Percentage word problems with step-by-step solutions

📚 Understanding Percentage Word Problems

Percentage word problems can seem daunting, but they become much easier when broken down into manageable steps. A percentage is simply a way of expressing a number as a fraction of 100. The word "percent" means "per hundred." The key to solving these problems is to identify the 'whole', the 'part', and the 'percentage'.

📜 A Brief History of Percentages

The concept of percentages dates back to ancient Rome. As the Roman Empire expanded, taxation became essential. Romans often calculated taxes as fractions of goods. However, the modern percentage symbol (%) emerged much later, evolving from abbreviations used in 15th-century commercial arithmetic. The widespread adoption of the decimal system further solidified the use of percentages.

🔑 Key Principles for Solving Percentage Problems

• 🔍 Identify the Whole, Part, and Percentage: Most percentage problems involve these three components. The 'whole' is the total amount, the 'part' is a portion of the whole, and the 'percentage' is the ratio of the part to the whole, expressed as a percentage.
• ✍️ Translate the Words into an Equation: Look for keywords such as 'is' (equals), 'of' (multiplication), and 'what' (unknown variable). For example, '15% of 200 is what?' translates to $0.15 \times 200 = x$.
• 🧮 Solve for the Unknown: Use basic algebraic principles to isolate the unknown variable.
• 🔄 Convert Percentages to Decimals or Fractions: Before performing calculations, convert the percentage to either a decimal (by dividing by 100) or a fraction (by placing it over 100 and simplifying).
• ✅ Check Your Answer: Make sure your answer makes sense in the context of the problem. If the percentage is small, the part should be significantly smaller than the whole.

💡 Real-World Examples with Step-by-Step Solutions

Let's explore some common types of percentage word problems:

Example 1: Finding a Percentage of a Number

Problem: What is 20% of 75?

Solution:

1. Convert the percentage to a decimal: 20% = 0.20
2. Multiply the decimal by the number: $0.20 \times 75 = 15$
3. Answer: 20% of 75 is 15.

Example 2: Finding What Percentage One Number Is of Another

Problem: 12 is what percent of 48?

Solution:

1. Set up the equation: $\frac{12}{48} = x$
2. Solve for $x$: $x = 0.25$
3. Convert the decimal to a percentage: $0.25 = 25\%$
4. Answer: 12 is 25% of 48.

Example 3: Finding the Whole When a Percentage is Known

Problem: 30 is 60% of what number?

Solution:

1. Set up the equation: $30 = 0.60 \times x$
2. Solve for $x$: $x = \frac{30}{0.60} = 50$
3. Answer: 30 is 60% of 50.

Example 4: Percentage Increase

Problem: A price increased from $20 to $25. What is the percentage increase?

Solution:

1. Calculate the amount of the increase: $25 - 20 = $5
2. Divide the increase by the original amount: $\frac{5}{20} = 0.25$
3. Convert the decimal to a percentage: $0.25 = 25\%$
4. Answer: The price increased by 25%.

Example 5: Percentage Decrease

Problem: A price decreased from $50 to $40. What is the percentage decrease?

Solution:

1. Calculate the amount of the decrease: $50 - 40 = $10
2. Divide the decrease by the original amount: $\frac{10}{50} = 0.20$
3. Convert the decimal to a percentage: $0.20 = 20\%$
4. Answer: The price decreased by 20%.

Example 6: Discount Problems

Problem: A shirt costs $30, and it's on sale for 15% off. What is the sale price?

Solution:

1. Calculate the amount of the discount: $0.15 \times 30 = $4.50
2. Subtract the discount from the original price: $30 - 4.50 = $25.50
3. Answer: The sale price is $25.50.

Example 7: Commission Problems

Problem: A salesperson earns a 5% commission on sales. If they sell $5000 worth of merchandise, what is their commission?

Solution:

1. Calculate the commission: $0.05 \times 5000 = $250
2. Answer: The salesperson's commission is $250.

📝 Practice Quiz

Test your understanding with these practice problems:

1. What is 35% of 80?
2. 24 is what percent of 60?
3. 45 is 90% of what number?
4. A store marks up a product by 20%. If the original cost was $50, what is the selling price?
5. After a 30% discount, a jacket costs $70. What was the original price?
6. A student scored 85 out of 100 on a test. What is their percentage score?
7. If a population increases by 10% each year, and the initial population is 1000, what is the population after 2 years?

🎉 Conclusion

By understanding the basic principles and practicing with different types of problems, you can master percentage word problems. Remember to identify the key components, translate the words into equations, and double-check your answers!