This article will guide you through the process of solving for the principal amount and the rate of interest in a simple interest scenario. We will use a specific problem to illustrate the steps and ensure that the solution is clear and comprehensible. Whether you are preparing for a math test or doing financial planning, understanding simple interest is crucial. Let's dive into the problem and see how we can find the principal amount and the rate.

Problem Recap

The given problem states that a sum of money at simple interest amounted to P7800 in 5 years and to P8840 in 9 years. The challenge is to determine the principal amount and the rate of interest.

Solving for the Principal Amount and Rate of Interest

Let's first restate the information given:

Amount in 5 years (A5) P7800 Amount in 9 years (A9) P8840 Time for A5 5 years Time for A9 9 years

We know the formula for simple interest is given by:

A P (Ptimes;Rtimes;T)

Where:

A Total amount after interest P Principal amount R Rate of interest (as a decimal) T Time in years

From the given information, we can write the equations for the two scenarios:

For A57800: 7800 P (Ptimes;Rtimes;5)

For A98840: 8840 P (Ptimes;Rtimes;9)

Step-by-Step Solution

We can simplify the equations:

7800 P(1 5R) ... (1)

8840 P(1 9R) ... (2)

By subtracting equation (1) from equation (2), we get:

8840 - 7800 P(1 9R) - P(1 5R)

1040 P(4R)

Dividing both sides by 4, we get:

260 P(R)

Solving for R, we find:

R 260/P

Substituting R in equation (1), we get:

7800 P(1 5(260/P))

7800 P(1 1300/P)

7800 P 1300

P 7800 - 1300 6500

Now, using P 6500 in the equation for R, we get:

R 260/6500 0.04 or 4%

Conclusion

The principal amount was P6500, and the rate of interest is 4%. This method can be applied to any similar problem involving simple interest to find the principal amount and the rate of interest.

Additional Insights

Understanding simple interest is fundamental for managing personal finances, understanding loan agreements, and making investment decisions. It is crucial when dealing with short-term investments or debts, as it helps in calculating the interest that is accumulated over a period. By knowing the principal amount and the rate, you can better plan your financial strategies and make informed decisions.

Final Thoughts

Solving for the principal amount and the rate of interest using simple interest formulas can be done systematically. This article has provided a step-by-step guide through a real-world problem, making the process clear and straightforward. Whether you are a student, a financial planner, or an investor, mastering these calculations can significantly impact your financial decisions. Happy calculating!