Problem 54 - Beginning and Intermediate Algebra

Question

Solve each problem using a quadratic equation. Use the formula \(A=P(1+r)^{2}\) to find the interest rate \(r\) at which a principal \(P\) of \(\$ 10,000\) will increase to \(\$ 10,920.25\) in 2 yr.

Step-by-Step Solution

Answer

The interest rate is 4.5\%.

Step 1 - Write down the formula

The given formula is: \[ A = P(1 + r)^2 \] Here, A is the amount after 2 years, P is the principal amount, and r is the interest rate.

Step 2 - Substitute known values

Substitute the given values into the formula: \[ 10,920.25 = 10,000(1 + r)^2 \] We need to solve for r.

Step 3 - Divide both sides by the principal P

Divide both sides of the equation by 10,000 to isolate the term with r: \[ \frac{10,920.25}{10,000} = (1 + r)^2 \] This simplifies to: \[ 1.092025 = (1 + r)^2 \]

Step 4 - Take the square root of both sides

To solve for r, take the square root of both sides of the equation: \[ \sqrt{1.092025} = 1 + r \] The square root of 1.092025 is approximately 1.045:

Step 5 - Solve for r

Isolate r by subtracting 1 from both sides: \[ 1.045 - 1 = r \] This simplifies to: \[ r = 0.045 \]

Step 6 - Write the interest rate as a percentage

To express the interest rate as a percentage, multiply r by 100: \[ 0.045 \times 100 = 4.5\text{\%} \] The interest rate is 4.5\%.

Key Concepts

Interest Rate Calculation

Interest rate calculation is a fundamental part of financial mathematics. It helps us understand how much our money will grow over time. The interest rate is usually represented as a percentage. It shows the amount of interest that will be earned on the principal (initial investment) over a specific period.
In our exercise, we found the interest rate using a given formula. First, we substituted the known values into the formula and then isolated the term with the interest rate to solve for it.
To make the calculation easier, it is important to follow each step methodically and ensure that the arithmetic is performed correctly. This will help avoid common mistakes and accurately determine the interest.

Compound Interest Formula

The compound interest formula is crucial in finance because it calculates the interest on both the initial principal and the interest that has been added over previous periods. This formula is different from simple interest, where you only earn interest on the initial principal.
The formula used in our exercise is \(A = P(1 + r)^2\). Here, \(A\) stands for the amount of money accumulated after n (in this case, 2) years, including interest. \(P\) is the original amount of money (the principal), and \(r\) is the annual interest rate.
By using the compound interest formula, you can see how much the investment will grow over time and understand the effect of compounding, which helps your money increase faster than simple interest.

Solving Quadratic Equations

Solving quadratic equations is an essential skill in algebra. A quadratic equation typically looks like \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants. In our example, we simplified the formula to \( (1 + r)^2 = 1.092025\) .
We began by isolating the quadratic term. Then we took the square root of both sides to simplify further, enabling us to solve for the variable \(r\). After this, isolating \(r\) helped us find the exact interest rate.
To tackle quadratic equations effectively, always make sure you simplify where possible and use algebraic techniques such as factoring, completing the square, or using the quadratic formula, depending on what best fits the specific problem.