Problem 24 - Algebra and Trigonometry

Question

Investments Jack invests \(\$ 1000\) at a certain annual interest rate, and he invests another \(\$ 2000\) at an annual rate that is one-half percent higher. If he receives a total of \(\$ 190\) interest in 1 year, at what rate is the \(\$ 1000\) invested?

Step-by-Step Solution

Answer

The \$1000 is invested at an annual interest rate of 6%.

Step 1: Identify Variables

Let the annual interest rate for the first investment (\(1000) be denoted by \( r \) in decimal form. Then, the annual interest rate for the second investment (\)2000) is \( r + 0.005 \).

Step 2: Set Up the Interest Equations

Calculate the interest from each investment separately. The interest from the \(\(1000\) investment is \( 1000 \times r \). The interest from the \(\)2000\) investment is \( 2000 \times (r + 0.005) \).

Step 3: Write the Total Interest Equation

According to the problem, the total interest received is \($190\). So, the equation is: \[ 1000 \times r + 2000 \times (r + 0.005) = 190 \].

Step 4: Simplify the Equation

Distribute and simplify the equation: \[ 1000r + 2000r + 10 = 190 \]. Combine like terms to get \[ 3000r + 10 = 190 \].

Step 5: Solve for \( r \)

Subtract 10 from both sides to isolate terms involving \( r \): \[ 3000r = 180 \]. Divide both sides by 3000: \[ r = \frac{180}{3000} = 0.06 \].

Step 6: Convert \( r \) to Percentage

Convert \( r \) from a decimal to a percentage by multiplying by 100. Thus, \( r = 0.06 \times 100 = 6\% \).

Key Concepts

Understanding Annual Interest Rate

When you deposit money into an account or invest it, you're often entitled to earn interest. The annual interest rate indicates how much money you will earn over a year, expressed as a percentage. In simpler terms, it's the rate at which your investment grows annually.
Imagine you have an investment of $1000 and an annual interest rate of 5%. By the end of the year, you would earn $50, because 5% of 1000 is $50.

• An annual interest rate is crucial for comparing the potential earnings from different investments.
• It is expressed as "r" in equations, and usually converted to decimal form (for example, 5% becomes 0.05).
• When interest compounds annually, the percentage is applied to your initial amount at the end of each year.

Understanding this rate helps in making informed investment decisions, ensuring you know how quickly your money can grow.

Basics of Investment Calculation

Investment calculation is all about figuring out how much you will earn over a period, based on the principal amount and the interest rate. In Jack's scenario, he invests \( \\( 1000 \) at a certain rate and \( \\) 2000 \) at a different rate. Calculating the annual interest from these investments involves multiplying the principal by the interest rate.
For instance, if the interest rate for his first investment is "r", the interest earned is \( 1000 \times r \). For the second investment with a slightly higher rate, the calculation is \( 2000 \times (r + 0.005) \).

• Each investment separately earns interest based on its own rate.
• The total interest from both is the sum of these individual interests.

Investment calculations are vital in predicting earnings and making strategic financial choices.

Solving Algebraic Equations in Investment Problems

Algebraic equations come into play when figuring out things like unknown interest rates. In Jack's example, we set up an equation to find out the rate of his first investment. The goal is to solve for the variable "r".
The equation for the total interest Jack receives is: \( 1000r + 2000(r + 0.005) = 190 \).
Here's the step-by-step process:

• Simplify the equation using distribution: \( 1000r + 2000r + 10 = 190 \)
• Combine like terms to get: \( 3000r + 10 = 190 \)
• Isolate "r" by subtracting 10: \( 3000r = 180 \)
• Ultimately, solve for "r" by dividing: \( r = \frac{180}{3000} = 0.06 \), which is 6% when converted to a percentage.

Algebraic equation solving is essential for breaking down complex financial situations and finding unknown variables.