Finding Interest Rate and Time in Compound Interest

Cracking the Code: Finding Interest Rate and Time in Compound Interest

In previous lessons, we've focused on calculating the compound interest, maturity value, and present value. Now, let's flip the script! In Lesson 5, we'll learn how to solve for the missing pieces of the puzzle: the interest rate and the time period in compound interest scenarios.

Get ready to:

• Manipulate compound interest formulas to solve for the interest rate and time.

• Apply these formulas to real-world situations, such as determining the interest rate needed to reach a savings goal or calculating the time it takes for an investment to double.

By the end of this lesson, you'll be a master at deciphering the mysteries of compound interest, even when some key variables are unknown. Let's unlock the secrets!

Activity: Unveiling the Unknowns

Scenario 1: Finding the Interest Rate

You invest ₱20,000 in an account that compounds interest annually. After 5 years, your investment grows to ₱25,000. What annual interest rate did you earn?

Scenario 2: Finding the Time

You invest ₱15,000 in an account that earns 5% interest compounded annually. How long will it take for your investment to double?

This lesson equips you with the tools to determine the time or interest rate required to achieve specific financial goals using compound interest. Let's explore!

Example 1: Time to Reach a Savings Goal

Problem: How long will it take P3,000 to accumulate to P3,500 in a bank savings account at 0.25% compounded monthly?

Solution:

• P = P3,000

• F = P3,500

• i = 0.25%/12 = 0.0025/12 (monthly interest rate)

n = (log F/P) / log(1 + i) = (log 3500/3000) / log(1 + 0.0025/12) ≈ 77.4 months

Therefore, it will take approximately 77.4 months, or about 6 years and 5 months.

Example 2: Time to Earn a Specific Interest

Problem: How long will it take P1,000 to earn P500 in interest if the interest rate is 12% compounded semi-annually?

Solution:

• P = P1,000

• F = P1,500 (P1,000 + P500 interest)

• i = 12%/2 = 0.12/2 = 0.06 (semi-annual interest rate)

n = (log F/P) / log(1 + i) = (log 1500/1000) / log(1 + 0.06) ≈ 7.2 semi-annual periods

Since compounding is semi-annual, it will take approximately 3.6 years (7.2 periods / 2 periods per year).

Example 3: Finding the Nominal Interest Rate

Problem: At what nominal rate compounded semi-annually will P10,000 accumulate to P15,000 in 10 years?

Solution:

• P = P10,000

• F = P15,000

• n = 10 years * 2 periods per year = 20 periods

Using a financial calculator or iterative methods, we find that the semi-annual interest rate (i) is approximately 0.0205.

Nominal rate (r) = i * m = 0.0205 * 2 = 0.041 or 4.1%

Therefore, the nominal interest rate compounded semi-annually is approximately 4.1%.

Example 4: Doubling Money

Problem: At what interest rate compounded quarterly will money double itself in 10 years?

Solution:

• Let P = any initial amount

• F = 2P (double the initial amount)

• n = 10 years * 4 periods per year = 40 periods

Using the formula and solving for 'i', we get a quarterly interest rate of approximately 0.0175.

Nominal rate (r) = i * m = 0.0175 * 4 = 0.07 or 7%

Therefore, the interest rate compounded quarterly to double money in 10 years is approximately 7%.

Time to turn the tables! This assessment challenges you to find the interest rate and time in compound interest scenarios. You'll use your problem-solving skills and formula knowledge to uncover these key variables.

Solved for interest rate and time? Get ready to compare rates like a pro in Lesson 6!