To construct a wall, sand and cement are mixed in the ratio of 3:1. The cost of sand and that of cement are in the ratio of 1:2. If the total cost of the mixture is Rs. 1000, what is the cost of cement used?
Options:
(A) Rs. 400
(B) Rs. 600
(C) Rs. 500
(D) Rs. 300
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2023 Q.7)**
**1. Governing Concept**:
- **Topic**: Quantitative Aptitude & Weighted Cost Ratios
- Quantity ratio of sand to cement $= 3 : 1$ (let quantities be $3x$ and $1x$).
- Unit cost ratio of sand to cement $= 1 : 2$ (let unit costs be $1y$ and $2y$).
- Total cost of mixture:
$\text{Cost} = (3x)(1y) + (1x)(2y) = 3xy + 2xy = 5xy = \text{Rs. } 1000 \implies xy = 200$
- Cost of cement used:
$\text{Cost of cement} = 1x \cdot 2y = 2xy = 2(200) = \text{Rs. } 400$
**2. Option Analysis**:
- **Option (A) [Rs. 400] is CORRECT**: Directly matches $2 \times 200 = \text{Rs. } 400$.
- **Option (B) [Rs. 600] is INCORRECT**: Cost of sand ($3xy = 3 \times 200 = \text{Rs. } 600$).
- **Option (C) [Rs. 500] is INCORRECT**: Assumes equal expenditure.
- **Option (D) [Rs. 300] is INCORRECT**: Miscalculated ratio factor.
**3. Final Answer**:
- **Correct Choice**: **A**
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