Multiple choice

Two equal sums of money are lent at the same time at $8\%$ and $7\%$ per annum simple interest. The former is recovered $6$ months earlier than the later and the amount in each case is Rs. $2560$. The sum and time for which the sums of money are lent out are

  1. Rs. $1500, 3.5$ year and $4$ year
  2. Rs. $2000, 3.5$ years and $4$ years
  3. Rs. $2000, 4$ years and $5.5$ years
  4. Rs. $3000, 4$ years and $4.5$ years
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the sum be P and time be T. Amount = P + P*R*T/100. 2560 = P(1 + 0.08T) and 2560 = P(1 + 0.07(T+0.5)). Solving these equations: P(1 + 0.08T) = P(1 + 0.07T + 0.035) => 0.01T = 0.035 => T = 3.5 years. P = 2560 / (1 + 0.08*3.5) = 2560 / 1.28 = 2000.

AI explanation

Let the principal sum be P and the time for the 8% loan be t years, making the time for the 7% loan (t + 0.5) years. Using the simple interest formula for the total amount, we set up the equation 2560 = P + (P * 8 * t / 100) and 2560 = P + (P * 7 * (t + 0.5) / 100). Subtracting the second equation from the first and simplifying gives P * t / 100 = 3.5P / 100, so t equals 3.5 years; substituting t back into the first equation yields P + 0.28P = 2560. Solving 1.28P = 2560 gives the principal sum P as Rs. 2000, with the respective times being 3.5 and 4 years.

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