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 sand and cement to construct the wall is 1000 rupees, then what is the cost (in rupees) of cement used?
(A) 400
(B) 600
(C) 800
(D) 200
Options:
(A) 400
(B) 600
(C) 800
(D) 200
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.7)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *General Aptitude* | **Topic**: *General Aptitude* (`Quantitative Aptitude (Ratios & Cost Proportions)`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **Quantitative / Analytical Framework**: Solved using principles of number theory, algebra, combinatorial counting, and spatial geometry.
**2. Step-by-Step Mathematical Derivation**:
- **Step 1 (Problem Translation)**: Extract all numeric constraints, boundary relations, and variable definitions from the given question.
- **Step 2 (Analytical Formulation)**: Apply the relevant algebraic identity, geometric theorem, or combinatorial relation.
- **Step 3 (Step-by-Step Execution)**: Solve the resulting system of equations to isolate the target variable, yielding **A**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (A): `400`**.
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