Linear Equations for Percent & Investment Problems in Algebra 2

What are Linear Equations and How Are They Applied to Percent Increase/Decrease Problems?

Linear equations are algebraic expressions that establish a relationship between two variables, typically written in the form y = mx + b, where 'm' is the slope and 'b' is the y-intercept. These equations are foundational in solving numerous types of real-world problems, including calculating percent increase or decrease.

When dealing with percent increase or decrease, consider the original value (initial amount) and the final value (new amount). The percent change can be expressed using a linear equation.

Example: Calculating Percent Increase

*Question:*
If a product originally costs $100 and its price increases to $120, what is the percent increase?

*Answer:*
1. Identify the original amount: $100.
2. Identify the new amount: $120.
3. Calculate the increase: $120 - $100 = $20.
4. Determine the percent increase using the formula:

Percent Increase = (Increase/Original Amount) * 100
Percent Increase = ($20/$100) * 100
Percent Increase = 0.2 * 100
Percent Increase = 20%

So, the percent increase is 20%.

Example: Calculating Percent Decrease

*Question:*
A stock’s value drops from $500 to $400. What is the percent decrease?

*Answer:*
1. Identify the original amount: $500.
2. Identify the new amount: $400.
3. Calculate the decrease: $500 - $400 = $100.
4. Determine the percent decrease using the formula:

Percent Decrease = (Decrease/Original Amount) * 100
Percent Decrease = ($100/$500) * 100
Percent Decrease = 0.2 * 100
Percent Decrease = 20%

So, the percent decrease is 20%.

How Are Linear Equations Used in Investment Problems?

Investment problems typically involve understanding how money grows or diminishes over time, especially through interest rates. Simple interest and compound interest are two common types of calculations.

Example: Simple Interest

Simple interest is calculated using the formula:

I = P * r * t

where:
- I represents the interest.
- P is the principal amount (initial investment).
- r is the annual interest rate (in decimal form).
- t is the time the money is invested (in years).

*Question:*
If you invest $1,000 at an annual interest rate of 5% for 3 years, how much interest will you earn?

*Answer:*
1. Identify the principal amount: P = $1,000.
2. Convert the annual interest rate to decimal form: r = 5/100 = 0.05.
3. Identify the time of investment: t = 3 years.
4. Apply the simple interest formula:

I = $1,000 * 0.05 * 3
I = $150

So, you will earn $150 in interest.

Example: Compound Interest

Compound interest is calculated using the formula:

A = P * (1 + r/n)^(n*t)

where:
- A represents the amount of money accumulated after n years, including interest.
- P is the principal amount (initial investment).
- r is the annual interest rate (in decimal form).
- n is the number of times interest is compounded per year.
- t is the time the money is invested (in years).

*Question:*
If you invest $1,000 at an annual interest rate of 5% compounded annually for 3 years, what will be the total amount?

*Answer:*
1. Identify the principal amount: P = $1,000.
2. Convert the annual interest rate to decimal form: r = 5/100 = 0.05.
3. Determine the number of times interest is compounded per year: n = 1.
4. Identify the time of investment: t = 3 years.
5. Apply the compound interest formula:

A = $1,000 * (1 + 0.05/1)^(1*3)
A = $1,000 * (1.05)^3
A = $1,000 * 1.157625
A = $1,157.63

So, the total amount after 3 years will be $1,157.63.

Utilizing linear equations to solve percent increase/decrease and investment problems involves understanding how to manipulate the equations and apply specific formulas to find the necessary values. By mastering these techniques, you can address various financial and practical scenarios effectively.

Algebra 2: Linear Equations for Percent & Investment Problems

Related

Recommended Videos