GATE PH 2022 • Q3
MCQ • 1 Mark
General Aptitude
Quantitative Aptitude
Subtopic: Percentages, Ratios & Proportions
Question Statement:
If
p ∶ q = 1 ∶ 2
q ∶ r = 4 ∶ 3
r ∶ s = 4 ∶ 5
and u is 50% more than s, what is the ratio p ∶ u?
Options:
-
(A) 2 ∶ 15
-
(B) 16 ∶ 15
-
(C) 1: 5
-
(D) 16: 45
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.3)**
**1. Mathematical Formulation**:
- **Given Ratios**:
- $\frac{p}{q} = \frac{1}{2}$
- $\frac{q}{r} = \frac{4}{3}$
- $\frac{r}{s} = \frac{4}{5}$
- $u$ is $50\%$ more than $s$: $u = s + 0.5s = 1.5s = \frac{3}{2}s \implies \frac{s}{u} = \frac{2}{3}$.
- **Objective**: Find the compound ratio $p : u$.
**2. Detailed Step-by-Step Derivation**:
- Multiply the chain of ratios:
$\frac{p}{u} = \left(\frac{p}{q}\right) \times \left(\frac{q}{r}\right) \times \left(\frac{r}{s}\right) \times \left(\frac{s}{u}\right)$
- Substitute the numerical values:
$\frac{p}{u} = \left(\frac{1}{2}\right) \times \left(\frac{4}{3}\right) \times \left(\frac{4}{5}\right) \times \left(\frac{2}{3}\right)$
- Simplify the factors:
$\frac{p}{u} = \frac{1 \times 4 \times 4 \times 2}{2 \times 3 \times 5 \times 3} = \frac{32}{90} = \frac{16}{45}$
- Thus, $p : u = 16 : 45$.
**3. Official Verification**:
The verified official IIT Master Answer is **Option (D)** (`16: 45`).
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