Lesson 24 Percent And Rates Per 100

Let's explore the concepts of percentages and rates per 100, providing a solid understanding of how these mathematical tools are used in everyday scenarios.

Understanding Percentages

A percentage is a way of expressing a number as a fraction of 100. The word "percent" comes from the Latin per centum, meaning "out of one hundred." Percentages are used to represent proportions, ratios, or fractions in a standardized way, making it easier to compare different quantities.

The Basics of Percentages

• Definition: A percentage is a ratio or fraction expressed as a part of 100.
• Symbol: The symbol for percent is %.
• Conversion:
  • To convert a fraction to a percentage, multiply the fraction by 100.
  • To convert a decimal to a percentage, multiply the decimal by 100.
  • To convert a percentage to a fraction, divide the percentage by 100 and simplify.
  • To convert a percentage to a decimal, divide the percentage by 100.

Calculating Percentages

Calculating percentages involves several fundamental operations:

1. Finding a Percentage of a Number:

   To find a percentage of a number, convert the percentage to a decimal or fraction and then multiply it by the number.

   Example: What is 25% of 200?

   Solution:

   • Convert 25% to a decimal: 25% = 25/100 = 0.25
   • Multiply the decimal by the number: 0.25 * 200 = 50
   • Therefore, 25% of 200 is 50.

2. Finding What Percentage One Number Is of Another:

   To find what percentage one number is of another, divide the first number by the second number and then multiply by 100.

   Example: What percentage is 30 of 120?

   Solution:

   • Divide the first number by the second number: 30 / 120 = 0.25
   • Multiply the result by 100: 0.25 * 100 = 25%
   • Therefore, 30 is 25% of 120.

3. Finding the Whole When Given a Percentage and a Part:

   To find the whole when given a percentage and a part, divide the part by the percentage (expressed as a decimal).

   Example: 40% of what number is 80?

   Solution:

   • Convert 40% to a decimal: 40% = 40/100 = 0.40
   • Divide the part by the decimal: 80 / 0.40 = 200
   • Therefore, 40% of 200 is 80.

Common Applications of Percentages

1. Sales and Discounts:

   Retailers often use percentages to advertise discounts. For example, a "20% off" sale means that the price of each item is reduced by 20%.

   Example: A shirt is priced at $50, and there is a 20% discount. What is the final price?

   Solution:

   • Calculate the discount amount: 20% of $50 = 0.20 * $50 = $10
   • Subtract the discount from the original price: $50 - $10 = $40
   • The final price of the shirt is $40.

2. Interest Rates:

   Interest rates on loans and investments are often expressed as percentages. For example, a savings account might offer an annual interest rate of 2%.

   Example: You deposit $1000 in a savings account with a 2% annual interest rate. How much interest will you earn in one year?

   Solution:

   • Calculate the interest earned: 2% of $1000 = 0.02 * $1000 = $20
   • You will earn $20 in interest in one year.

3. Taxes:

   Tax rates are often expressed as percentages of income or sales. For example, a sales tax of 7% means that an additional 7% is added to the price of goods and services.

   Example: You buy a product for $100, and the sales tax is 7%. What is the total cost?

   Solution:

   • Calculate the sales tax amount: 7% of $100 = 0.07 * $100 = $7
   • Add the sales tax to the original price: $100 + $7 = $107
   • The total cost of the product is $107.

4. Statistics:

   Percentages are used to present statistical data in a clear and understandable way. For example, a survey might report that 60% of respondents prefer a certain product.

5. Grades:

   In education, grades are often expressed as percentages. For example, scoring 85 out of 100 on a test is equivalent to a grade of 85%.

Tips for Working with Percentages

1. Convert Percentages to Decimals:

   When performing calculations with percentages, it is often easier to convert the percentage to a decimal first. This avoids confusion and simplifies the arithmetic.

2. Understand the Context:

   Pay attention to the context in which percentages are used. Make sure you understand what the percentage is referring to.

3. Check Your Work:

   Always double-check your calculations to ensure accuracy. A small error in percentage calculations can lead to significant discrepancies.

4. Use Estimation:

   Before performing calculations, estimate the answer to get a sense of the expected result. This can help you identify errors and ensure that your answer is reasonable.

Rates Per 100

Rates per 100 are a specific way of expressing ratios or proportions, similar to percentages, but with a focus on the number of occurrences per 100 units. This method is commonly used in various fields, including healthcare, demographics, and finance, to provide standardized and easily comparable measures.

Understanding Rates Per 100

• Definition: A rate per 100 is the number of occurrences of an event or characteristic for every 100 units in a population or sample.
• Purpose: To standardize rates for easy comparison across different populations or time periods.
• Calculation: Divide the number of occurrences by the total population or sample size, then multiply by 100.

Common Applications of Rates Per 100

1. Healthcare:

   In healthcare, rates per 100 are used to express various health-related statistics, such as disease prevalence, incidence rates, and mortality rates.

   Example: A study finds that 5 out of every 100 people in a town have diabetes. The prevalence rate of diabetes is 5 per 100.

2. Demographics:

   Demographers use rates per 100 to express birth rates, death rates, and other population characteristics.

   Example: A country has 2 births per 100 people each year. The birth rate is 2 per 100.

3. Finance:

   In finance, rates per 100 can be used to express returns on investment or interest rates on loans. While percentages are more common, rates per 100 provide a similar standardized measure.

   Example: An investment yields $8 in profit for every $100 invested. The return rate is 8 per 100.

4. Crime Statistics:

   Crime rates are often expressed as the number of crimes per 100 people, providing a standardized measure of crime prevalence across different regions.

   Example: A city reports 3 crimes per 100 residents. The crime rate is 3 per 100.

5. Education:

   Dropout rates or graduation rates can be expressed per 100 students to compare educational outcomes across different schools or districts.

   Example: A school has 95 students graduating for every 100 enrolled. The graduation rate is 95 per 100.

Calculating Rates Per 100

To calculate a rate per 100, use the following formula:

Rate per 100 = (Number of Occurrences / Total Population or Sample Size) * 100

Example: In a town of 5,000 people, there are 150 reported cases of a particular disease. What is the rate of the disease per 100 people?

Solution:

• Number of Occurrences = 150
• Total Population = 5,000
• Rate per 100 = (150 / 5,000) * 100 = 3
• The rate of the disease is 3 per 100 people.

Advantages of Using Rates Per 100

1. Standardization:

   Rates per 100 provide a standardized measure that allows for easy comparison across different populations or time periods.

2. Clarity:

   Expressing rates per 100 makes it easier to understand the prevalence or incidence of a particular event or characteristic.

3. Comparability:

   Rates per 100 allow for meaningful comparisons between groups with different sizes.

4. Ease of Interpretation:

   The rate per 100 is straightforward to interpret, making it accessible to a wide audience.

Comparison of Percentages and Rates Per 100

While percentages and rates per 100 are both used to express proportions, there are subtle differences in their application:

• Percentages: Are generally used to express a part of a whole, often in the context of proportions, discounts, and changes.
• Rates Per 100: Are typically used to express the occurrence of an event or characteristic within a population, emphasizing standardization and comparability.

In many cases, percentages and rates per 100 can be used interchangeably, but the choice often depends on the context and the specific information being conveyed.

Practical Examples and Exercises

To reinforce your understanding of percentages and rates per 100, consider the following examples and exercises:

Example 1: Calculating a Discount

A store is offering a 30% discount on a TV that is originally priced at $800. What is the sale price of the TV?

Solution:

• Convert the percentage to a decimal: 30% = 0.30
• Calculate the discount amount: 0.30 * $800 = $240
• Subtract the discount from the original price: $800 - $240 = $560
• The sale price of the TV is $560.

Example 2: Calculating Sales Tax

You purchase a computer for $1200, and the sales tax rate is 6%. What is the total cost of the computer?

Solution:

• Convert the percentage to a decimal: 6% = 0.06
• Calculate the sales tax amount: 0.06 * $1200 = $72
• Add the sales tax to the original price: $1200 + $72 = $1272
• The total cost of the computer is $1272.

Example 3: Calculating Disease Prevalence

In a city of 25,000 people, there are 500 reported cases of influenza. What is the rate of influenza per 100 people?

Solution:

• Number of Occurrences = 500
• Total Population = 25,000
• Rate per 100 = (500 / 25,000) * 100 = 2
• The rate of influenza is 2 per 100 people.

Exercise 1

A company's revenue increased from $500,000 to $600,000. What is the percentage increase in revenue?

Exercise 2

If 15% of a number is 45, what is the number?

Exercise 3

A school has 1,500 students, and 120 of them participate in the debate club. What is the participation rate in the debate club per 100 students?

Advanced Applications

As you become more comfortable with percentages and rates per 100, you can explore more advanced applications:

1. Compound Interest:

   Compound interest involves earning interest on both the principal amount and the accumulated interest. Understanding percentages is crucial for calculating compound interest accurately.

2. Statistical Analysis:

   Percentages and rates per 100 are fundamental tools in statistical analysis, used for calculating confidence intervals, hypothesis testing, and data interpretation.

3. Financial Planning:

   Percentages are used extensively in financial planning for budgeting, investment analysis, and retirement planning.

4. Risk Assessment:

   Rates per 100 are used in risk assessment to evaluate the likelihood of adverse events, such as accidents, illnesses, or financial losses.

Conclusion

Mastering percentages and rates per 100 is essential for various real-world applications, from calculating discounts and taxes to interpreting statistical data and making informed financial decisions. By understanding the basics, practicing calculations, and exploring advanced applications, you can enhance your mathematical literacy and problem-solving skills. These tools provide a standardized and accessible way to express proportions and rates, making them invaluable in a wide range of fields.