Understanding the Simple Interest Problem

This problem asks us to first find the initial sum of money (principal) based on the simple interest earned over a specific period at a given rate. Then, using this principal, we need to calculate the total amount payable (principal plus interest) for a different time period and interest rate.

Step 1: Calculate the Principal Sum

We are given the simple interest (SI), the time period (T), and the annual interest rate (R) for the first scenario. We can use the simple interest formula to find the principal sum (P).

The formula for Simple Interest is:

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

Where:

• SI = Simple Interest = ₹1,188
• R = Annual Interest Rate = 7.7% per annum
• T = Time Period = $2\frac{6}{7}$ years
• P = Principal Sum (what we need to find)

First, let's convert the time period into an improper fraction:

$ T = 2\frac{6}{7} = \frac{(2 \times 7) + 6}{7} = \frac{14 + 6}{7} = \frac{20}{7} \text{ years} $

Now, we rearrange the formula to solve for P:

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

Substitute the given values:

$ P = \frac{1188 \times 100}{7.7 \times \frac{20}{7}} $

Calculate the denominator ($R \times T$):

$ R \times T = 7.7 \times \frac{20}{7} = \frac{77}{10} \times \frac{20}{7} = \frac{11 \times 7}{10} \times \frac{2 \times 10}{7} = 11 \times 2 = 22 $

Now, calculate the Principal (P):

$ P = \frac{118800}{22} $

$ P = 5400 $

So, the Principal Sum is ₹5,400.

Step 2: Calculate the Simple Interest for the Second Scenario

Now we use the principal sum calculated in Step 1 (P = ₹5,400) to find the simple interest for the second scenario. The new rate and time period are given.

Given for the second scenario:

• P = ₹5,400
• R = 8.5% per annum
• T = $4\frac{5}{9}$ years

Convert the time period into an improper fraction:

$ T = 4\frac{5}{9} = \frac{(4 \times 9) + 5}{9} = \frac{36 + 5}{9} = \frac{41}{9} \text{ years} $

Using the simple interest formula again:

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

Substitute the values:

$ SI = \frac{5400 \times 8.5 \times \frac{41}{9}}{100} $

Simplify the calculation:

$ SI = \frac{5400 \times \frac{85}{10} \times \frac{41}{9}}{100} $

$ SI = \frac{54 \times 85 \times 41}{9 \times 10} $

$ SI = \frac{6 \times 85 \times 41}{10} \quad (\text{since } 54 \div 9 = 6) $

$ SI = \frac{510 \times 41}{10} $

$ SI = 51 \times 41 $

$ SI = 2091 $

The Simple Interest for the second scenario is ₹2,091.

Step 3: Calculate the Total Amount Payable

The total amount payable is the sum of the principal amount and the simple interest calculated for the second scenario.

Amount (A) = Principal (P) + Simple Interest (SI)

$ A = P + SI $

Substitute the values we found:

$ A = 5400 + 2091 $

$ A = 7491 $

Therefore, the total amount payable is ₹7,491.