Understanding Simple Interest Calculation

This problem involves calculating the final amount on a sum of money based on simple interest. First, we need to determine the rate of simple interest from the initial information provided. Then, using this rate, we calculate the simple interest and the final amount for the second sum over a different period.

Step 1: Calculate the Simple Interest Rate

The first part of the question gives us the initial principal amount, the final amount after a certain time, and the time period. We can use this to find the simple interest earned and then the rate of interest per annum.

• Initial Principal ($\text{P}_1$): Rs. 640
• Amount after 2 years ($\text{A}_1$): Rs. 832
• Time period ($\text{T}_1$): 2 years

The simple interest ($\text{SI}_1$) earned is the difference between the amount and the principal:

\text{SI}_1 = \text{A}_1 - \text{P}_1

\text{SI}_1 = 832 - 640

\text{SI}_1 = 192 \text{ Rs.}

The formula for simple interest is:

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

Where P is Principal, R is the Rate of Interest per annum, and T is the Time in years.

We can rearrange this formula to find the rate (R):

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

Substituting the values from the first scenario:

\text{R} = \frac{192 \times 100}{640 \times 2}

\text{R} = \frac{19200}{1280}

\text{R} = \frac{1920}{128}

Let's simplify this fraction:

\text{R} = \frac{1920 \div 64}{128 \div 64} = \frac{30}{2} = 15

So, the simple interest rate is 15% per annum.

Step 2: Calculate the Amount for the Second Sum

Now we use the rate (R = 15%) calculated in Step 1 to find the amount for the second principal over the given time period.

• New Principal ($\text{P}_2$): Rs. 860
• Time period ($\text{T}_2$): 4 years
• Rate ($\text{R}$): 15% per annum

First, calculate the simple interest ($\text{SI}_2$) for the second sum:

\text{SI}_2 = \frac{\text{P}_2 \times \text{R} \times \text{T}_2}{100}

\text{SI}_2 = \frac{860 \times 15 \times 4}{100}

\text{SI}_2 = \frac{860 \times 60}{100}

\text{SI}_2 = \frac{86 \times 60}{10}

\text{SI}_2 = 86 \times 6

\text{SI}_2 = 516 \text{ Rs.}

The final amount ($\text{A}_2$) will be the sum of the new principal and the simple interest earned:

\text{A}_2 = \text{P}_2 + \text{SI}_2

\text{A}_2 = 860 + 516

\text{A}_2 = 1376 \text{ Rs.}

Summary of Calculations

Therefore, Rs. 860 will become Rs. 1,376 in 4 years at the same rate of simple interest.

Revision Table: Key Concepts

Additional Information: Simple Interest vs. Compound Interest

It's important to understand the difference between simple interest and compound interest.

• Simple Interest: Interest is calculated only on the original principal amount throughout the entire term. This means the interest earned does not get added back to the principal to earn further interest. The interest amount is the same for each period (assuming constant principal and rate).
• Compound Interest: Interest is calculated on the initial principal and also on the accumulated interest from previous periods. Interest is added to the principal at the end of each period, and the next period's interest is calculated on this new, larger principal. This leads to faster growth of the amount compared to simple interest.

This question specifically deals with simple interest, where the calculation is straightforward based on the original principal amount only.