A ratio comparing a number to 100 is called a percentage. compare.edu.vn provides comprehensive comparisons to help you understand the nuances of percentages and their applications. Percentages are widely used in various fields, offering a standardized way to express proportions and rates, aiding informed decision-making. Explore insightful comparisons on our site to master the language of proportions, including ratios, rates, and percentage points.
1. Understanding Ratios: A Comprehensive Guide
A ratio is a way to compare two quantities. The basic formula for a ratio is:
Ratio = Number or rate of events, items, persons, etc. in one group / Number or rate of events, items, persons, etc. in another group
When you divide the numerator by the denominator, the result is often expressed as “to one” (e.g., x to 1) or written as “:1” (e.g., x:1).
Ratios are versatile and can compare different categories of the same variable (like males and females) or completely different variables (like the number of hospitals and the population size in a city).
1.1. Example: Calculating a Ratio with Different Categories of the Same Variable
Consider a study from the National Health and Nutrition Examination Survey (NHANES) conducted between 1971 and 1975, which followed 7,381 individuals aged 40–77 years. Participants were classified as either having diabetes or not. Follow-up data from 1982–1984 documented whether the enrollees were alive or deceased.
| Participant | Original Enrollment (1971–1975) | Dead at Follow-Up (1982–1984) |
|---|---|---|
| Diabetic men | 189 | 100 |
| Nondiabetic men | 3,151 | 811 |
| Diabetic women | 218 | 72 |
| Nondiabetic women | 3,823 | 511 |
To calculate the ratio of non-diabetic to diabetic men:
Ratio = 3,151 / 189 × 1 = 16.7:1
This means there were 16.7 non-diabetic men for every diabetic man in the study.
1.2. Properties and Uses of Ratios
- Ratios are commonly used descriptive measures in various fields, including epidemiology, where they serve as both descriptive measures and analytic tools.
- As descriptive measures, ratios can describe the male-to-female ratio in a study or the ratio of controls to cases (e.g., two controls per case).
- As analytic tools, ratios can be calculated for the occurrence of illness, injury, or death between two groups. These include measures like risk ratio (relative risk), rate ratio, and odds ratio.
- Ratios can compare related or unrelated numerators and denominators, such as comparing the number of males to females or the number of residents to the number of hospitals.
- Typically, either the numerator or denominator is set to 1.0 for easier interpretation, like reporting the ratio of non-diabetics to diabetics as 16.7:1 rather than 3,151:189.
1.3. Examples: Calculating Ratios for Different Variables
Example A: A city with 4,000,000 residents has 500 clinics. Calculate the ratio of clinics per person.
Ratio = 500 / 4,000,000 × 10n = 0.000125 clinics per person
To make this more understandable, set 10n = 10,000:
0. 000125 × 10,000 = 1.25 clinics per 10,000 persons
Alternatively, divide each value by 1.25 to express the ratio as 1 clinic for every 8,000 persons.
Example B: In 2001, Delaware’s infant mortality rate was 10.7 per 1,000 live births, while New Hampshire’s was 3.8 per 1,000 live births. Calculate the ratio of Delaware’s infant mortality rate to New Hampshire’s.
Ratio = 10.7 / 3.8 × 1 = 2.8:1
This indicates that Delaware’s infant mortality rate was 2.8 times higher than New Hampshire’s in 2001.
1.4. A Commonly Used Epidemiologic Ratio: Death-to-Case Ratio
The death-to-case ratio is calculated by dividing the number of deaths attributed to a specific disease during a period by the number of new cases identified during the same period. It serves as a measure of illness severity. For example, rabies has a death-to-case ratio close to 1 (almost everyone who gets rabies dies), while the common cold has a ratio close to 0.
In the United States in 2002, there were 15,075 new cases of tuberculosis reported, and 802 deaths were attributed to it. The death-to-case ratio for tuberculosis in 2002 is:
Ratio = 802 / 15,075
Dividing both numerator and denominator by the numerator yields 1 death per 18.8 new cases. Dividing both by the denominator and multiplying by 100 yields 5.3 deaths per 100 new cases. Both expressions are correct.
It’s important to note that many who died in 2002 may have contracted tuberculosis years earlier, making the death-to-case ratio a ratio but not a proportion.
2. Proportions: Understanding the Part-to-Whole Relationship
A proportion is a comparison of a part to the whole. It is a specific type of ratio where the numerator is included in the denominator. Proportions are used to describe what fraction of a group has a particular characteristic. For instance, the proportion of clinic patients who tested positive for HIV or the percentage of a population under 25 years old. A proportion can be expressed as a decimal, a fraction, or a percentage.
2.1. Method for Calculating a Proportion
The formula for calculating a proportion is:
Proportion = (Number of persons or events with a particular characteristic / Total number of persons or events, of which the numerator is a subset) × 10n
Typically, 10n is 100, and the proportion is expressed as a percentage.
2.2. Example: Calculating a Proportion
Example A: Calculate the proportion of men in the NHANES follow-up study who had diabetes.
- Numerator = 189 diabetic men
- Denominator = Total number of men = 189 + 3,151 = 3,340
Proportion = (189 / 3,340) × 100 = 5.66%
Example B: Calculate the proportion of deaths among men.
- Numerator = deaths in men = 100 (diabetic men) + 811 (nondiabetic men) = 911 deaths
Note that the numerator (911 deaths in men) is a subset of the denominator. - Denominator = all deaths = 911 (men) + 72 (diabetic women) + 511 (nondiabetic women) = 1,494 deaths
Proportion = (911 / 1,494) × 100 = 60.98% ≈ 61%
Your Turn: What proportion of all study participants were men?
(Answer = 45.25%)
2.3. Properties and Uses of Proportions
- Proportions are common descriptive measures used across various fields. In epidemiology, they are primarily used as descriptive measures. Examples include the proportion of eligible individuals enrolled in a study (the “participation rate”), the proportion of children in a village vaccinated against measles, or the proportion of passengers on a cruise ship who developed an illness.
- Proportions are also used to describe the amount of disease that can be attributed to a particular exposure. Public health officials, based on studies of smoking and lung cancer, estimate that over 90% of lung cancer cases are attributable to cigarette smoking.
- In a proportion, the numerator must be included in the denominator. For instance, the number of apples divided by the number of oranges is not a proportion, but the number of apples divided by the total number of fruits is a proportion.
- A proportion can be expressed as a fraction, a decimal, or a percentage. The statements “one-fifth of the residents became ill” and “twenty percent of the residents became ill” are equivalent.
- Proportions can be easily converted to ratios. If 179 women attended a clinic out of 341 attendees, the proportion of women is 179 / 341, or 52%. To convert this to a ratio, subtract the numerator from the denominator to find the number of men (341 − 179 = 162).
Ratio = 179 / (341 − 179) = 179 / 162 = 1.1:1 female-to-male ratio
Conversely, if a ratio’s numerator and denominator together make up a whole population, the ratio can be converted to a proportion by adding the ratio’s numerator and denominator to form the denominator of the proportion.
2.4. A Specific Type of Epidemiologic Proportion: Proportionate Mortality
Proportionate mortality refers to the proportion of deaths in a specified population during a period that are attributable to different causes. Each cause is expressed as a percentage of all deaths, and the sum of the causes adds up to 100%. These proportions are not rates because the denominator is all deaths, not the size of the population in which the deaths occurred.
Table 3.1 presents the primary causes of death in the United States in 2003 for persons of all ages and for those aged 25–44 years, including the number of deaths, proportionate mortality, and rank.
Table 3.1: Number, Proportionate Mortality, and Ranking of Deaths for Leading Causes of Death, All Ages and 25–44 Year Age Group — United States, 2003
| Cause of Death | All Ages | Ages 25–44 Years |
|---|---|---|
| Number | Percentage | |
| All causes | 2,443,930 | 100 |
| Diseases of heart | 684,462 | 28.0 |
| Malignant neoplasms | 554,643 | 22.7 |
| Accidents (unintentional injuries) | 105,695 | 4.3 |
| Intentional self-harm (suicide) | 30,642 | 1.3 |
| Assault (homicide) | 17,096 | 0.7 |
| HIV disease | 13,544 | 0.5 |
* Not among top-ranked causes
Data Sources: Centers for Disease Control and Prevention, 2003; National Vital Statistics Reports, 2005.
For example, the proportionate mortality for HIV was 0.5% among all age groups and 5.3% among those aged 25–44 years. This means that HIV infection accounted for 0.5% of all deaths and 5.3% of deaths among 25–44 year olds.
3. Rates: Measuring Frequency Over Time
In epidemiology, a rate is a measure of the frequency with which an event occurs in a defined population over a specified period. Rates are crucial for comparing disease frequency across different locations, times, or groups of people with varying population sizes, making it a measure of risk.
3.1. Understanding Rates
To a non-epidemiologist, rate implies how fast something is happening or going. Like a car’s speedometer indicates speed in miles or kilometers per hour. Epidemiologists often use the term rate for measures expressed per unit of time, such as 70 new breast cancer cases per 1,000 women per year. This is an incidence rate, describing how quickly disease occurs in a population, implying the pattern is ongoing.
However, some epidemiologists use the term rate more loosely for proportions with case counts in the numerator and population size in the denominator. For example, an attack rate is the proportion of the population that develops illness during an outbreak (also known as incidence proportion). A prevalence rate is the proportion of the population with a health condition at a point in time. A case-fatality rate is the proportion of persons with the disease who die from it.
3.2. Common Epidemiologic Measures
Table 3.2 summarizes some common epidemiologic measures categorized as ratios, proportions, or rates.
Table 3.2: Epidemiologic Measures Categorized as Ratio, Proportion, or Rate
| Category | Ratio | Proportion | Rate |
|---|---|---|---|
| Morbidity (Disease) | Risk ratio (Relative risk), Rate ratio, Odds ratio, Period prevalence | Attack rate (Incidence proportion), Secondary attack rate, Point prevalence, Attributable proportion | Person-time incidence rate |
| Mortality (Death) | Death-to-case ratio | Proportionate mortality | Crude mortality rate, Case-fatality rate, Cause-specific mortality rate, Age-specific mortality rate, Maternal mortality rate, Infant mortality rate |
| Natality (Birth) | Crude birth rate, Crude fertility rate |
4. A Ratio Comparing a Number to 100 Name: Percentage
A ratio that compares a number to 100 is most commonly known as a percentage. The term “percent” comes from the Latin “per centum,” meaning “out of one hundred.” Therefore, a percentage is a way of expressing a number as a fraction of 100. It is a standardized way to represent proportions, making it easy to understand relative sizes and changes.
4.1. How Percentages Are Calculated
To express a ratio as a percentage, you multiply the ratio by 100.
Percentage = (Value / Total Value) × 100
For example, if you have 20 apples out of a total of 50 fruits, the percentage of apples is:
(20 / 50) × 100 = 40%
This means 40% of the fruits are apples.
4.2. Uses and Importance of Percentages
Percentages are widely used in various fields, including:
- Finance: Interest rates, investment returns, and discounts are often expressed as percentages.
- Statistics: Used to represent data and compare different groups or categories.
- Retail: Discounts, sales tax, and profit margins are typically given as percentages.
- Health: Used to report statistics on disease prevalence, vaccination rates, and treatment success.
- Education: Grades and test scores are frequently expressed as percentages.
4.3. Converting Between Ratios, Proportions, and Percentages
- Ratio to Percentage: To convert a ratio to a percentage, divide the first number by the second number and multiply by 100. For example, the ratio 1:4 can be converted to a percentage as (1/4) × 100 = 25%.
- Percentage to Ratio: To convert a percentage to a ratio, divide the percentage by 100 and simplify the fraction. For example, 75% can be written as 75/100, which simplifies to the ratio 3:4.
- Proportion to Percentage: A proportion is already a part-to-whole comparison, so you simply multiply the proportion by 100 to express it as a percentage. For example, if the proportion is 0.6, then the percentage is 0.6 × 100 = 60%.
4.4. Practical Examples of Percentages
- Sales Discount: A store offers a 20% discount on a $100 item. The discount amount is 20% of $100, which is $20.
- Test Scores: A student scores 85 out of 100 on a test. The student’s score is 85%.
- Market Share: A company has a 30% market share. This means the company controls 30% of the total sales in that market.
- Loan Interest: A loan has an annual interest rate of 5%. This means the borrower will pay 5% of the loan amount each year as interest.
- Population Growth: A city’s population increased by 10% in the last decade. This means the population is now 10% larger than it was a decade ago.
4.5. Common Mistakes to Avoid When Working with Percentages
- Misunderstanding the Base: Always be clear about what the percentage is “of.” For example, a 10% increase followed by a 10% decrease does not return the original value because the base changes.
- Adding or Subtracting Percentages Incorrectly: You cannot simply add or subtract percentages unless they are based on the same total.
- Confusing Percentage Points with Percentages: A percentage point is the simple difference between two percentages. For example, if a rate increases from 10% to 12%, that’s a 2 percentage point increase.
5. Decimals: Bridging the Gap Between Fractions and Percentages
Decimals are another way to represent numbers that are not whole numbers. They are closely related to both fractions and percentages, providing a seamless transition between these forms of numerical representation. Understanding decimals is crucial for performing calculations and interpreting data in various contexts.
5.1. Understanding Decimals
A decimal is a number expressed in the base-10 numeral system, using a decimal point to separate the whole number part from the fractional part. Each digit to the right of the decimal point represents a fraction with a denominator of a power of 10 (e.g., tenths, hundredths, thousandths).
For example, the decimal 0.75 represents seventy-five hundredths, which is equivalent to the fraction 75/100 or 3/4.
5.2. Converting Between Decimals, Fractions, and Percentages
- Decimal to Fraction: To convert a decimal to a fraction, write the decimal as a fraction with a denominator of 10, 100, 1000, etc., depending on the number of decimal places. Then, simplify the fraction if possible. For example, 0.65 can be written as 65/100, which simplifies to 13/20.
- Fraction to Decimal: To convert a fraction to a decimal, divide the numerator by the denominator. For example, the fraction 1/4 can be converted to a decimal by dividing 1 by 4, resulting in 0.25.
- Decimal to Percentage: To convert a decimal to a percentage, multiply the decimal by 100 and add the percent sign (%). For example, 0.45 becomes 0.45 × 100 = 45%.
- Percentage to Decimal: To convert a percentage to a decimal, divide the percentage by 100. For example, 80% becomes 80 / 100 = 0.80.
5.3. Uses and Importance of Decimals
Decimals are essential in various fields, including:
- Science: Used for precise measurements and calculations.
- Engineering: Employed in designing and constructing structures and machines.
- Finance: Used in calculating interest rates, currency exchange rates, and financial analysis.
- Everyday Life: Used in shopping, cooking, and measuring.
5.4. Examples of Decimal Conversions
- Convert 0.8 to a fraction: 0.8 = 8/10 = 4/5
- Convert 3/8 to a decimal: 3 ÷ 8 = 0.375
- Convert 0.15 to a percentage: 0.15 × 100 = 15%
- Convert 92% to a decimal: 92 ÷ 100 = 0.92
5.5. Adding, Subtracting, Multiplying, and Dividing Decimals
-
Adding and Subtracting Decimals: Align the decimal points, then add or subtract as you would with whole numbers. For example:
2.56 + 1.32 ------- 3.88 -
Multiplying Decimals: Multiply the numbers as if they were whole numbers, then count the total number of decimal places in the factors and apply that many decimal places in the product. For example:
1.25 (2 decimal places) × 0.3 (1 decimal place) ------- 0.375 (3 decimal places) -
Dividing Decimals: Make the divisor a whole number by moving the decimal point to the right. Move the decimal point in the dividend the same number of places. Then divide as usual. For example:
Divide 4.5 by 0.5: 0. 5 ) 4.5 -> 5 ) 45 45 ÷ 5 = 9
5.6. Common Mistakes to Avoid When Working with Decimals
- Misplacing the Decimal Point: Always double-check the placement of the decimal point in calculations.
- Rounding Errors: Be mindful of rounding rules to avoid significant errors in calculations.
- Incorrect Conversion: Ensure you are using the correct method when converting between decimals, fractions, and percentages.
6. Percentage Points: Measuring the Change in Percentages
Percentage points are used to describe the absolute difference between two percentages. While percentages express a value as a fraction of 100, percentage points indicate how much a percentage has changed. This distinction is crucial for clear communication and accurate analysis.
6.1. Understanding Percentage Points
A percentage point is simply the arithmetic difference between two percentages. It is a straightforward way to express the magnitude of change or difference between two percentage values.
Percentage Point Difference = New Percentage – Old Percentage
For example, if a country’s unemployment rate rises from 5% to 7%, the increase is 2 percentage points.
6.2. Uses and Importance of Percentage Points
Percentage points are particularly useful in:
- Economics: Describing changes in interest rates, inflation rates, and unemployment rates.
- Politics: Illustrating shifts in poll results, voter turnout, and election outcomes.
- Statistics: Communicating changes in data sets and experimental results.
- Finance: Analyzing changes in investment returns and market trends.
6.3. Examples of Percentage Point Calculations
- Interest Rates: If the interest rate on a loan increases from 4% to 6%, the increase is 2 percentage points.
- Election Results: If a candidate’s support in a poll rises from 45% to 52%, the gain is 7 percentage points.
- Sales Tax: If the sales tax rate decreases from 8% to 7.5%, the decrease is 0.5 percentage points.
- Unemployment Rate: If the unemployment rate drops from 10% to 8%, the decrease is 2 percentage points.
- Investment Returns: If an investment’s return increases from 12% to 15%, the increase is 3 percentage points.
6.4. Distinguishing Between Percentage Points and Percentages
It is essential to differentiate between percentage points and percentages to avoid misinterpretation. A change expressed in percentage points is an absolute difference, while a change expressed as a percentage is a relative difference.
For example, consider an increase from 5% to 10%:
- Percentage Point Increase: The increase is 5 percentage points (10% – 5% = 5%).
- Percentage Increase: The increase is 100% because the new value is double the old value [(10% – 5%) / 5% = 1 = 100%].
Using the correct term ensures that the information is conveyed accurately.
6.5. Common Mistakes to Avoid When Using Percentage Points
- Confusing Percentage Points with Percentages: Always use the term “percentage points” when describing the arithmetic difference between two percentages.
- Misinterpreting the Context: Understand the context to determine whether percentage points or percentages are more appropriate for describing a change.
- Incorrect Calculations: Double-check your calculations to ensure the percentage point difference is accurate.
7. Applying Ratios, Proportions, Rates, and Percentages in Real-World Scenarios
Understanding ratios, proportions, rates, and percentages is essential for analyzing and interpreting data in various real-world scenarios. These concepts are fundamental in fields like finance, healthcare, marketing, and public policy. Here are some practical applications:
7.1. Financial Analysis
- Profit Margins: Businesses use percentages to calculate profit margins, which represent the percentage of revenue remaining after deducting costs. This helps assess the profitability of products or services.
- Return on Investment (ROI): Investors use ROI to measure the percentage return on their investments, allowing them to evaluate the performance of different investment opportunities.
- Debt-to-Equity Ratio: Companies use this ratio to assess their financial leverage, indicating the proportion of debt and equity used to finance assets.
- Interest Rates: Banks and lenders use percentages to express interest rates on loans and savings accounts, influencing borrowing and saving decisions.
7.2. Healthcare
- Mortality Rates: Public health officials use mortality rates (deaths per population) to assess the health status of a community and track the effectiveness of interventions.
- Prevalence and Incidence: Epidemiologists use proportions and rates to measure the prevalence (existing cases) and incidence (new cases) of diseases, informing public health strategies.
- Vaccination Coverage: Healthcare providers use percentages to track vaccination coverage rates, helping to prevent outbreaks and protect vulnerable populations.
- Treatment Success Rates: Researchers use percentages to report the success rates of medical treatments, allowing patients and healthcare professionals to make informed decisions.
7.3. Marketing and Sales
- Market Share: Companies use percentages to measure their market share, indicating their portion of total sales in a given market.
- Conversion Rates: Marketers use conversion rates (percentage of website visitors who complete a desired action) to assess the effectiveness of their marketing campaigns.
- Customer Satisfaction: Businesses use percentages to measure customer satisfaction levels, helping them identify areas for improvement and enhance customer loyalty.
- Sales Growth: Companies use percentages to track sales growth over time, indicating the performance of their products or services.
7.4. Public Policy and Governance
- Unemployment Rate: Governments use the unemployment rate (percentage of the labor force without a job) to assess the health of the economy and inform policy decisions.
- Poverty Rate: Policymakers use the poverty rate (percentage of the population below the poverty line) to understand income inequality and develop social welfare programs.
- Literacy Rate: Governments use the literacy rate (percentage of the population that can read and write) to measure educational attainment and promote literacy initiatives.
- Budget Allocation: Governments use percentages to allocate funds to different sectors, such as education, healthcare, and infrastructure, reflecting policy priorities.
7.5. Scientific Research
- Error Rates: Scientists use percentages to express error rates in experiments, indicating the accuracy and reliability of their findings.
- Statistical Significance: Researchers use p-values (expressed as percentages) to determine the statistical significance of their results, helping to validate their conclusions.
- Growth Rates: Biologists use rates to measure the growth rates of populations or the spread of diseases, informing conservation and control efforts.
- Composition Analysis: Chemists use percentages to analyze the composition of substances, determining the proportion of different elements or compounds.
By understanding how to apply ratios, proportions, rates, and percentages in these real-world scenarios, individuals can make more informed decisions, analyze data effectively, and communicate information clearly.
8. Tips and Tricks for Mastering Ratios, Proportions, Rates, and Percentages
Mastering ratios, proportions, rates, and percentages can significantly enhance your analytical and problem-solving skills. Here are some tips and tricks to help you understand and apply these concepts effectively:
8.1. Understand the Basics
- Ratios: Grasp the concept of comparing two quantities. Understand how to simplify ratios and express them in different forms (e.g., a:b, a/b).
- Proportions: Know that a proportion is a type of ratio where the numerator is part of the denominator. Recognize how to set up and solve proportions.
- Rates: Understand that a rate measures how frequently an event occurs over time. Differentiate between different types of rates (e.g., incidence rate, mortality rate).
- Percentages: Understand that a percentage is a ratio expressed as a fraction of 100. Know how to convert between percentages, decimals, and fractions.
8.2. Practice Regularly
- Solve Problems: Regularly practice solving problems involving ratios, proportions, rates, and percentages. Start with simple problems and gradually move to more complex ones.
- Real-World Examples: Apply these concepts to real-world scenarios. Look for opportunities to use ratios, proportions, rates, and percentages in your daily life.
8.3. Visualize the Concepts
- Diagrams and Charts: Use diagrams and charts to visualize ratios, proportions, rates, and percentages. This can help you understand the relationships between different quantities.
- Bar Graphs and Pie Charts: Use bar graphs to compare different categories and pie charts to show the proportion of each category to the whole.
8.4. Memorize Key Formulas
- Ratio Formula: Ratio = a/b
- Proportion Formula: a/b = c/d
- Percentage Formula: Percentage = (Part / Whole) × 100
- Rate Formula: Rate = (Number of events / Population size) × 10n
8.5. Use Online Resources
- Online Calculators: Use online calculators to check your answers and speed up calculations.
- Educational Websites: Explore educational websites and online courses to learn more about ratios, proportions, rates, and percentages.
- Practice Quizzes: Take practice quizzes to test your understanding and identify areas where you need improvement.
8.6. Pay Attention to Units
- Consistent Units: Ensure that you are using consistent units when working with ratios, proportions, rates, and percentages.
- Unit Conversions: Know how to convert between different units (e.g., converting meters to kilometers, seconds to hours).
8.7. Understand the Context
- Problem Context: Pay attention to the context of the problem to determine which concept (ratio, proportion, rate, or percentage) is most appropriate.
- Word Problems: Practice solving word problems that involve ratios, proportions, rates, and percentages. This will help you develop your problem-solving skills.
8.8. Avoid Common Mistakes
- Misinterpreting the Base: Always be clear about what the percentage is “of.” For example, a 10% increase followed by a 10% decrease does not return the original value because the base changes.
- Adding or Subtracting Percentages Incorrectly: You cannot simply add or subtract percentages unless they are based on the same total.
- Confusing Percentage Points with Percentages: A percentage point is the simple difference between two percentages. For example, if a rate increases from 10% to 12%, that’s a 2 percentage point increase.
8.9. Teach Someone Else
- Explain the Concepts: Teaching someone else about ratios, proportions, rates, and percentages can help reinforce your understanding.
- Answer Questions: Answering questions from others can help you identify areas where you need to improve your knowledge.
8.10. Review and Reinforce
- Regular Review: Regularly review the concepts of ratios, proportions, rates, and percentages to keep your knowledge fresh.
- Practice Tests: Take practice tests to assess your overall understanding and identify areas where you need to focus your efforts.
By following these tips and tricks, you can master ratios, proportions, rates, and percentages and use them effectively in various aspects of your life.
9. Frequently Asked Questions (FAQ) About Ratios, Proportions, Rates, and Percentages
9.1. What is a ratio?
A ratio is a comparison of two quantities, indicating how much of one thing there is compared to another. It can be expressed as a fraction, a decimal, or using a colon (e.g., 3:4).
9.2. How do you simplify a ratio?
To simplify a ratio, divide all parts of the ratio by their greatest common divisor (GCD). For example, the ratio 12:16 can be simplified by dividing both numbers by 4, resulting in the simplified ratio 3:4.
9.3. What is a proportion?
A proportion is an equation stating that two ratios are equal. It is used to show that two sets of numbers are related in the same way.
9.4. How do you solve a proportion?
To solve a proportion, use cross-multiplication. If you have a/b = c/d, then ad = bc. Solve for the unknown variable.
9.5. What is a rate?
A rate is a ratio that compares two quantities with different units, usually measured per unit of time. Examples include speed (miles per hour) and population density (people per square mile).
9.6. How do you calculate a rate?
To calculate a rate, divide the quantity by the time period. For example, if you travel 150 miles in 3 hours, your rate (speed) is 150 miles / 3 hours = 50 miles per hour.
9.7. What is a percentage?
A percentage is a ratio expressed as a fraction of 100. It is a way of expressing a number as a part of a whole.
9.8. How do you convert a fraction to a percentage?
To convert a fraction to a percentage, multiply the fraction by 100. For example, 1/4 = (1/4) * 100 = 25%.
