A sum of money is invested at a certain rate of compound interest. If the amount

A sum of money is invested at a certain rate of compound interest. If the amount becomes three times in 10 years, what is the annual interest rate?

A sum of money is invested at a certain rate of compound interest. If the amount becomes three times in 10 years, what is the annual interest rate?

Step 1: Understand that we are using the formula for compound interest, which is A = P(1 + r)^n.
Step 2: Identify what each symbol means: A is the final amount, P is the initial amount (principal), r is the annual interest rate, and n is the number of years.
Step 3: In this problem, we know that the amount becomes three times the initial amount after 10 years. So we can write A as 3P.
Step 4: Substitute A in the formula: 3P = P(1 + r)^10.
Step 5: Divide both sides of the equation by P (assuming P is not zero): 3 = (1 + r)^10.
Step 6: To find r, we need to isolate it. Take the 10th root of both sides: 1 + r = 3^(1/10).
Step 7: Calculate 3^(1/10) using a calculator or estimation. This gives approximately 1.116.
Step 8: Now, subtract 1 from both sides to find r: r = 1.116 - 1.
Step 9: This simplifies to r ≈ 0.116, which is about 0.15 when rounded.
Step 10: Convert r to a percentage by multiplying by 100: r ≈ 15%.

Compound Interest – Understanding how compound interest works and how to apply the formula A = P(1 + r)^n to calculate the future value of an investment.
Exponential Growth – Recognizing that the relationship between time and growth in compound interest is exponential, which can lead to non-linear results.
Algebraic Manipulation – Ability to manipulate equations to isolate variables and solve for the interest rate.