Expressing One Quantity as a Percentage of Another

Introduction

Key Concepts

Definition of Percentage

A percentage represents a part of a whole expressed out of 100. It is a way to describe proportions, making it easier to compare different quantities. The term "percent" originates from the Latin "per centum," meaning "by the hundred."

Calculating Percentage

To calculate the percentage of one quantity relative to another, use the following formula:

For example, if there are 25 apples in a basket and 5 are red, the percentage of red apples is:

Converting Between Percentages and Decimals

Percentages can be converted to decimals by dividing by 100, and vice versa. This is useful in various mathematical calculations and real-life applications.

For instance:

• Convert 45% to a decimal: $45 \div 100 = 0.45$
• Convert 0.75 to a percentage: $0.75 \times 100 = 75\%$

Percentage Increase and Decrease

Percentage increase and decrease measure how much a quantity has grown or shrunk in comparison to its original value.

The formulas are:

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

For example, if a product's price increases from $50 to $60:

There is a 20% increase in price.

Percentage Change Formula

The general formula to determine percentage change, whether increase or decrease, is:

Where:

• Change = New Value - Original Value
• If the result is positive, it's a percentage increase.
• If the result is negative, it's a percentage decrease.

Applications of Percentages

Percentages are widely used in various fields such as finance (interest rates, discounts), statistics (probability, data analysis), and everyday life (sales tax, tips). Understanding how to express one quantity as a percentage of another is crucial for making informed decisions and interpreting information accurately.

Percentage Problems and Solutions

Let's explore a few examples to illustrate how to apply these concepts:

1. Example 1: A class has 40 students, and 10 of them play basketball. What percentage of the class plays basketball?

   Solution:

   $$ \left( \frac{10}{40} \right) \times 100 = 25\% $$

   25% of the class plays basketball.

2. Example 2: A company's revenue increased from $200,000 to $250,000. Calculate the percentage increase.

   Solution:

   $$ \left( \frac{250,000 - 200,000}{200,000} \right) \times 100 = 25\% $$

   The revenue increased by 25%.

3. Example 3: The population of a town decreased from 5,000 to 4,500. What is the percentage decrease?

   Solution:

   $$ \left( \frac{5,000 - 4,500}{5,000} \right) \times 100 = 10\% $$

   The population decreased by 10%.

Understanding Ratios and Percentages

Ratios and percentages are both ways to compare quantities, but they are used differently. A ratio compares two quantities directly, whereas a percentage represents the ratio as a part of 100. Converting ratios to percentages can simplify comparisons and make them more intuitive.

For example, the ratio of 2:3 can be expressed in percentages as approximately 40% to 60%.

Percentage in Real-Life Contexts

Understanding percentages aids in interpreting data in real-life scenarios such as calculating taxes, determining discounts during sales, analyzing statistical data, and managing personal finances. For instance, knowing how to calculate the percentage of a discount can help determine the final price of an item during a sale.

Advanced Concepts

Theoretical Foundations of Percentages

Percentages are deeply rooted in the concept of proportionality and are a specific case of fractions where the denominator is 100. This relationship is foundational in various areas of mathematics, including algebra, calculus, and statistics.

In algebra, percentages are used to solve equations involving relative change and scaling. In calculus, the concept of rates of change and derivatives often involves percentage-based interpretations. In statistics, percentages are crucial for representing data distributions, probabilities, and in the creation of charts and graphs.

Mathematically, the percentage is a dimensionless quantity, allowing it to be universally applicable across different units and measures. This universality makes percentages a versatile tool in comparing diverse datasets.

Mathematical Derivations and Proofs

Consider the derivation of the basic percentage formula from the concept of ratios. Let’s denote:

• Part: The quantity being compared.
• Whole: The total or reference quantity.

The ratio of Part to Whole can be expressed as:

To convert this ratio into a percentage:

This derivation shows how percentages are a direct scaling of the ratio to a base of 100, facilitating easier interpretation and comparison.

Complex Percentage Problems

Advanced percentage problems often involve multiple steps, require the integration of various mathematical concepts, and may involve indirect reasoning. Let’s explore a few such problems:

1. Example 4: A store offers a 15% discount on an item and then applies a 5% sales tax on the discounted price. If the original price of the item is $200, calculate the final price after discount and tax.

   Solution:

   First, calculate the discount:

   $$ \text{Discount} = 200 \times 0.15 = 30 $$

   Discounted price:

   $$ 200 - 30 = 170 $$

   Next, calculate the sales tax:

   $$ \text{Sales Tax} = 170 \times 0.05 = 8.5 $$

   Final price:

   $$ 170 + 8.5 = 178.5 $$

   The final price is $178.50.

2. Example 5: The population of a city grows by 8% annually. If the current population is 50,000, what will be the population after 3 years, assuming a constant growth rate?

   Solution:

   Use the compound growth formula:

   $$ P = P_0 \times (1 + r)^t $$

   Where:

   • P = future population
   • P₀ = initial population (50,000)
   • r = growth rate (0.08)
   • t = time in years (3)

   Calculating:

   $$ P = 50,000 \times (1 + 0.08)^3 \approx 50,000 \times 1.2597 \approx 62,985 $$

   The population after 3 years is approximately 62,985.

3. Example 6: A student scored 72 out of 90 in a test. The teacher decides to curve the scores by increasing each score by 10%. What is the student's new score?

   Solution:

   Calculate the percentage increase:

   $$ \text{Increase} = 72 \times 0.10 = 7.2 $$

   New score:

   $$ 72 + 7.2 = 79.2 $$

   Therefore, the student's new score is 79.2 out of 90.

Percentage in Financial Mathematics

In financial mathematics, percentages are pivotal in understanding interest rates, investment growth, and loan repayments. Concepts such as simple interest, compound interest, and annual percentage rates (APR) all rely on percentage calculations.

For example, calculating the future value of an investment using compound interest involves applying the percentage growth rate over multiple periods:

Where:

• A = the amount of money accumulated after n years, including interest.
• P = the principal amount (initial investment).
• r = annual interest rate (decimal).
• n = number of times that interest is compounded per year.
• t = time the money is invested for in years.

Interdisciplinary Connections

Understanding percentages connects to various academic disciplines:

• Economics: Analyzing market trends, inflation rates, and economic growth.
• Science: Calculating concentrations in chemistry, growth rates in biology.
• Social Studies: Interpreting demographic data, election results.
• Health: Assessing statistical data in epidemiology, nutrition percentages.

These connections demonstrate the versatility and importance of mastering percentage calculations across different fields.

Challenging Theorems and Proofs Involving Percentages

While percentages are primarily a tool for practical calculations, they also play a role in more abstract mathematical theorems and proofs. For instance, in probability theory, percentages can represent probabilities expressed out of 100, linking percentages to the foundational principles of combinatorics and probability distributions.

Another example is in geometric series, where percentages can represent the rate of convergence or divergence of the series. Understanding the interplay between percentages and geometric sequences can lead to deeper insights into series summation and limits.

Advanced Problem-Solving Techniques

Advanced problem-solving involving percentages may require the application of algebraic manipulation, understanding of proportional reasoning, and the ability to set up and solve equations based on percentage relationships. For example, solving for an unknown percentage given multiple related quantities demands a structured approach to identify knowns and unknowns and apply relevant formulas effectively.

Consider the following problem:

• Problem: If 30% of a number is equal to 45, find the number.

Solution:

Let the number be x. According to the problem:

Solving for x:

The number is 150.

Comparison Table

Summary and Key Takeaways