The formula will automatically result in a negative percentage, indicating a percent decrease.

Q: Can I use this formula for any type of data?

A: Yes, as long as you have an original and a new value to compare. It's commonly used for prices, populations, sales figures, and various other types of quantitative data.

Q: How do I handle percentages greater than 100%?

A: While less common in decrease calculations, an increase greater than 100% simply indicates that the new value is more than double the original value. The formula works perfectly for such scenarios.

Q: Are there any alternative methods to calculate percent change?

A: While the standard formula is the most efficient, you can conceptually break the problem down into smaller steps; calculate the absolute change, then express that change as a fraction or decimal of the original value, and finally multiply by 100% to obtain the percentage. However, the standard formula directly provides the result, optimizing the calculation.

Conclusion

Mastering percent increase and decrease is a crucial skill that significantly enhances your ability to interpret data, analyze trends, and solve real-world problems. By understanding the formula, practicing with various examples, and utilizing the provided worksheet, you will be well-equipped to confidently tackle percent change calculations in diverse contexts. Remember to always clearly identify the original and new values before applying the formula and interpret the results carefully—paying close attention to the sign of the result (positive for increase, negative for decrease). Through consistent practice and application, you'll transform this essential mathematical concept from a challenge into a strength, enabling you to approach quantitative data with greater accuracy and confidence.