P invested ₹ 5000 per month for 6 months of a year and Q invested ₹ x per
month for 8 months of the year in a partnership business. The profit is shared in
proportion to the total investment made in that year.
If at the end of that investment year, Q receives
4
of the total profit, what is the
value of x (in ₹)?
Options:
(A) 2500
(B) 3000
(C) 4687
(D) 8437
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.7)**
**1. Mathematical Formulation**:
- **P's Investment**: ₹ $5000$ per month for $6$ months:
$I_P = 5000 \times 6 = ₹\, 30,000$
- **Q's Investment**: ₹ $x$ per month for $8$ months:
$I_Q = x \times 8 = 8x$
- **Profit Sharing**: The profit is divided in direct proportion to total annual investment:
$\frac{\text{Profit}_Q}{\text{Total Profit}} = \frac{I_Q}{I_P + I_Q} = \frac{4}{9}$
**2. Detailed Step-by-Step Derivation**:
- Substitute $I_P$ and $I_Q$ into the profit ratio equation:
$\frac{8x}{30000 + 8x} = \frac{4}{9}$
- Cross-multiply to solve for $x$:
$9 \times (8x) = 4 \times (30000 + 8x)$
$72x = 120,000 + 32x$
$72x - 32x = 120,000$
$40x = 120,000$
$x = \frac{120,000}{40} = 3000$
- Therefore, $Q$ invested ₹ $3000$ per month.
**3. Official Verification**:
The verified official IIT Master Answer is **Option (B)** (`3000`).
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