A, B, and C started a business by investing Rs 5y, Rs z, and Rs (y+z+100). After six months, A withdrew 50% of his initial investment, B added Rs 500, and C added Rs 200. At the en... A, B, and C started a business by investing Rs 5y, Rs z, and Rs (y+z+100). After six months, A withdrew 50% of his initial investment, B added Rs 500, and C added Rs 200. At the end of one year, profit share of A and B is the same and profit share of C is 20% more than that of B. What is the initial investment of A and C together? Sum of initial investment of A and average initial investment of B and C together. Read more

Steps to Solve

1. Define Investments

   Let the initial investments of A, B, and C be:

   • A = $5y$
   • B = $z$
   • C = $(y + z + 100)$

2. Calculate A and C's initial investment

   The combined initial investment of A and C is: $$ A + C = 5y + (y + z + 100) = 6y + z + 100 $$

3. Calculate the average investment of B and C

   Average investment of B and C is: $$ \text{Average} = \frac{B + C}{2} = \frac{z + (y + z + 100)}{2} = \frac{y + 2z + 100}{2} $$

4. Set up the equation for their profit shares

   Given that A and B's profit share is the same and C's profit share is 20% more than B's: Let profit share of B = $P$. Then,

   • Profit share of C = $P + 0.2P = 1.2P$

5. Set up equations of investments to find relationships

   The amount of money earned by each party from their investments over time must be adjusted based on their initial investments:

   • Since A and B have the same profit share: $$ \text{Profit share of A} = \text{Profit share of B} $$

6. Compare Quantities

   Now we compare Quantity I and Quantity II:

   • Quantity I = Initial investment of A and C ($6y + z + 100$)
   • Quantity II = Sum of initial investment of A and the average initial investment of B and C.

   Which is: $$ \text{Quantity II} = 5y + \frac{y + 2z + 100}{2} $$

7. Analyze and simplify the equations

   Simplifying Quantity II gives: $$ \text{Quantity II} = 5y + \frac{y + 2z + 100}{2} = 5y + \frac{y}{2} + z + 50 $$

8. Final comparison of quantities

   By equating and simplifying both quantities and noting their dependencies on variables $y$ and $z$, determine whether Quantity I is greater, less or equal to Quantity II.