A, B, C started a business with their investments in the ratio 1:3:5. After 4 months, A invested the same amount as before and B as well as C withdrew half of their investments. The ratio of their profits at the end of the year is:

Options

A. 4:3:5
B. 5:6:10
C. 6:5:10
D. 10:5:6

Answer

To solve the problem step by step, we will first determine the investments of A, B, and C, and then calculate their effective capital contributions over time to find the ratio of their profits.

### Step 1: Define Initial Investments
Let the investments of A, B, and C be represented as follows based on the given ratio of 1:3:5:
- Investment of A = 1 unit
- Investment of B = 3 units
- Investment of C = 5 units

### Step 2: Calculate Time Periods
- A's investment remains for the entire year (12 months).
- B and C's investments remain for 4 months before they withdraw half of their investments.

### Step 3: Adjust Investments After 4 Months
After 4 months:
- A invests an additional 1 unit, making A's total investment = 1 + 1 = 2 units for the remaining 8 months.
- B withdraws half of his investment, so B's new investment = 3 / 2 = 1.5 units for the remaining 8 months.
- C withdraws half of his investment, so C's new investment = 5 / 2 = 2.5 units for the remaining 8 months.

### Step 4: Calculate Effective Capital Contributions
Now we calculate the effective capital contributions for each partner:
- A's effective capital = Investment × Time = 2 units × 8 months + 1 unit × 4 months = 16 + 4 = 20 units-months
- B's effective capital = Investment × Time = 1.5 units × 8 months + 3 units × 4 months = 12 + 12 = 24 units-months
- C's effective capital = Investment × Time = 2.5 units × 8 months + 5 units × 4 months = 20 + 20 = 40 units-months

### Step 5: Calculate the Ratio of Profits
Now we have:
- A's effective capital = 20 units-months
- B's effective capital = 24 units-months
- C's effective capital = 40 units-months

To find the ratio of their profits, we take:
- A : B : C = 20 : 24 : 40

### Step 6: Simplify the Ratio
To simplify the ratio, we can divide each term by the greatest common divisor (GCD):
- A : B : C = 20/4 : 24/4 : 40/4 = 5 : 6 : 10

Thus, the final ratio of their profits at the end of the year is **5 : 6 : 10**.