Calculating Simple Interest and Ratio

This problem involves calculating the simple interest earned on two different investments and then finding the ratio of these interests. Simple interest is calculated using the formula:

$$ \text{Simple Interest (SI)} = \frac{\text{Principal (P)} \times \text{Rate (R)} \times \text{Time (T)}}{100} $$

Where:

• P is the principal amount (the initial investment).
• R is the annual rate of interest in percent.
• T is the time period in years.

Simple Interest Calculation for Scenario A

In scenario A:

• Principal (P\(_A\)) = Rs. 6000
• Annual Rate (R\(_A\)) = 15 percent
• Time (T\(_A\)) = 5 years

Using the simple interest formula:

$$ \text{SI}_A = \frac{P_A \times R_A \times T_A}{100} $$

$$ \text{SI}_A = \frac{6000 \times 15 \times 5}{100} $$

$$ \text{SI}_A = \frac{6000 \times 75}{100} $$

$$ \text{SI}_A = 60 \times 75 $$

$$ \text{SI}_A = 4500 $$

So, the simple interest earned in scenario A is Rs. 4500.

Simple Interest Calculation for Scenario B

In scenario B:

• Principal (P\(_B\)) = Rs. 5000
• Annual Rate (R\(_B\)) = 20 percent
• Time (T\(_B\)) = 2 years

Using the simple interest formula:

$$ \text{SI}_B = \frac{P_B \times R_B \times T_B}{100} $$

$$ \text{SI}_B = \frac{5000 \times 20 \times 2}{100} $$

$$ \text{SI}_B = \frac{5000 \times 40}{100} $$

$$ \text{SI}_B = 50 \times 40 $$

$$ \text{SI}_B = 2000 $$

So, the simple interest earned in scenario B is Rs. 2000.

Finding the Ratio of Interests Earned

We need to find the ratio of the interests earned in A and B respectively, which is SI\(_A\) : SI\(_B\).

$$ \text{Ratio} = \text{SI}_A : \text{SI}_B $$

$$ \text{Ratio} = 4500 : 2000 $$

To simplify the ratio, we can divide both numbers by their greatest common divisor. Both numbers can be divided by 100:

$$ 4500 \div 100 = 45 $$

$$ 2000 \div 100 = 20 $$

The ratio becomes 45 : 20. Both 45 and 20 are divisible by 5:

$$ 45 \div 5 = 9 $$

$$ 20 \div 5 = 4 $$

The simplified ratio is 9 : 4.

Therefore, the ratio of interests earned in A and B respectively is 9 ∶ 4.

Revision Table: Simple Interest Calculation Summary

Additional Information: Understanding Simple Interest

Simple interest is a basic and quick method of calculating the interest charge on a loan or investment. It is calculated only on the principal amount, not on any accumulated interest. This is in contrast to compound interest, where interest is calculated on the initial principal and also on the accumulated interest from previous periods. Simple interest is often used for short-term loans or specific types of investments where the interest is paid out periodically rather than reinvested.

• Principal: The initial sum borrowed or invested.
• Interest Rate: The percentage charged or earned on the principal per period (usually annually).
• Time: The duration for which the principal is borrowed or invested.
• Simple Interest Amount: The total interest earned or paid over the time period.
• Total Amount: The sum of the principal and the simple interest earned. Total Amount = P + SI.

Understanding simple interest is fundamental before moving on to more complex concepts like compound interest or annuities.