Solve each problem. See Example 4. Zhu inherited \$200,000 from her grandmother. She first gave 30% to her favorite charity. She invested some of the rest at 1.5% and some at 4%, earning \$4350 interest per year. How much did she invest at each rate?

Verified step by step guidance

1. Identify the total amount Zhu inherited: \$200,000.
2. Calculate the amount given to charity by finding 30% of \$200,000 using the formula \(\(\text{Charity}\) = 0.30 \(\times\) 200,000\).
3. Determine the remaining amount after charity by subtracting the charity amount from the total inheritance: \(\text{Remaining} = 200,000 - \text{Charity}\).
4. Let \(x\) be the amount invested at 1.5%, and \(y\) be the amount invested at 4%. Write the equation for the total invested: \(x + y = \text{Remaining}\).
5. Write the interest equation based on the rates and total interest earned: \(0.015x + 0.04y = 4350\). Use the system of equations to solve for \(x\) and \(y\).

Key Concepts

Percentage Calculations

Understanding how to calculate percentages is essential to determine the amount Zhu gave to charity and the remaining amount she invested. For example, 30% of $200,000 is found by multiplying 200,000 by 0.30, which helps in finding the leftover principal for investment.

Systems of Linear Equations

This problem requires setting up and solving a system of linear equations to find the amounts invested at each interest rate. One equation represents the total invested amount, and the other represents the total interest earned, allowing simultaneous solution for the two unknowns.

Simple Interest Calculation

Simple interest is calculated by multiplying the principal by the interest rate. Knowing how to apply this formula helps in expressing the total interest earned from each investment, which is crucial for forming the equations needed to solve the problem.