Problem 73 - Algebra for College Students

Question

Solve \(A=P+P r\) for \(P\), given that \(A=\$ 2173.75\), \(r=8 \frac{3}{4} \%\), and \(t=2\) years.

Step-by-Step Solution

Answer

The value of \(P\) is approximately \$1849.57.

Step 1: Convert Percentage Rate to Decimal

First, convert the interest rate from a percentage to a decimal. The given rate is \(8\frac{3}{4}\%\) or \(8.75\%\). To convert this to a decimal, divide by 100: \(8.75\div 100 = 0.0875\).

Step 2: Substitute Known Values into Formula

Now, substitute the known values for \(A\), \(r\), and \(t\) into the equation. The formula becomes \(2173.75 = P + P \times 0.0875 \times 2\).

Step 3: Simplify the Equation

Simplify the equation on the right-hand side. This becomes \(2173.75 = P (1 + 0.175)\) or \(2173.75 = P \times 1.175\).

Step 4: Solve for P

To solve for \(P\), divide both sides of the equation by \(1.175\): \(P = \frac{2173.75}{1.175}\). Calculating this gives \(P \approx 1849.57\).

Step 5: Verify the Solution

Check the solution by substituting \(P\) back into the original formula: \(P + P \times 0.0875 \times 2 = 2173.75\). With \(P = 1849.57\), the left side calculates to approximately \(2173.75\), confirming the solution is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding an unknown variable by rearranging and simplifying an equation. In the given exercise, the task is to find the value of \( P \) in the equation \( A = P + P r \). The solution involves substituting known values, then simplifying and rearranging the equation to solve for \( P \). Steps include isolating \( P \) by dividing both sides by a common factor, which helps in clearly identifying the unknown variable's value. Mastering this process can simplify complex problems and help in deriving the solution accurately.

Percentage to Decimal Conversion

Percentage to decimal conversion is a useful tool in dealing with interest rate problems. In mathematical terms, converting a percentage to a decimal involves dividing the percentage value by 100. For example, the interest rate of \(8\frac{3}{4}\%\) given in the problem is converted by calculating \(\frac{8.75}{100} = 0.0875\). This decimal form is essential, as it directly aligns with how the rate is used in calculations in formulas. By converting percentages to decimals, you ensure consistency and correctness when performing mathematical operations, such as multiplication in equations.

Verification of Solutions

Verification of solutions is a crucial step to ensure that the answer obtained is correct. Once the value of \( P \) is calculated, checking the solution helps confirm its accuracy and detect any potential errors. In the exercise, after finding \( P \approx 1849.57 \), it is inserted back into the original equation: \( P + P \times 0.0875 \times 2 = 2173.75 \). This step verifies that both sides of the equation are equal, reinforcing the correctness of the solution. Ensuring solution verification prevents mistakes and strengthens understanding of the problem-solving process.

Algebraic Manipulation

Algebraic manipulation involves rearranging terms and simplifying expressions to isolate variables. In this exercise, manipulating the equation \( 2173.75 = P (1 + 0.175) \) to \( 2173.75 = P \times 1.175 \) simplifies the calculation process. By rearranging, you can solve for \( P \) more efficiently. Using algebraic manipulation helps to solve equations by breaking them down into simpler steps, making complex problems more manageable. Practicing these techniques in different scenarios builds proficiency and confidence in handling algebraic expressions across various mathematical contexts.