Understanding the Simple Interest Problem

This problem involves simple interest calculations. We are given the amounts to which a certain sum grows after a specific number of years and asked to find the time it takes for the original sum to double itself at a modified interest rate.

Let the principal sum be \(P\) and the original rate of simple interest be \(x\)% per annum.

The formula for Simple Interest (SI) is:

\(SI = \frac{P \times R \times T}{100}\)

where:

• \(P\) is the Principal amount
• \(R\) is the Rate of interest per annum
• \(T\) is the Time period in years

The Amount (\(A\)) after \(T\) years is given by:

\(A = P + SI = P + \frac{P \times R \times T}{100}\)

Step 1: Finding the Simple Interest Earned

We are given the amount after 3 years and the amount after 5 years under simple interest. The difference between these amounts is the simple interest earned during the period from the end of the 3rd year to the end of the 5th year, which is a period of \(5 - 3 = 2\) years.

• Amount after 5 years = Rs. 92400
• Amount after 3 years = Rs. 81840

Simple Interest earned in 2 years = Amount after 5 years - Amount after 3 years

\(SI_{2 \text{ years}} = \text{Rs. } 92400 - \text{Rs. } 81840\)

\(SI_{2 \text{ years}} = \text{Rs. } 10560\)

Since simple interest is constant for every year on the original principal, the simple interest for 1 year is:

\(SI_{1 \text{ year}} = \frac{SI_{2 \text{ years}}}{2} = \frac{\text{Rs. } 10560}{2}\)

\(SI_{1 \text{ year}} = \text{Rs. } 5280\)

Step 2: Calculating the Principal Sum

We know the amount after 3 years and the simple interest earned in 3 years. The amount after 3 years is the principal plus the simple interest for 3 years.

Simple Interest earned in 3 years = \(3 \times SI_{1 \text{ year}}\)

\(SI_{3 \text{ years}} = 3 \times \text{Rs. } 5280 = \text{Rs. } 15840\)

Principal \(P\) = Amount after 3 years - Simple Interest for 3 years

\(P = \text{Rs. } 81840 - \text{Rs. } 15840\)

\(P = \text{Rs. } 66000\)

The original principal sum is Rs. 66000.

Step 3: Determining the Original Rate of Interest (x%)

We can find the original rate \(x\)% using the simple interest for 1 year, the principal, and the time (1 year).

\(SI = \frac{P \times R \times T}{100}\)

\(5280 = \frac{66000 \times x \times 1}{100}\)

\(5280 = 660 \times x\)

\(x = \frac{5280}{660}\)

\(x = \frac{528}{66}\)

\(x = 8\)

So, the original rate of interest is 8% per annum.

Step 4: Calculating the New Rate of Interest

The problem states that the new rate of interest is \((x + 2)\)% per annum.

New Rate \(R_{\text{new}} = (x + 2)\%\)

\(R_{\text{new}} = (8 + 2)\%\)

\(R_{\text{new}} = 10\%\) per annum

Step 5: Finding the Time for the Sum to Double at the New Rate

We need to find the time \(T\) it takes for the original sum \(P\) to double itself at the new rate of 10% per annum. When the sum doubles, the amount will be \(2P\).

Amount = Principal + Simple Interest

\(2P = P + SI\)

This means the simple interest earned must be equal to the principal amount:

\(SI = P\)

Now, use the simple interest formula with \(SI = P\), \(R = 10\%\), and the principal \(P\).

\(SI = \frac{P \times R_{\text{new}} \times T}{100}\)

\(P = \frac{P \times 10 \times T}{100}\)

Assuming the principal \(P\) is not zero, we can divide both sides by \(P\):

\(1 = \frac{10 \times T}{100}\)

\(1 = \frac{T}{10}\)

Now, solve for \(T\):

\(T = 1 \times 10\)

\(T = 10\)

It will take 10 years for the same sum to double itself at the new rate of 10% per annum simple interest.

Summary of Calculations

Revision Table: Simple Interest Concepts

Additional Information: Doubling Sums

When a sum of money doubles itself under simple interest, it means the total simple interest earned is equal to the original principal amount (\(SI = P\)).

Using the formula \(SI = \frac{P \times R \times T}{100}\), if \(SI = P\), we get:

\(P = \frac{P \times R \times T}{100}\)

Dividing both sides by \(P\) (assuming \(P \neq 0\)):

\(1 = \frac{R \times T}{100}\)

This gives a direct relationship between the rate and the time required for a sum to double under simple interest:

\(R \times T = 100\)

or

\(T = \frac{100}{R}\)

and

\(R = \frac{100}{T}\)

In our problem, the new rate is 10%. Using this relationship:

\(T = \frac{100}{10} = 10\)

This confirms our calculated time of 10 years for the sum to double at a 10% simple interest rate.