**X**, **Y** and **Z** started a business with investments of Rs. $(a-1200)$, Rs. $a$, and Rs. $(a+1800)$ respectively. The profit of **Y** is invested in a scheme offering simple interest at 18\% p.a. for 5 years, and the interest received is Rs. 3600. If the total profit in the business is Rs. 4800 more than twice the profit of **Y**, then which of the following statements is/are correct?

(A) The value of $a$ is a multiple of 12.

(B) **Z** gets 37.5\% of the total profit.

(C) The sum of the investments of **X** and **Y** is completely divisible by 8.

Choose the correct option:

Question

**X**, **Y** and **Z** started a business with investments of Rs. $(a-1200)$, Rs. $a$, and Rs. $(a+1800)$ respectively. The profit of **Y** is invested in a scheme offering simple interest at 18\% p.a. for 5 years, and the interest received is Rs. 3600. If the total profit in the business is Rs. 4800 more than twice the profit of **Y**, then which of the following statements is/are correct?

(A) The value of $a$ is a multiple of 12.

(B) **Z** gets 37.5\% of the total profit.

(C) The sum of the investments of **X** and **Y** is completely divisible by 8.

Choose the correct option:

Accepted Answer

**Solution:**

1. **Calculate profit earned by Y from the interest scheme:**
   - Simple Interest (SI) = Rs. 3600
   - Rate = 18\% p.a.
   - Time = 5 years

Using SI formula:
   $$ SI = \frac{P 	imes R 	imes T}{100} $$

Rearranged to find principal (profit of Y):
   $$ P = \frac{SI 	imes 100}{R 	imes T} = \frac{3600 	imes 100}{18 	imes 5} = 4000 $$

2. **Calculate total profit:**
   - Given total profit = Rs. 4800 more than twice the profit of Y
   - So,
   $$ 	ext{Total profit} = 2 	imes 4000 + 4800 = 12800 $$

3. **Find value of $a$ using investment ratios and profit sharing:**
   - Investments: X = $(a-1200)$, Y = $a$, Z = $(a+1800)$
   - Profit shares are proportional to investments:

Profit of Y = Rs. 4000
   Profit of X + Z = Total profit - Profit of Y = 12800 - 4000 = 8800

So,
   $$ \frac{a}{(a - 1200) + (a + 1800)} = \frac{4000}{8800} $$

Simplify denominator:
   $$ (a - 1200) + (a + 1800) = 2a + 600 $$

So,
   $$ \frac{a}{2a + 600} = \frac{4000}{8800} = \frac{10}{22} = \frac{5}{11} $$

Cross multiply:
   $$ 11a = 5(2a + 600) $$
   $$ 11a = 10a + 3000 $$
   $$ 11a - 10a = 3000 $$
   $$ a = 3000 $$

4. **Check statements:**

(A) Is $a$ a multiple of 12?
   - $3000 \div 12 = 250$ (exact)
   - So, $a$ **is** a multiple of 12.

(B) Calculate percentage of profit Z gets:
   - Investment ratio: X : Y : Z = $(a-1200) : a : (a+1800)$
   - Substitute $a=3000$:
     - X = 1800
     - Y = 3000
     - Z = 4800

Ratio = 1800 : 3000 : 4800 = 3 : 5 : 8

Total parts = 3 + 5 + 8 = 16

Profit of Z = $\frac{8}{16} 	imes 12800 = 6400$

Percentage of total profit:
   $$ \frac{6400}{12800} 	imes 100 = 50\% $$

Statement (B) claims 37.5\%, which is false.

(C) Sum of investments of X and Y:
   $$ 1800 + 3000 = 4800 $$

Check divisibility by 8:
   $$ 4800 \div 8 = 600 $$ (exact)

So, (C) is true.

**Answer:** Both (A) & (C) are correct, so option (e).

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