Solving Word Problems Involving Percentages

Introduction

Key Concepts

Understanding Percentages

A percentage represents a fraction of 100 and is denoted using the symbol "%". It is a way to express proportions and comparisons, making it easier to understand and communicate numerical information. In mathematical terms, a percentage can be converted to a decimal by dividing by 100. For example, $25\% = \frac{25}{100} = 0.25$.

Converting Between Percentages, Decimals, and Fractions

To effectively solve percentage problems, it's essential to convert between percentages, decimals, and fractions:

• Percentage to Decimal: Divide by 100. Example: $45\% = 0.45$.
• Decimal to Percentage: Multiply by 100. Example: $0.75 = 75\%$.
• Percentage to Fraction: Simplify the fraction formed by placing the percentage number over 100. Example: $60\% = \frac{60}{100} = \frac{3}{5}$.

Finding the Percentage of a Number

To find p% of a number N, multiply the number by the percentage expressed as a decimal:

**Example:** What is $20\%$ of $150$?

So, $20\%$ of $150$ is $30$.

Calculating Percentage Increase and Decrease

Percentage increase and decrease are used to express the change in a value relative to its original amount. The formulas are:

• Percentage Increase: $$\text{Percentage Increase} = \left(\frac{\text{Increase}}{\text{Original Value}}\right) \times 100\%$$
• Percentage Decrease: $$\text{Percentage Decrease} = \left(\frac{\text{Decrease}}{\text{Original Value}}\right) \times 100\%$$

**Example:** If a price increases from $80$ to $100$, the increase is $20$. The percentage increase is:

Finding the Original Number When Given the Percentage

If you know the percentage and the part, you can find the original number using the formula:

**Example:** $30$ is $25\%$ of what number?

So, the original number is $120$.

Percentage Change Problems

Percentage change problems involve calculating how much a value has increased or decreased relative to its original value. These problems often require setting up equations based on the percentage change formulas.

**Example:** A shirt originally costs $50$ dollars but is now sold at a $20\%$ discount. What is the sale price?

First, calculate the discount:

Then, subtract the discount from the original price:

So, the sale price is $40$ dollars.

Complex Percentage Word Problems

Complex percentage problems may involve multiple steps, such as successive percentage changes or combining different percentage calculations within a single problem.

**Example:** A population of a town increases by $10\%$ in the first year and then decreases by $5\%$ in the second year. If the original population was $2000$, what is the population after two years?

First, calculate the increase in the first year:

New population after the first year:

Next, calculate the decrease in the second year:

Population after the second year:

So, the population after two years is $2090$.

Real-Life Applications of Percentage Problems

Percentage problems are prevalent in various real-life contexts, including finance, statistics, and everyday transactions. Understanding how to solve these problems enhances financial literacy, aids in data interpretation, and supports informed decision-making.

• Financial Literacy: Calculating interest rates, discounts, taxes, and profit margins.
• Data Analysis: Interpreting statistical data and understanding trends.
• Everyday Transactions: Determining tips, budgeting, and comparing prices.

Strategies for Solving Percentage Word Problems

Effective problem-solving strategies can simplify percentage word problems:

• Identify What is Given and What is Asked: Determine the known values and what needs to be found.
• Convert Percentages to Decimals: Simplify calculations by converting percentages to their decimal form.
• Use Relevant Formulas: Apply the appropriate formulas based on the problem context.
• Set Up Equations: Translate the word problem into mathematical equations.
• Check Your Work: Verify calculations to ensure accuracy.

Common Mistakes to Avoid

Being aware of common pitfalls can improve accuracy in solving percentage problems:

• Incorrect Conversion: Failing to accurately convert percentages to decimals or fractions.
• Misunderstanding the Base: Confusing what value the percentage is based on.
• Calculation Errors: Making arithmetic mistakes during multiplication or division.
• Ignoring Units: Overlooking the context, such as money or population counts.

Practice Problems

Engaging with practice problems reinforces understanding and application of percentage concepts:

1. If a jacket costs $80$ dollars and is on sale for $25\%$ off, what is the sale price?
2. A school has $500$ students. If $40\%$ are involved in extracurricular activities, how many students are participating?
3. A laptop's price increases from $600$ dollars to $660$ dollars. What is the percentage increase?
4. During a sale, an item is first discounted by $15\%$ and then an additional $10\%$ off the reduced price. What is the final price of an item originally priced at $200$ dollars?
5. A population decreases by $5\%$ each year. If the current population is $10,000$, what will it be after two years?

**Solutions:**

1. Sale price = $80 - (80 \times 0.25) = 80 - 20 = 60$ dollars.
2. Number of students = $500 \times 0.40 = 200$ students.
3. Percentage increase = $\left(\frac{660 - 600}{600}\right) \times 100\% = \frac{60}{600} \times 100\% = 10\%$.
4. First discount: $200 \times 0.15 = 30$ dollars. New price = $200 - 30 = 170$ dollars. Second discount: $170 \times 0.10 = 17$ dollars. Final price = $170 - 17 = 153$ dollars.
5. After first year: $10,000 \times 0.05 = 500$ decrease. New population = $10,000 - 500 = 9,500$. After second year: $9,500 \times 0.05 = 475$ decrease. New population = $9,500 - 475 = 9,025$.

Comparison Table

Summary and Key Takeaways