Problem 62 - Precalculus

Question

Toby split his savings into two different investments, one earning \(5 \%\) and the other earning \(7 \% .\) He put twice as much in the investment earning the higher rate. In one year, he earned \(\$ 665\) in interest. How much money did he invest in each account?

Step-by-Step Solution

Answer

Toby invested $3500 at 5% and $7000 at 7%.

Step 1: Define Variables

Let's denote the amount invested in the 5% account as \( x \). Since Toby invested twice as much into the 7% account as the 5% account, let the amount invested in the 7% account be \( 2x \).

Step 2: Write the Interest Equations

The interest earned from the 5% account is \( 0.05x \), and the interest earned from the 7% account is \( 0.07(2x) \). The total interest from both accounts is \$665, so we set up the equation: \( 0.05x + 0.14x = 665 \).

Step 3: Simplify and Solve the Equation

Combine the terms in the equation: \( 0.19x = 665 \). Solving for \( x \), divide both sides by 0.19: \( x = \frac{665}{0.19} \approx 3500 \).

Step 4: Determine Investment Amounts

From Step 1, we know that \( x \) is the amount in the 5% account, so \( x = 3500 \). The amount in the 7% account is \( 2x = 2 \times 3500 = 7000 \).

Key Concepts

Interest Calculation

Interest calculation is a key concept when it comes to investment word problems. In these problems, interest is typically the money earned from the amount invested over a certain period. There are different types of interests, like simple and compound, but we'll focus on simple interest here since it's what the problem involves.

For Toby's problem, the interest formula used is for simple interest, which is:

• Interest = Principal × Rate × Time

The principal is the initial investment, the rate is the percentage earned per year, and time is the period of investment, usually in years.

In Toby's case, the interest from each account is calculated with this formula. For the 5% investment, his interest is calculated as such:

• Interest from 5% = 0.05 × Principal (x in this case)
• Interest from 7% = 0.07 × Principal (which is 2x for the higher rate)

Adding up these interests gives the total interest Toby earned. Understanding this calculation helps us see how different interest rates affect the total earnings over the same period.

Linear Equations

Linear equations are mathematical expressions that describe a straight line when graphed. They are crucial in solving problems where variables need to be determined. In Toby's investment scenario, the solution involves setting up and solving a linear equation to find out the amounts invested.

Here, the linear equation represents the total interest earned from both investments. This is built from the two individual interest expressions for each account:

• For the 5% account: Interest = 0.05x
• For the 7% account: Interest = 0.07(2x)

Adding these expressions, the total interest equation is:\(0.05x + 0.14x = 665\)This equation includes the term for both investments, reflecting Toby's total earnings from the interest. Setting up this equation shows how different rates and amounts affect the total return and helps to unravel the initial amounts invested.

Solving Equations

The final step in finding how much Toby invested in each account is solving the equation we set up. Solving equations is about isolating the variable to find its value. The key here is to perform operations that make the equation simpler without changing its equality.

In this problem, we start with the equation:\(0.05x + 0.14x = 665\)We combine like terms to simplify it:

• \(0.19x = 665\)

The objective is to isolate \(x\), the amount invested at 5%. We do this by dividing both sides of the equation by 0.19:

• \(x = \frac{665}{0.19}\)

After calculating, we find \(x \approx 3500\). This implies Toby invested $3500 in the 5% account. For the 7% account, since it is double, he invested \(2 \times 3500 = 7000\). Understanding each step in solving equations ensures the solution makes sense and is correct.