Percentage Change Word Problems
Core formulas:
% change = \(((\text{New} - \text{Old})/\text{Old}) \times 100\%\)
Apply change: New = Base × (1 ± p/100), where “+” is increase and “−” is decrease
Find original: Original = Final ÷ (1 ± p/100)
Successive changes: Factor = ∏(1 ± pᵢ/100)
Percentage Change Word Problems Calculator
Percentage change word problems are a common way to apply math to real-life situations. They focus on finding how much a value has increased or decreased relative to its original amount, expressed as a percentage.
Whether it’s analyzing discounts, price increases, population growth, or test score improvements, these problems give us insight into change over time or between two values. Mastering percentage change word problems is essential for students, professionals, and anyone who wants to understand everyday financial and numerical information.
What Is Percentage Change?
Percentage change measures the relative difference between an original value and a new value. It tells us how much a quantity has increased or decreased compared to its starting point, expressed as a percent.
The general formula is:
Percentage Change = (New Value − Original Value) ÷ Original Value × 100%
- If the result is positive, it represents a percentage increase.
- If the result is negative, it represents a percentage decrease.
Steps to Solve Percentage Change Word Problems
- Identify the original value and the new value.
- Subtract the original value from the new value.
- Divide the difference by the original value.
- Multiply by 100% to convert into a percentage.
- Determine whether it is an increase or a decrease.
Common Scenarios in Word Problems
1. Price Increases and Decreases
Stores often raise or reduce prices. Word problems help calculate how much more or less customers pay after a change.
2. Salary Changes
Workers track raises or pay cuts through percentage change calculations.
3. Population Growth or Decline
Demographers use percentage change to study how populations change across years.
4. Academic Performance
Students calculate percentage increases in test scores to measure improvement.
5. Business Profits
Companies use percentage change to analyze growth or decline in sales or profits.
Worked Examples of Percentage Change Word Problems
Example 1: Price Increase
The price of a movie ticket increased from $10 to $12. Find the percentage increase.
New Value − Original Value = 12 − 10 = 2
2 ÷ 10 = 0.2 → 0.2 × 100 = 20%
Answer: The price increased by 20%.
Example 2: Price Decrease
A dress originally cost $80 but is now on sale for $60. Find the percentage decrease.
New Value − Original Value = 60 − 80 = −20
−20 ÷ 80 = −0.25 → −0.25 × 100 = −25%
Answer: The price decreased by 25%.
Example 3: Salary Increase
Sam’s monthly salary increased from $2,000 to $2,400. What is the percentage change?
2400 − 2000 = 400
400 ÷ 2000 = 0.2 → 0.2 × 100 = 20%
Answer: Sam’s salary increased by 20%.
Example 4: Salary Decrease
Maria’s pay dropped from $1,500 to $1,200. Find the percentage decrease.
1200 − 1500 = −300
−300 ÷ 1500 = −0.2 → −0.2 × 100 = −20%
Answer: Maria’s salary decreased by 20%.
Example 5: Population Growth
A town’s population grew from 5,000 to 6,500 in ten years. Find the percentage increase.
6500 − 5000 = 1500
1500 ÷ 5000 = 0.3 → 0.3 × 100 = 30%
Answer: The population increased by 30%.
Example 6: Exam Scores
A student’s score improved from 72 to 90. Find the percentage increase.
90 − 72 = 18
18 ÷ 72 = 0.25 → 0.25 × 100 = 25%
Answer: The score increased by 25%.
Example 7: Business Profits
A company’s profit decreased from $50,000 to $42,000. Find the percentage change.
42,000 − 50,000 = −8,000
−8,000 ÷ 50,000 = −0.16 → −0.16 × 100 = −16%
Answer: The profit decreased by 16%.
Tips for Solving Word Problems
- Always identify the “original value.” The percentage change is based on this number, not the new one.
- Be careful with decreases. The answer will be negative, but word problems usually ask for the magnitude (absolute percentage).
- Double-check arithmetic. Errors in subtraction or division lead to wrong answers.
- Practice real-world examples. The more you connect problems to daily life, the easier they become.
Applications of Percentage Change
1. Finance
Percentage change is used to track investment growth, stock market performance, and interest rates.
2. Economics
Governments use percentage change to measure inflation, GDP growth, and unemployment shifts.
3. Retail
Shoppers and businesses calculate percentage changes to understand discounts, markdowns, and seasonal price changes.
4. Science
Researchers use percentage change in experiments to measure how much a quantity increases or decreases under specific conditions.
Common Mistakes in Word Problems
- Using the new value instead of the original value when dividing.
- Forgetting to multiply by 100 to convert to a percentage.
- Not identifying whether the problem is asking for increase or decrease.
- Mixing up subtraction order (new − original).
Practice Word Problems
- A car originally worth $15,000 is now worth $12,000. What is the percentage decrease?
- The number of students in a class increased from 40 to 52. What is the percentage increase?
- A phone’s price rose from $600 to $750. Find the percentage change.
- A company’s expenses dropped from $20,000 to $17,500. What is the percentage change?
- A person’s weight went from 180 pounds to 165 pounds. What is the percentage decrease?
Conclusion
Percentage change word problems are everywhere—in shopping, salaries, business, finance, and science. By learning to identify the original and new values, subtracting carefully, and dividing correctly, you can confidently solve these problems.
They not only sharpen mathematical skills but also prepare you to make informed decisions in everyday life. With consistent practice, percentage change calculations become second nature, transforming complex scenarios into simple, logical steps.
Frequently Asked Questions
What is the difference between percentage increase and percentage decrease?
A percentage increase happens when the new value is greater than the original, while a percentage decrease occurs when the new value is smaller.
Why do we divide by the original value?
The original value serves as the reference point to measure how large or small the change is compared to where it started.
Do negative percentages always mean decrease?
Yes, a negative percentage change indicates that the value went down compared to the original.
Can percentage change exceed 100%?
Yes, if the new value is more than double the original value, the percentage change will exceed 100%.
How do multiple changes affect the result?
If there are two consecutive increases or decreases, you must calculate them step by step, since percentage changes are not directly additive.