Solved problems: Finding the percentage part explained

📚 Understanding Percentage Parts

Finding the percentage part is a fundamental skill in mathematics with wide-ranging applications, from calculating discounts to understanding statistics. It involves determining a specific portion of a whole, expressed as a percentage. This guide provides a comprehensive overview of the concept, its history, underlying principles, and practical applications.

📜 History and Background

The concept of percentages has ancient roots, tracing back to the Roman Empire. Roman emperors often levied taxes expressed as fractions of assets. As trade and commerce flourished during the Middle Ages, the need for a standardized way to express proportions grew. The symbol '%' evolved from the Italian term 'per cento,' meaning 'out of one hundred.'

🔑 Key Principles

• 🔢 Definition: The percentage part is the amount that results from applying a percentage to a given whole. It answers the question: "What is X% of Y?"
• ➗ Formula: The formula to find the percentage part is: $\text{Percentage Part} = \frac{\text{Percentage}}{100} \times \text{Whole}$.
• ⚖️ Understanding Proportions: Percentages are essentially proportions. X% is equivalent to the fraction $\frac{X}{100}$.
• 🔄 Converting Percentages to Decimals: To easily use percentages in calculations, convert them to decimals by dividing by 100. For example, 25% becomes 0.25.

🌍 Real-World Examples

Here are some examples to illustrate how to calculate the percentage part:

1. Discount Calculation: A store offers a 20% discount on a shirt priced at $25. What is the amount of the discount?
   • 💰 Identify the whole: $25 (original price)
   • 📈 Identify the percentage: 20%
   • 🧮 Calculate the percentage part: $\frac{20}{100} \times 25 = 0.20 \times 25 = $5$. The discount is $5.
2. Sales Tax: A purchase of $150 is subject to a 6% sales tax. What is the amount of the sales tax?
   • 🛍️ Identify the whole: $150 (purchase price)
   • 🧾 Identify the percentage: 6%
   • 📊 Calculate the percentage part: $\frac{6}{100} \times 150 = 0.06 \times 150 = $9$. The sales tax is $9.
3. Test Score: A student scores 80% on a test with 50 questions. How many questions did the student answer correctly?
   • 📝 Identify the whole: 50 (total questions)
   • 💯 Identify the percentage: 80%
   • ✅ Calculate the percentage part: $\frac{80}{100} \times 50 = 0.80 \times 50 = 40$. The student answered 40 questions correctly.

💡 Tips and Tricks

• 🧠 Estimation: Before calculating, estimate the answer to check if your final result is reasonable. For instance, 10% of a number is easy to calculate (just move the decimal one place to the left).
• 🔍 Breaking Down Percentages: Sometimes, it's easier to break down the percentage into smaller, manageable parts. For example, to find 15% of a number, you can find 10% and 5% (half of 10%) and add them together.
• 📝 Practice Regularly: The key to mastering percentage calculations is consistent practice.

📝 Practice Quiz

Solve the following problems to test your understanding:

1. What is 25% of 80?
2. What is 75% of 120?
3. What is 10% of 350?
4. What is 50% of 92?
5. What is 1% of 1000?
6. What is 30% of 60?
7. What is 90% of 200?

✅ Conclusion

Understanding how to find the percentage part is a valuable skill that can be applied in numerous real-life scenarios. By mastering the basic formula and practicing regularly, you can confidently solve percentage problems and make informed decisions in various situations.