Directions : Avni and Beena started a business by investing Rs. P and Rs.1.2P…

2021

Directions : Avni and Beena started a business by investing Rs. P and Rs.1.2P respectively. After ‘x’ months, Avni withdrew her entire amount and Chetna entered into business. After end of 9 months, Beena increased her initial investment by 25%. On completion of one year, share of Avni and Beena in entire profit was Rs.14200. If Beena would have increased her investment after ‘x’ months, then ratio of share of Avni and Beena in entire profit would have been 10: 27 respectively.

If a man invested Rs. 1.2P for x years on simple interest at the rate of 2.5x % p.a. and received total interest of Rs. 6480, then find P?

  1. A.

    5400

  2. B.

    8000

  3. C.

    7200

  4. D.

    6000

  5. E.

    12000

Show answer & explanation

Correct answer: D

Concept

In a partnership, each partner's share of profit is proportional to their capital multiplied by the time it stays invested (capital-months). When investments change mid-year, sum each partner's capital over every distinct period. Simple Interest is given by SI = (Principal x Rate x Time) / 100.

Application

Set the yearly period to 12 months and let the unknown switch-point be x months. The trick is to first pin down x using the hypothetical ratio, then feed x into the simple-interest part.

  1. Avni invests Rs. P for the full count of x months, so her capital-time = x (the same in both the real and the hypothetical case).

  2. Actual case: Beena keeps Rs. 1.2P for 9 months, then raises it by 25% to Rs. 1.5P for the last 3 months. Beena's capital-time = 1.2P(9) + 1.5P(3) = 10.8P + 4.5P = 15.3P.

  3. Hypothetical case: Beena raises to Rs. 1.5P after x months, so Beena = 1.2P(x) + 1.5P(12 - x) = 18P - 0.3Px, while Avni stays at Px.

  4. Given hypothetical ratio Avni : Beena = 10 : 27, write Px / (18P - 0.3Px) = 10/27. Cancel P: x / (18 - 0.3x) = 10/27.

  5. Cross-multiply: 27x = 180 - 3x, so 30x = 180, giving x = 6.

  6. Now the simple-interest part: Principal = 1.2P, Time = x = 6 years, Rate = 2.5x = 2.5(6) = 15% per annum. SI = (1.2P x 15 x 6) / 100 = 108P/100 = 1.08P.

  7. Set SI = 6480: 1.08P = 6480, so P = 6480 / 1.08 = 6000.

Cross-check

Plug P = 6000 back: principal = 1.2(6000) = 7200, rate = 15%, time = 6 years. SI = 7200 x 15 x 6 / 100 = 7200 x 0.9 = 6480, which matches the given interest exactly. So P = 6000.

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