Question Statement:
If
πβΆπ= 1 βΆ2
πβΆπ= 4 βΆ3
πβΆπ = 4 βΆ5
and π’ is 50% more than π , what is the ratio πβΆπ’?
Options:
-
(A) 2 βΆ15
-
(B) 16 βΆ15
-
(C) 1: 5
-
(D) 16: 45
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2022 Q.3)**
**1. Governing Concept**:
- **Topic**: Quantitative Aptitude & Compound Ratios
- Given ratios:
$\frac{p}{q} = \frac{1}{2}, \quad \frac{q}{r} = \frac{4}{3}, \quad \frac{r}{s} = \frac{4}{5}$
- Finding the compound ratio $\frac{p}{s}$:
$\frac{p}{s} = \frac{p}{q} \times \frac{q}{r} \times \frac{r}{s} = \left(\frac{1}{2}\right) \times \left(\frac{4}{3}\right) \times \left(\frac{4}{5}\right) = \frac{16}{30} = \frac{8}{15}$
- Given that $u$ is $50\%$ more than $s$:
$u = s + 0.5s = 1.5s = \frac{3}{2}s \implies \frac{s}{u} = \frac{2}{3}$
- Therefore, the ratio $\frac{p}{u}$ is:
$\frac{p}{u} = \frac{p}{s} \times \frac{s}{u} = \frac{8}{15} \times \frac{2}{3} = \frac{16}{45}$
**2. Option Analysis**:
- **Option (A) [2 : 15] is INCORRECT**: Arithmetic error in compound reduction.
- **Option (B) [16 : 15] is INCORRECT**: Missing denominator factor 3.
- **Option (C) [1 : 5] is INCORRECT**: Oversimplified ratio.
- **Option (D) [16 : 45] is CORRECT**: Derived precisely as $\frac{8}{15} \times \frac{2}{3} = \frac{16}{45}$.
**3. Final Answer**:
- **Correct Choice**: **D**
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