Find the Amount Invested at Each Rate Calculator

Use this calculator to solve algebra "mixture" problems where a total sum is split between two investments with different interest rates, yielding a specific total return. Enter the total amount, the total interest earned, and the two specific interest rates to determine how much was allocated to each.

Understanding Investment Mixture Problems

One of the most common applications of linear equations in finance and algebra involves determining how a specific principal amount was divided between two different accounts or investment vehicles. This is often phrased as: "A total amount was invested, part at Rate X and the rest at Rate Y. The total interest earned was Z. How much was invested at each rate?"

This calculator automates the system of linear equations required to solve this problem, providing instantaneous results for students, investors, and financial planners.

The Mathematical Formula

To find the amounts invested at each rate, we use a system of two equations with two variables. Let x be the amount invested at the first rate (Rate 1), and y be the amount invested at the second rate (Rate 2).

The variables are defined as:

• P: Total Principal (Total Investment Amount)
• I: Total Interest Earned
• r₁: First Interest Rate (in decimal form)
• r₂: Second Interest Rate (in decimal form)

The system of equations is:

1. x + y = P
2. (x * r₁) + (y * r₂) = I

By solving for x through substitution, we derive the direct formula used in this calculator:

Amount at Rate 1 (x) = (I – (P * r₂)) / (r₁ – r₂)

Once x is found, the amount at Rate 2 (y) is simply the remainder:

Amount at Rate 2 (y) = P – x

Real-World Example

Consider a scenario where an investor has $10,000 to invest. They put some of it into a conservative bond yielding 5% and the rest into a high-yield account earning 10%. At the end of the year, the total interest earned is $700.

Using the logic above:

• Total (P) = 10,000
• Interest (I) = 700
• Rate 1 (r₁) = 0.05
• Rate 2 (r₂) = 0.10

Calculation:
x = (700 – (10,000 * 0.10)) / (0.05 – 0.10)
x = (700 – 1000) / -0.05
x = -300 / -0.05
x = $6,000 (Amount at 5%)

The remaining amount is $10,000 – $6,000 = $4,000 (Amount at 10%).

Why Am I Getting Negative Numbers?

If the calculator returns negative values, it means the scenario is mathematically impossible. For example, if you invest $10,000 between accounts earning 5% and 10%, the minimum possible interest is $500 (all at 5%) and the maximum is $1,000 (all at 10%). If you enter a Total Interest of $1,200, the math will break because that return cannot be achieved with the specified rates.

(Warning: Negative values indicate the Total Interest entered is not mathematically possible given the two rates and principal provided.)