Calculate Profit Shares

Let M's profit share be P_M and N's profit share be P_N. The total profit is ₹7,696.

• The sum of their shares is the total profit:

  $ P_M + P_N = 7696 $

• M's share is ₹962 more than N's:

  $ P_M = P_N + 962 $

• Substitute the second equation into the first:

  $ (P_N + 962) + P_N = 7696 $

  $ 2P_N + 962 = 7696 $

  $ 2P_N = 7696 - 962 = 6734 $

  $ P_N = \frac{6734}{2} = 3367 $

• Calculate M's share:

  $ P_M = 3367 + 962 = 4329 $

M's profit share is ₹4,329 and N's profit share is ₹3,367.

Establish Profit Ratio Equation

Let the capital invested by N be C_N and by M be C_M.

• M invested ₹24,000 more than N:

  $ C_M = C_N + 24000 $

• M invested for 3 months and N for 4 months. Profit is shared based on the product of capital and duration (Investment $\times$ Time).
• The ratio of their profit shares is equal to the ratio of their investment products:

  $ \frac{P_M}{P_N} = \frac{C_M \times 3}{C_N \times 4} $

• Substitute the calculated profit shares and the relation between capitals:

  $ \frac{4329}{3367} = \frac{(C_N + 24000) \times 3}{C_N \times 4} $

Simplify Ratio and Solve for N's Capital

First, simplify the profit ratio $\frac{4329}{3367}$. Both numbers are divisible by 481:

• $ \frac{4329}{481} = 9 $
• $ \frac{3367}{481} = 7 $
• So, the ratio is $\frac{9}{7}$.

The equation becomes:

• $ \frac{9}{7} = \frac{3(C_N + 24000)}{4C_N} $
• Cross-multiply:

  $ 9 \times 4C_N = 7 \times 3(C_N + 24000) $

  $ 36C_N = 21(C_N + 24000) $

  $ 36C_N = 21C_N + 21 \times 24000 $

  $ 36C_N = 21C_N + 504000 $

• Isolate C_N:

  $ 36C_N - 21C_N = 504000 $

  $ 15C_N = 504000 $

  $ C_N = \frac{504000}{15} = 33600 $

N's capital contribution is ₹33,600.

Calculate M's Capital Investment

Use the relationship between M's and N's capital:

• $ C_M = C_N + 24000 $
• $ C_M = 33600 + 24000 $
• $ C_M = 57600 $

M's capital contribution is ₹57,600.