Understanding Simple and Compound Interest Rates

This problem asks us to find the ratio of two different interest rates: one for simple interest and one for compound interest, based on how an initial sum of money grows over time under each system.

Calculating Simple Interest Rate

Let the initial sum of money be P. At simple interest, the sum becomes two times itself in 8 years. This means the final amount is 2P. The interest earned is the final amount minus the principal, which is $\text{2P} - \text{P} = \text{P}$.

The formula for simple interest (SI) is:

$\text{SI} = \frac{\text{P} \times \text{R}_S \times \text{T}}{100}$

Where:

• SI = Simple Interest
• P = Principal amount
• $\text{R}_S$ = Rate of simple interest per annum
• T = Time in years

Given that SI = P and T = 8 years, we can substitute these values into the formula:

$\text{P} = \frac{\text{P} \times \text{R}_S \times 8}{100}$

We can divide both sides by P (assuming P is not zero):

$1 = \frac{\text{R}_S \times 8}{100}$

Now, we solve for $\text{R}_S$:

$\text{R}_S = \frac{100}{8} = 12.5\%$ per annum.

So, the simple interest rate is 12.5%.

Calculating Compound Interest Rate

For compound interest, the same sum of money P becomes four times itself in 2 years, compounded annually. This means the final amount is 4P.

The formula for the amount (A) after compound interest is:

$\text{A} = \text{P}\left(1 + \frac{\text{R}_C}{100}\right)^\text{T}$

Where:

• A = Amount after T years
• P = Principal amount
• $\text{R}_C$ = Rate of compound interest per annum
• T = Time in years

Given that A = 4P and T = 2 years, we substitute these values:

$4\text{P} = \text{P}\left(1 + \frac{\text{R}_C}{100}\right)^2$

Divide both sides by P:

$4 = \left(1 + \frac{\text{R}_C}{100}\right)^2$

Take the square root of both sides:

$\sqrt{4} = 1 + \frac{\text{R}_C}{100}$

$2 = 1 + \frac{\text{R}_C}{100}$

Subtract 1 from both sides:

$2 - 1 = \frac{\text{R}_C}{100}$

$1 = \frac{\text{R}_C}{100}$

Solve for $\text{R}_C$:

$\text{R}_C = 100\%$ per annum.

So, the compound interest rate is 100%.

Finding the Ratio of Interest Rates

We need to find the ratio of the simple interest rate ($\text{R}_S$) to the compound interest rate ($\text{R}_C$).

Ratio = $\text{R}_S : \text{R}_C$

Ratio = $12.5\% : 100\%$

To express this ratio in its simplest form, we can write it as a fraction and simplify:

$\frac{12.5}{100} = \frac{125}{1000}$ (multiplying numerator and denominator by 10)

Now, simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 125:

$\frac{125 \div 125}{1000 \div 125} = \frac{1}{8}$

So, the ratio of the simple interest rate to the compound interest rate is $1 : 8$.

Revision Table: Interest Concepts

Additional Information: Comparing Interest Types

Simple interest is a straightforward way to calculate interest, where the interest amount is fixed over the investment or loan period, depending only on the initial principal. It is often used for short-term loans or simple savings accounts.

Compound interest is often called "interest on interest". The interest earned in each period is added to the principal for calculating the interest in the next period. This leads to exponential growth. It is commonly used for long-term investments, loans, and mortgages.

In this problem, we saw how different interest types and time periods result in significantly different interest rates to achieve a certain growth multiple of the principal.