Math Word Problem: Solving Investment Amounts with Interest Rates

In this article, we will walk you through the process of solving a common math word problem involving investments and interest rates. You will learn how to set up equations, solve for unknown variables, and apply the unitary method to find the amounts invested at different interest rates. Let's dive into this problem and the detailed steps involved.

Problem Statement

A woman invests 4 times as much at 5% as she did at 6%. The total amount of interest she earns in 1 year from both accounts is 520 dollars. How much did she invest at 5% and at 6% in dollars?

Solution Approach

To solve this problem, we can use algebraic methods and the unitary method. We will break it down step-by-step to find the correct solution.

Step 1: Define the Variables

Let n be the amount invested at 6%. Then, the amount invested at 5% would be 4n since it is 4 times as much as the amount invested at 6%.

Step 2: Set Up the Equation

Using the interest formula, we know the interest earned from the 6% investment is 0.06n and from the 5% investment is 0.05(4n).

The total interest earned is:

0.06n 0.05(4n) 520

Simplifying this equation, we have:

0.06n 0.2n 520

0.26n 520

n 520 / 0.26 2000

Therefore, the amount invested at 6% is n 2000, and the amount invested at 5% is 4n 8000.

Alternative Solution

Another method to solve this problem is by representing the amounts as x and 4x. Let:

Amount invested at 6% x

Amount invested at 5% 4x

The interest from the investment at 6% is 0.06x and at 5% is 0.05(4x) 0.2x.

The total interest earned is 0.06x 0.2x 0.26x, and this is equal to 520 dollars.

Thus:

0.26x 520

x 520 / 0.26 2000

So, the amount invested at 6% is x 2000, and the amount invested at 5% is 4x 8000.

Conclusion

In this thorough analysis, we have solved the math word problem involving investments and interest rates. We found that the woman invested 2000 dollars at 6% and 8000 dollars at 5%. The solution process can be applied to similar problems by defining variables, setting up equations, simplifying them, and solving for the unknowns.

Additional Information

To further understand the concept of interest rates and investments, you might be interested in exploring:

Investment Strategies and Techniques Understanding Different Interest Rates Common Word Problems Involving Mathematics

The step-by-step approach detailed in this article should help you tackle similar math word problems with ease. Whether you are a student, a math enthusiast, or someone looking to improve your problem-solving skills, understanding these concepts is crucial.