In the summer, water consumption is known to decrease overall by 25%. A Water Board official
states that in the summer household consumption decreases by 20%. while other consumption
increases by 70%.
Which of the following statements is correct?
Options:
(A) The ratio of household to other consumption is 8/17
(B) The ratio of household to other consumption is 1/17
(C) The ratio of household to other consumption is 17/8
(D) There are errors in the official's statement.
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2017 Q.58)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (1 Mark)
- **Topic**: Quantitative Aptitude & Data Consistency (Weighted Averages & System of Equations)
**2. Detailed Step-by-Step Derivation**:
- Let $H$ be the household water consumption and $O$ be the other water consumption during normal conditions ($H > 0, O > 0$).
- Total initial consumption is $T = H + O$.
- According to the overall data:
- In summer, total water consumption decreases by $25\%$:
$T_{\text{summer}} = 0.75 T = 0.75(H + O)$
- According to the Water Board official:
- Household consumption decreases by $20\% \implies H_{\text{summer}} = 0.80 H$.
- Other consumption increases by $70\% \implies O_{\text{summer}} = 1.70 O$.
- Therefore, the official's total summer consumption is:
$T_{\text{summer}} = 0.80 H + 1.70 O$
- Equating the two expressions for summer consumption:
$0.80 H + 1.70 O = 0.75(H + O)$
$0.80 H + 1.70 O = 0.75 H + 0.75 O$
$0.80 H - 0.75 H + 1.70 O - 0.75 O = 0$
$0.05 H + 0.95 O = 0$
$\frac{H}{O} = -\frac{0.95}{0.05} = -19$
- Since physical water consumptions $H$ and $O$ must be strictly positive real numbers ($H > 0$ and $O > 0$), their ratio cannot be negative!
- This mathematical impossibility proves that the official's stated figures are self-contradictory.
- Therefore, **there are errors in the official's statement**.
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: Yields a positive ratio that mathematically contradicts the equations.
- **Option (B) [INCORRECT]**: Erroneous ratio.
- **Option (C) [INCORRECT]**: Erroneous ratio.
- **Option (D) [CORRECT]**: The system of linear equations yields a strictly negative consumption ratio ($-19$), proving that the official's statement is mathematically inconsistent.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **D**
- **Scoring Criteria**: +1.0 Mark (Negative marking: -0.33 marks).
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.