A, B and C invested in a business in the ratio $6 : 8 : 9$. If B invested for a period whose numerical value is 112.5\% of B’s investment but A and C invested for one year. If profit of B at the end of the year is 16750 then what is the share of profit of C?

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Accepted Answer

**Solution:**

1. Given investment ratio of A, B, and C is $6 : 8 : 9$.

2. A and C invested for 1 year each.

3. B invested for a period whose numerical value is 112.5\% of B’s investment amount.

- 112.5\% = $\frac{112.5}{100} = \frac{9}{8}$

- So, B's investment period = $\frac{9}{8}$ years.

4. Profit sharing is proportional to the product of investment and time.

- A's share = $6 	imes 1 = 6$

- B's share = $8 	imes \frac{9}{8} = 9$

- C's share = $9 	imes 1 = 9$

5. So, the ratio of profits = $6 : 9 : 9 = 2 : 3 : 3$ (dividing by 3).

6. Given B's profit = 16750, which corresponds to ratio part 3.

7. Find the value of one part:

$$	ext{One part} = \frac{16750}{3} = 5583.33$$

8. C's profit = $3 	imes 5583.33 = 16750$ (Wait, this contradicts the original solution. Let's re-examine.)

**Re-examining the solution:**

The original solution states:

- Profit ratio = $12 	imes 6 : 8 	imes \left(\frac{9}{8}ight) : 9 	imes 12$

- Then simplified to $12 	imes 6 : 8 	imes 9 : 9 	imes 12$

- Then to $2 : 2 : 3$

This seems inconsistent because the time for A and C is 1 year, but the solution uses 12 for A and C.

Let's clarify the time units:

- If B's investment time is $\frac{9}{8}$ years, then to avoid fractions, multiply all times by 12 (months):

- A's time = 12 months

- B's time = $\frac{9}{8} 	imes 12 = 13.5$ months

- C's time = 12 months

- Investment amounts: A=6, B=8, C=9

- So, profit shares:

- A = $6 	imes 12 = 72$

- B = $8 	imes 13.5 = 108$

- C = $9 	imes 12 = 108$

- Ratio = $72 : 108 : 108 = 2 : 3 : 3$

Given B's profit corresponds to 3 parts = 16750

One part = $\frac{16750}{3} = 5583.33$

C's profit = $3 	imes 5583.33 = 16750$

This contradicts the original answer.

**Original question states B's profit is 16750 and the answer for C's profit is 25125.**

Let's check the original solution's ratio:

- The original solution uses 12 for A and C's time, and for B it uses $8 	imes \frac{9}{8} = 9$ (investment times time).

- So ratio is $12 	imes 6 : 9 : 12 	imes 9 = 72 : 9 : 108$

- This is $8 : 1 : 9$ after dividing by 9, which is inconsistent.

**Correct approach:**

- Let time for A and C be 1 year.

- B's time = 112.5\% of B's investment amount (numerical value), meaning time = $112.5\%$ of investment amount.

- B's investment amount = 8 units.

- B's time = $112.5\%$ of 8 = $8 	imes \frac{112.5}{100} = 9$ units (time).

- So, B's time = 9 (units of time).

- So, profit shares:

- A = $6 	imes 12 = 72$ (assuming 12 months)

- B = $8 	imes 9 = 72$

- C = $9 	imes 12 = 108$

- Ratio = $72 : 72 : 108 = 2 : 2 : 3$

- Given B's profit corresponds to 2 parts = 16750

- One part = $\frac{16750}{2} = 8375$

- C's profit = $3 	imes 8375 = 25125$

**Answer:** 25125

This matches the original solution.

**Answer:** 25125

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