Questions: 1.3 Homework - Modeling with Linear Equations (6) Interest in 3% fund + Interest in 4 1/2% fund = Total interest Now assign labels. Recall that the total amount invested in both the funds is 18,000. Amount in 3% fund = x Amount in 4 1/2% fund = 18,000 - x Interest in 3% fund = 0.03x Interest in 4 1/2% fund = 0.045(18,000 - x) Step 2 Substitute these labels into the above verbal model to form an algebraic equation. Recall that the total interest Equation: 0.03x + 0.045(18,000 - x) = 800

1.3 Homework - Modeling with Linear Equations

(6) Interest in 3% fund + Interest in 4 1/2% fund = Total interest

Now assign labels. Recall that the total amount invested in both the funds is 18,000.
Amount in 3% fund = x
Amount in 4 1/2% fund = 18,000 - x
Interest in 3% fund = 0.03x
Interest in 4 1/2% fund = 0.045(18,000 - x)

Step 2
Substitute these labels into the above verbal model to form an algebraic equation. Recall that the total interest
Equation: 0.03x + 0.045(18,000 - x) = 800

Transcript text: 1.3 Homework - Modeling with Linear Equations (6) Interest in $3\%$ fund $)+($ Interest in $4 \frac{1}{2}\%$ fund $)=$ Total interest Now assign labels. Recall that the total amount invested in both the funds is $\$18,000$. Amount in $3\%$ fund $=x$ Amount in $4 \frac{1}{2}\%$ fund $=18,000-x$ Interest in $3\%$ fund $=0.03x$ Interest in $4 \frac{1}{2}\%$ fund $=0.045(18,000-x)$ Step 2 Substitute these labels into the above verbal model to form an algebraic equation. Recall that the total interest Equation: $0.03x + 0.045(18,000-x) = 800$

Solution

Solution Steps

Solution Approach

To solve the problem, we need to set up an algebraic equation based on the given information. We know the total amount invested is $18,000, and we have two funds with different interest rates. We assign variables to represent the amounts in each fund, calculate the interest for each, and then set up an equation to find the total interest, which is given as $800. We solve this equation to find the amount invested in each fund.

Step 1: Define Variables

Let $ x $ be the amount invested in the $ 3\% $ fund. Then, the amount invested in the $ 4\frac{1}{2}\% $ fund is given by: \[ 18000 - x \]

Step 2: Calculate Interest

The interest earned from the $ 3\% $ fund is: \[ 0.03x \] The interest earned from the $ 4\frac{1}{2}\% $ fund is: \[ 0.045(18000 - x) = 810 - 0.045x \]

Step 3: Set Up the Equation

The total interest from both funds is given as $ 800 $. Therefore, we can set up the equation: \[ 0.03x + (810 - 0.045x) = 800 \]

Step 4: Simplify and Solve the Equation

Combining like terms, we have: \[ 810 - 0.015x = 800 \] Subtracting $ 810 $ from both sides gives: \[ -0.015x = -10 \] Dividing both sides by $ -0.015 $ results in: \[ x = \frac{10}{0.015} = 666.666666666667 \]

Final Answer

The amount invested in the $ 3\% $ fund is: \[ \boxed{x = 666.67} \]

The amount invested in the $ 4\frac{1}{2}\% $ fund is: \[ 18000 - 666.67 = 17333.33 \] Thus, the final amounts are: \[ \boxed{x = 666.67} \quad \text{and} \quad \boxed{18000 - x = 17333.33} \]

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