Chapter Notes: Rates And Percentages
Rates and percentages are everywhere in daily life. You see them when shopping for discounts, comparing phone plans, calculating tips at restaurants, or measuring how fast you travel. Understanding rates helps you compare different quantities that are measured in different units, like miles per hour or dollars per pound. Percentages help you understand parts of a whole, especially when comparing data or calculating changes. This chapter will teach you how to work confidently with both rates and percentages, solving real-world problems step by step.
Understanding Rates
A rate is a special kind of ratio that compares two quantities with different units. When you say a car travels at 60 miles per hour, you are expressing a rate. The word "per" is a key signal that you are working with a rate-it means "for each" or "for every." Rates help us answer questions like "How much does this cost for one item?" or "How far can I go in one hour?"
Rates are written as fractions with different units in the numerator and denominator. For example:
- \( \frac{120 \text{ miles}}{2 \text{ hours}} \) - distance per time
- \( \frac{\$15}{3 \text{ pounds}} \) - cost per weight
- \( \frac{300 \text{ calories}}{2 \text{ servings}} \) - calories per serving
Unit Rates
A unit rate is a rate where the denominator is 1. It tells you the quantity of the first measurement for exactly one unit of the second measurement. Unit rates make comparisons much easier because everything is expressed "per one unit."
To find a unit rate, divide the numerator by the denominator:
\[ \text{Unit Rate} = \frac{\text{Total Amount}}{\text{Total Units}} \]where the result tells you how much per one unit.
Example: A package of 8 granola bars costs $12.
What is the unit rate in dollars per granola bar?
Solution:
We need to find the cost for one granola bar.
Unit rate = Total cost ÷ Number of bars
Unit rate = $12 ÷ 8
Unit rate = $1.50 per granola bar
Each granola bar costs $1.50.
Example: Maria drove 270 miles in 4.5 hours.
What was her average speed in miles per hour?
Solution:
We need to find how many miles she drove in one hour.
Unit rate = Total miles ÷ Total hours
Unit rate = 270 ÷ 4.5
Unit rate = 60 miles per hour
Maria's average speed was 60 miles per hour.
Comparing Unit Rates
Unit rates are especially useful when you need to compare two or more options to decide which is the better deal. By converting each option to a unit rate, you can easily see which gives you more value.
Example: Store A sells 5 notebooks for $7.50.
Store B sells 3 notebooks for $4.80.Which store offers a better price per notebook?
Solution:
Find the unit rate for Store A:
Unit rate for Store A = $7.50 ÷ 5 = $1.50 per notebook
Find the unit rate for Store B:
Unit rate for Store B = $4.80 ÷ 3 = $1.60 per notebook
Compare the two unit rates: $1.50 <>
Store A offers the better price at $1.50 per notebook.
Understanding Percentages
The word percent comes from the Latin phrase "per centum," which means "per hundred." A percentage is a ratio that compares a number to 100. The symbol for percent is %, which represents "out of 100." Percentages make it easy to compare fractions and ratios because they all use the same denominator: 100.
When we write 25%, we mean 25 out of 100, which is the same as the fraction \( \frac{25}{100} \) or the decimal 0.25.
Converting Between Percentages, Decimals, and Fractions
Being able to move fluently between percentages, decimals, and fractions is essential for solving problems. Here are the conversion methods:
Percentage to Decimal
To convert a percentage to a decimal, divide by 100 (or move the decimal point two places to the left).
- 35% = 35 ÷ 100 = 0.35
- 8% = 8 ÷ 100 = 0.08
- 125% = 125 ÷ 100 = 1.25
Decimal to Percentage
To convert a decimal to a percentage, multiply by 100 (or move the decimal point two places to the right) and add the % symbol.
- 0.42 = 0.42 × 100 = 42%
- 0.07 = 0.07 × 100 = 7%
- 1.5 = 1.5 × 100 = 150%
Percentage to Fraction
Write the percentage as a fraction with denominator 100, then simplify if possible.
- 60% = \( \frac{60}{100} = \frac{3}{5} \)
- 25% = \( \frac{25}{100} = \frac{1}{4} \)
- 12.5% = \( \frac{12.5}{100} = \frac{125}{1000} = \frac{1}{8} \)
Fraction to Percentage
Divide the numerator by the denominator to get a decimal, then convert the decimal to a percentage.
- \( \frac{3}{4} = 0.75 = 75\% \)
- \( \frac{1}{5} = 0.2 = 20\% \)
- \( \frac{7}{8} = 0.875 = 87.5\% \)
Finding the Percentage of a Number
One of the most common problems involving percentages is finding a certain percent of a given number. For example, finding 20% of 80, or calculating a 15% tip on a $40 meal. There are several methods to solve these problems.
Method 1: Convert to Decimal and Multiply
Convert the percentage to a decimal, then multiply by the number.
\[ \text{Part} = \text{Percent as decimal} \times \text{Whole} \]Example: Find 30% of 150.
What is 30% of 150?
Solution:
Convert 30% to a decimal: 30% = 0.30
Multiply: 0.30 × 150
0.30 × 150 = 45
30% of 150 is 45.
Method 2: Use the Fraction Form
Convert the percentage to a fraction, then multiply by the number.
Example: Find 25% of 80.
What is 25% of 80?
Solution:
Convert 25% to a fraction: 25% = \( \frac{25}{100} = \frac{1}{4} \)
Multiply: \( \frac{1}{4} \times 80 \)
\( \frac{1}{4} \times 80 = \frac{80}{4} = 20 \)
25% of 80 is 20.
Finding What Percent One Number Is of Another
Sometimes you know two numbers and need to find what percentage one is of the other. For example, if you scored 18 out of 20 on a quiz, what percentage did you earn?
To find what percent one number is of another, use this formula:
\[ \text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\% \]Example: Ryan answered 45 questions correctly out of 60 total questions on a test.
What percent of the questions did he answer correctly?
Solution:
Part = 45, Whole = 60
Percent = \( \frac{45}{60} \times 100\% \)
First divide: 45 ÷ 60 = 0.75
Then multiply by 100: 0.75 × 100 = 75%
Ryan answered 75% of the questions correctly.
Finding the Whole When Given a Percentage
Sometimes you know what a certain percentage equals, and you need to find the original whole amount. For example, if 40% of a number is 32, what is the number?
Use this formula:
\[ \text{Whole} = \frac{\text{Part}}{\text{Percent as decimal}} \]Example: 15% of a number is 27.
What is the number?
Solution:
Convert 15% to a decimal: 15% = 0.15
Part = 27, Percent = 0.15
Whole = 27 ÷ 0.15
27 ÷ 0.15 = 180
The number is 180.
Percent Increase and Percent Decrease
A percent increase describes how much a quantity has grown compared to its original value. A percent decrease describes how much a quantity has shrunk. These are very common in real life: prices increase, populations grow, discounts decrease prices, and scores change.
Calculating Percent Increase
To find the percent increase, use this formula:
\[ \text{Percent Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\% \]The numerator is the amount of change, which is always the difference between the new and original values.
Example: The price of a video game increased from $40 to $50.
What is the percent increase?
Solution:
Original Value = $40, New Value = $50
Amount of change = $50 - $40 = $10
Percent Increase = \( \frac{10}{40} \times 100\% \)
10 ÷ 40 = 0.25
0.25 × 100 = 25%
The price increased by 25%.
Calculating Percent Decrease
To find the percent decrease, use this formula:
\[ \text{Percent Decrease} = \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100\% \]Example: A jacket originally priced at $80 is on sale for $60.
What is the percent decrease?
Solution:
Original Value = $80, New Value = $60
Amount of change = $80 - $60 = $20
Percent Decrease = \( \frac{20}{80} \times 100\% \)
20 ÷ 80 = 0.25
0.25 × 100 = 25%
The price decreased by 25%.
Applications of Rates and Percentages
Rates and percentages work together to solve many types of real-world problems. Here are some of the most common situations you will encounter.
Discounts and Sales Tax
When shopping, you often need to calculate the final price after a discount or after adding sales tax. A discount is a percent decrease in price, while sales tax is a percent increase added to the price.
Example: A pair of headphones costs $120.
The store offers a 25% discount.What is the sale price?
Solution:
Find the discount amount: 25% of $120
0.25 × 120 = $30
Subtract the discount from the original price:
$120 - $30 = $90
The sale price is $90.
Example: A meal at a restaurant costs $35.
The sales tax rate is 8%.What is the total cost including tax?
Solution:
Find the tax amount: 8% of $35
0.08 × 35 = $2.80
Add the tax to the original price:
$35 + $2.80 = $37.80
The total cost is $37.80.
Tips and Gratuities
When dining out, it is customary to leave a tip, which is a percentage of the bill. Common tip percentages are 15%, 18%, or 20%.
Example: A dinner bill is $48.
You want to leave an 18% tip.How much should the tip be, and what is the total amount?
Solution:
Find the tip amount: 18% of $48
0.18 × 48 = $8.64
Add the tip to the bill:
$48 + $8.64 = $56.64
The tip is $8.64, and the total amount is $56.64.
Speed, Distance, and Time
Speed is a rate that compares distance to time. The basic relationship between speed, distance, and time is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]You can rearrange this formula to solve for distance or time:
\[ \text{Distance} = \text{Speed} \times \text{Time} \] \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]Example: A cyclist rides at a constant speed of 15 miles per hour for 3 hours.
How far does the cyclist travel?
Solution:
Use the formula: Distance = Speed × Time
Distance = 15 miles/hour × 3 hours
Distance = 45 miles
The cyclist travels 45 miles.
Population and Data Changes
Percentages are frequently used to describe changes in populations, social media followers, test scores, and other data over time.
Example: The population of a town was 8,000 last year.
This year, the population increased by 12%.What is the current population?
Solution:
Find the increase: 12% of 8,000
0.12 × 8,000 = 960
Add the increase to the original population:
8,000 + 960 = 8,960
The current population is 8,960.
Summary of Key Formulas
| Concept | Formula |
|---|---|
| Unit Rate | \( \frac{\text{Total Amount}}{\text{Total Units}} \) |
| Part (Percentage of a number) | \( \text{Percent as decimal} \times \text{Whole} \) |
| Percent (What percent one number is of another) | \( \frac{\text{Part}}{\text{Whole}} \times 100\% \) |
| Whole (Given a percent and part) | \( \frac{\text{Part}}{\text{Percent as decimal}} \) |
| Percent Increase | \( \frac{\text{New} - \text{Original}}{\text{Original}} \times 100\% \) |
| Percent Decrease | \( \frac{\text{Original} - \text{New}}{\text{Original}} \times 100\% \) |
| Speed | \( \frac{\text{Distance}}{\text{Time}} \) |
Rates and percentages are powerful tools that help you make sense of numbers in everyday situations. Whether you are comparing prices, calculating discounts, understanding data changes, or planning travel, these concepts give you the skills to analyze information and make smart decisions. Practice converting between fractions, decimals, and percentages, and always take time to understand what each number represents in the context of the problem.
