Increasing And Decreasing Percentages Questions

Mastering Percentage Increase and Decrease: A Comprehensive Guide

Percentage increase and decrease are fundamental concepts in mathematics with wide-ranging applications in everyday life, from calculating sale discounts to understanding economic growth. This comprehensive guide will equip you with the knowledge and skills to confidently tackle any percentage increase or decrease problem, regardless of its complexity. We will explore the underlying principles, delve into practical examples, and address frequently asked questions. By the end, you'll not only be able to solve these problems but also understand the logic behind them.

Understanding the Fundamentals: Percentage Basics

Before diving into increase and decrease, let's solidify our understanding of percentages. A percentage is simply a fraction expressed as a part of 100. For example, 50% means 50 out of 100, or ½. To convert a percentage to a decimal, you divide the percentage by 100. For example, 25% becomes 0.25 (25/100). Conversely, to convert a decimal to a percentage, you multiply the decimal by 100. 0.75 becomes 75% (0.75 x 100).

Calculating Percentage Increase

A percentage increase represents the growth of a quantity. The formula for calculating percentage increase is:

Percentage Increase = [(New Value - Original Value) / Original Value] x 100%

Let's break this down with an example. Suppose the price of a product increased from $50 to $60.

1. Find the difference: $60 - $50 = $10
2. Divide the difference by the original value: $10 / $50 = 0.2
3. Multiply by 100% to express as a percentage: 0.2 x 100% = 20%

Therefore, the price increased by 20%.

Example 1: Real-World Application

Imagine a company's profits rose from $1 million to $1.2 million. To calculate the percentage increase:

1. Difference: $1.2 million - $1 million = $0.2 million
2. Divide by original: $0.2 million / $1 million = 0.2
3. Multiply by 100%: 0.2 x 100% = 20%

The company's profits increased by 20%.

Example 2: More Complex Scenario

A town's population increased from 15,000 to 18,750. What is the percentage increase?

1. Difference: 18,750 - 15,000 = 3,750
2. Divide by original: 3,750 / 15,000 = 0.25
3. Multiply by 100%: 0.25 x 100% = 25%

The town's population increased by 25%.

Calculating Percentage Decrease

A percentage decrease represents the reduction of a quantity. The formula is very similar to the increase formula:

Percentage Decrease = [(Original Value - New Value) / Original Value] x 100%

Notice that the order of subtraction is reversed compared to the increase formula.

Example 3: Sale Discount

An item originally priced at $100 is on sale for $80. What is the percentage decrease?

1. Difference: $100 - $80 = $20
2. Divide by original: $20 / $100 = 0.2
3. Multiply by 100%: 0.2 x 100% = 20%

The item is discounted by 20%.

Example 4: Population Decline

A city's population decreased from 200,000 to 180,000. What is the percentage decrease?

1. Difference: 200,000 - 180,000 = 20,000
2. Divide by original: 20,000 / 200,000 = 0.1
3. Multiply by 100%: 0.1 x 100% = 10%

The city's population decreased by 10%.

Finding the New Value After a Percentage Increase or Decrease

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

For Percentage Increase:

New Value = Original Value x (1 + Percentage Increase as a decimal)

For Percentage Decrease:

New Value = Original Value x (1 - Percentage Decrease as a decimal)

Example 5: Applying the New Value Formula (Increase)

A house is worth $250,000 and appreciates by 15%. What is its new value?

1. Convert percentage to decimal: 15% = 0.15
2. Apply the formula: $250,000 x (1 + 0.15) = $250,000 x 1.15 = $287,500

The new value of the house is $287,500.

Example 6: Applying the New Value Formula (Decrease)

A car originally costing $30,000 depreciates by 10% after one year. What is its value after one year?

1. Convert percentage to decimal: 10% = 0.10
2. Apply the formula: $30,000 x (1 - 0.10) = $30,000 x 0.90 = $27,000

The car's value after one year is $27,000.

Successive Percentage Changes

When dealing with successive percentage increases or decreases, you cannot simply add or subtract the percentages. Instead, you must apply the percentage change sequentially.

Example 7: Successive Percentage Increases

A shirt's price increases by 10% and then by another 5%. If the original price was $20, what is the final price?

1. First increase: $20 x (1 + 0.10) = $22
2. Second increase: $22 x (1 + 0.05) = $23.10

The final price of the shirt is $23.10. Notice that a simple addition of 10% and 5% (15%) would give an incorrect result.

Example 8: Successive Percentage Decreases

A store offers a 20% discount followed by an additional 15% discount on an item originally priced at $50. What is the final price?

1. First discount: $50 x (1 - 0.20) = $40
2. Second discount: $40 x (1 - 0.15) = $34

The final price after both discounts is $34.

Solving Word Problems Involving Percentage Increase and Decrease

Word problems often present percentage increase and decrease in real-world contexts. The key is to carefully identify the original value, the new value, and the percentage change. Always double-check your calculations and make sure your answer makes logical sense within the context of the problem.

Example 9: A Word Problem

A farmer's harvest increased from 1000 bushels to 1250 bushels. What is the percentage increase in his harvest?

1. Difference: 1250 - 1000 = 250
2. Divide by original: 250 / 1000 = 0.25
3. Multiply by 100%: 0.25 x 100% = 25%

The farmer's harvest increased by 25%.

Frequently Asked Questions (FAQ)

Q: Can I add percentages directly when dealing with multiple increases or decreases?

A: No, you cannot simply add or subtract percentages when dealing with multiple changes. You must apply each percentage change sequentially using the formulas provided.

Q: What if the percentage increase or decrease is more than 100%?

A: The formulas still apply. For example, a 150% increase means the new value is 2.5 times the original value (1 + 1.5 = 2.5).

Q: How can I check my work?

A: Always perform a reasonableness check. Does the answer make sense in the context of the problem? You can also work backward from the new value using the appropriate formula to verify the original value.

Q: What if I am given the new value and the percentage change, and I need to find the original value?

A: You can rearrange the formulas to solve for the original value:

• For Percentage Increase: Original Value = New Value / (1 + Percentage Increase as a decimal)
• For Percentage Decrease: Original Value = New Value / (1 - Percentage Decrease as a decimal)

Conclusion

Mastering percentage increase and decrease is crucial for navigating various aspects of daily life and professional endeavors. By understanding the fundamental formulas and practicing with different examples, you can build confidence in solving problems involving percentage changes, interpreting data, and making informed decisions based on numerical information. Remember to always break down the problem systematically, clearly identify the original and new values, and meticulously apply the appropriate formula. With consistent practice, you'll become proficient in handling any percentage increase or decrease calculation.