Understanding the Simple Interest Problem

This question involves calculating the time period after which the total amount owed by two individuals, A and B, becomes equal. They have borrowed different principal amounts at different simple interest rates.

Key Concepts: Simple Interest

The fundamental formula for calculating Simple Interest (SI) is:

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

Where:

• P is the Principal amount (the initial amount borrowed).
• R is the Rate of interest per annum (p.a.).
• T is the Time period in years.

The total Amount (A) owed after time T is the sum of the Principal and the Simple Interest:

$ A = P + SI $

$ A = P + \frac{P \times R \times T}{100} $

Setting Up the Equations

Let's define the variables for both borrowers:

• For Borrower A:
  • Principal ($P_A$) = Rs. 16,000
  • Rate ($R_A$) = 12% p.a.
• For Borrower B:
  • Principal ($P_B$) = Rs. 12,000
  • Rate ($R_B$) = 18% p.a.
• Let the time period in years be t.

We need to find the value of t when the total amount owed by A ($A_A$) is equal to the total amount owed by B ($A_B$).

Calculating Amounts Owed

The amount owed by A after t years is:

$ A_A = P_A + \frac{P_A \times R_A \times t}{100} $

$ A_A = 16000 + \frac{16000 \times 12 \times t}{100} $

$ A_A = 16000 + (160 \times 12)t $

$ A_A = 16000 + 1920t $

The amount owed by B after t years is:

$ A_B = P_B + \frac{P_B \times R_B \times t}{100} $

$ A_B = 12000 + \frac{12000 \times 18 \times t}{100} $

$ A_B = 12000 + (120 \times 18)t $

$ A_B = 12000 + 2160t $

Equating the Amounts and Solving for Time (t)

We need to find t when $A_A = A_B$.

$ 16000 + 1920t = 12000 + 2160t $

Now, we rearrange the equation to solve for t:

Subtract 12000 from both sides:

$ 16000 - 12000 + 1920t = 2160t $

$ 4000 + 1920t = 2160t $

Subtract 1920t from both sides:

$ 4000 = 2160t - 1920t $

$ 4000 = 240t $

Divide by 240 to find t:

$ t = \frac{4000}{240} $

Simplify the fraction:

$ t = \frac{400}{24} $

Divide both numerator and denominator by 8:

$ t = \frac{50}{3} $

Convert the improper fraction to a mixed number:

$ t = 16 \frac{2}{3} \text{ years} $

Conclusion

The amounts owed by A and B will be equal after $16 \frac{2}{3}$ years.