Problem 9 - Algebra for College Students

Question

9\. Solve \(A=P+P r\) for \(A\), given that \(P=\$ 1000\), \(r=12 \%\), and \(t=5\) years.

Step-by-Step Solution

Answer

The total amount, \( A \), after 1 year is \( \$1120 \).

Step 1: Understand the formula

The formula given is for simple interest: \( A = P + Prt \). Where \( A \) is the total amount including interest, \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time period in years. In this problem, the formula is simplified to \( A = P + Pr \) because \( r \) is given per year and we are told that \( t = 1 \).

Step 2: Assign the known values

Assign the values into the formula: principal \( P = 1000 \), rate of interest \( r = 0.12 \) (as 12% converted to decimal is 0.12), and time \( t = 1 \). The formula becomes: \( A = 1000 + 1000 \times 0.12 \).

Step 3: Calculate the interest

Calculate \( Pr \) by multiplying \( P \) and \( r \):\[ Pr = 1000 \times 0.12 = 120 \].

Step 4: Solve for A

Add \( P \) and \( Pr \) to find \( A \):\[ A = 1000 + 120 = 1120 \].

Key Concepts

Interest Formula

In the context of simple interest, the Interest Formula expresses how much money is charged or earned over a certain period. The basic formula is \( A = P(1 + rt) \). Here, \( A \) represents the future value or the total amount after interest, \( P \) refers to the principal, \( r \) is the rate of interest per time period, and \( t \) signifies the time the money is invested or borrowed for.

When applying this in real-life scenarios, we often know the principal, rate, and time, and need to determine the total amount \( A \). For example, if you invested a certain amount and are curious about the total returns after a specific period, this formula becomes your go-to solution. Make sure to convert percentage rates to decimal form, as this is often a common oversight for many learners.

Percentage Calculation

Percentage calculations convert a fraction or ratio into a percentage, making comparisons easier. In financial mathematics, this skill is essential when calculating or interpreting interest rates. To change a percentage into a usable form in formulas, divide by 100.

For instance, converting an interest rate of 12% into a decimal means you change 12% to 0.12 by dividing 12 by 100. Similarly, this process is reversed when expressing decimal results as percentages, by multiplying by 100. Understanding this conversion can clarify many calculations involving growth, interest, and other financial changes.

Basic Algebra

Basic Algebra involves manipulating equations to solve for unknown variables. It's a foundational math skill that allows us to rearrange and simplify expressions to uncover hidden values. In financial equations, you might need to solve for the total amount, principal, or rate given the rest of the variables.

For example, given the expression \( A = P(1 + r) \) for simple interest, we can rearrange if needed to solve for any of the other variables. Flexibility with algebraic expressions ensures you can adapt calculations to the specific requirements of a problem, enhancing problem-solving efficiency.

Financial Mathematics

Financial Mathematics applies mathematical methods to solve economic problems related to personal, business, and investment decisions. Simple interest calculations form a core part of this field, providing a straightforward way to assess the impact of single-period loans or investments.

By mastering simple financial mathematics concepts, you can estimate loan costs, investment returns, and savings growth. These analyses help individuals and businesses make informed decisions about borrowing, lending, and investing. Understanding how to apply mathematical formulas in financial contexts is thus a crucial skill for managing economic activities.