Percentages and Proportional Relationships Study Notes

Understanding percentages and proportional relationships is fundamental in 7th-grade mathematics. These concepts help us compare quantities, calculate changes, and solve real-world problems involving ratios.

🔢 Understanding Percentages

A percentage is a way to express a number as a fraction of 100. The word "percent" means "per hundred" or "out of 100". The symbol for percent is %. For example, 50% means 50 out of 100, or \( \frac{50}{100} \).

🔄 Converting Between Forms

You can express numbers as fractions, decimals, or percentages. Knowing how to convert between these forms is crucial.

• Decimal to Percent: Multiply the decimal by 100 and add the % symbol. (Move the decimal point two places to the right.)
  Example: \( 0.75 = 0.75 \times 100% = 75% \)
• Percent to Decimal: Divide the percentage by 100 and remove the % symbol. (Move the decimal point two places to the left.)
  Example: \( 25% = 25 \div 100 = 0.25 \)
• Fraction to Percent: Convert the fraction to a decimal first (divide the numerator by the denominator), then convert the decimal to a percent.
  Example: \( \frac{3}{4} = 3 \div 4 = 0.75 = 75% \)
• Percent to Fraction: Write the percentage as a fraction with a denominator of 100, then simplify.
  Example: \( 40% = \frac{40}{100} = \frac{2}{5} \)

Here's a quick reference table:

Fraction	Decimal	Percent
\( \frac{1}{2} \)	0.5	50%
\( \frac{1}{4} \)	0.25	25%
\( \frac{3}{4} \)	0.75	75%
\( \frac{1}{10} \)	0.1	10%
\( \frac{1}{5} \)	0.2	20%

➕ Solving Percentage Problems

Many percentage problems involve finding a part, a whole, or the percentage itself. There are two common methods:

1. Using the Percentage Formula

The basic formula is: Part = Percent \(\times\) Whole

Where the percent must be in decimal form.

• Finding the Part: What is 20% of 80?
  \( \text{Part} = 0.20 \times 80 = 16 \)
• Finding the Percent: 15 is what percent of 60?
  \( 15 = \text{Percent} \times 60 \)
  \( \text{Percent} = \frac{15}{60} = 0.25 = 25% \)
• Finding the Whole: 12 is 30% of what number?
  \( 12 = 0.30 \times \text{Whole} \)
  \( \text{Whole} = \frac{12}{0.30} = 40 \)

2. Using Proportions

You can set up a proportion: \( \frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100} \)

• Finding the Part: What is 20% of 80?
  \( \frac{x}{80} = \frac{20}{100} \)
  \( 100x = 20 \times 80 \)
  \( 100x = 1600 \)
  \( x = 16 \)
• Finding the Percent: 15 is what percent of 60?
  \( \frac{15}{60} = \frac{x}{100} \)
  \( 60x = 15 \times 100 \)
  \( 60x = 1500 \)
  \( x = 25% \)

🛒 Real-World Applications of Percentages

💰 Sales Tax, Tips, and Commissions

These are calculated as a percentage of a total amount and added to it.

• Sales Tax: An additional amount charged by the government on goods and services.
  Example: A shirt costs 25. Sales tax is 6%.
  Tax Amount = \( 0.06 \times \25 = \1.50 \)
  Total Cost = \( \25 + \1.50 = \26.50 \)
• Tips: An extra payment for service, usually to restaurant staff or service providers.
  Example: A dinner bill is 40. You want to leave a 15% tip.
  Tip Amount = \( 0.15 \times \40 = \6.00 \)
  Total Paid = \( \40 + \6.00 = \46.00 \)
• Commission: A percentage of sales paid to a salesperson.
  Example: A salesperson earns 10% commission on sales. They sell 500 worth of items.
  Commission Earned = \( 0.10 \times \500 = \50.00 \)

🏷️ Discounts and Markups

These involve changing an original price by a percentage.

• Discount: A reduction in price, calculated as a percentage of the original price.
  Example: A 60 item is on sale for 20% off.
  Discount Amount = \( 0.20 \times \60 = \12.00 \)
  Sale Price = \( \60 - \12.00 = \48.00 \)
• Markup: An increase in price, often by businesses to cover costs and make a profit.
  Example: A store buys a hat for 10 and marks it up by 40%.
  Markup Amount = \( 0.40 \times \10 = \4.00 \)
  Selling Price = \( \10 + \4.00 = \14.00 \)

📈 Percent Increase and Decrease

Used to describe how much a quantity has changed relative to its original amount.

\( \text{Percent Change} = \frac{\text{Amount of Change}}{\text{Original Amount}} \times 100% \)

• Percent Increase: A value goes up.
  Example: A price increases from 50 to 60.
  Amount of Change = \( \60 - \50 = \10 \)
  Percent Increase = \( \frac{\10}{\50} \times 100% = 0.20 \times 100% = 20% \)
• Percent Decrease: A value goes down.
  Example: A population decreases from 200 to 180.
  Amount of Change = \( 200 - 180 = 20 \)
  Percent Decrease = \( \frac{20}{200} \times 100% = 0.10 \times 100% = 10% \)

📏 Percent Error

Measures the accuracy of a measurement or estimation compared to the actual value.

\( \text{Percent Error} = \frac{|\text{Measured Value} - \text{Actual Value}|}{\text{Actual Value}} \times 100% \)

• Example: You estimate a book to be 250 pages. It's actually 240 pages.
  Amount of Error = \( |250 - 240| = 10 \)
  Percent Error = \( \frac{10}{240} \times 100% \approx 0.0416 \times 100% \approx 4.17% \)

🏦 Simple Interest

Simple interest is calculated only on the principal amount, or the initial amount of money.

The formula for simple interest is: I = Prt

• I = Interest earned
• P = Principal (the initial amount of money)
• r = Interest rate (as a decimal)
• t = Time (in years)

• Example: You deposit 500 into a savings account with a 3% annual simple interest rate. How much interest do you earn in 2 years?
  \( P = \500 \)
  \( r = 3% = 0.03 \)
  \( t = 2 \text{ years} \)
  \( I = \500 \times 0.03 \times 2 \)
  \( I = \30 \)
  You would earn 30 in interest.

🔗 Proportional Relationships

A proportional relationship exists between two quantities when one quantity is a constant multiple of the other. This means that as one quantity changes, the other quantity changes by a consistent factor.

🔍 Identifying Proportional Relationships

A relationship is proportional if:

• In a Table: The ratio \( \frac{y}{x} \) is constant for all pairs of values (except when \( x=0 \)).
• In a Graph: It is a straight line that passes through the origin \( (0,0) \).
• In an Equation: It can be written in the form \( y = kx \), where \( k \) is a constant.

✖️ Constant of Proportionality (k)

The constant of proportionality, denoted by \( k \), is the constant ratio between two proportional quantities \( y \) and \( x \).

It is found by the formula: \( k = \frac{y}{x} \)

• Meaning of \( k \): It represents the unit rate, or how much \( y \) changes for every one unit change in \( x \).
• Example: If a car travels 150 miles in 3 hours, the constant of proportionality (speed) is:
  \( k = \frac{150 \text{ miles}}{3 \text{ hours}} = 50 \text{ miles per hour} \)

✍️ Writing Equations for Proportional Relationships

Once you find the constant of proportionality \( k \), you can write an equation for the relationship in the form:

\[ y = kx \]

• Example: If the constant of proportionality for the cost of apples is 1.50 per pound, the equation relating cost \( (y) \) to pounds \( (x) \) is:
  \( y = 1.50x \)

🚫 Non-Proportional Relationships

A relationship is non-proportional if:

• The ratio \( \frac{y}{x} \) is not constant.
• The graph is a straight line but does not pass through the origin.
• The equation cannot be written in the form \( y = kx \) (e.g., \( y = kx + b \) where \( b \ne 0 \)).

Example of non-proportional: The cost of a taxi ride is 2.00 plus 1.50 per mile. The equation is \( y = 1.50x + 2 \). This is not proportional because of the initial 2.00 fee (it does not pass through the origin). The ratio \( \frac{y}{x} \) would not be constant.