Machines A, B, and C produced identical bottles at different constant

Let the production rates of machines A, B, and C be \(a, b,\) and \(c\) bottles per hour, respectively.

Total production in each phase:
- A operates for 2 hours → produces \(2a\) bottles.
- B operates for 3 hours → produces \(3b\) bottles.
- C operates for 4 hours → produces \(4c\) bottles.

Statement (1)
\(\frac{a}{b} = \frac{3}{2}\)
This implies \(a = \frac{3}{2}b\).
Substituting in the production formula:
\(2a = 2 \times \frac{3}{2}b = 3b\).
Thus, the production contributions become:
- A: \(3b\)
- B: \(3b\)
- C: \(4c\)
Not sufficient

Statement (2)
\(\frac{b}{c} = \frac{4}{3}\)
This implies \(b = \frac{4}{3}c\).
Substituting in the production formula:
\(3b = 3 \times \frac{4}{3}c = 4c\).
Thus, the production contributions become:
- A: \(2a\)
- B: \(4c\)
- C: \(4c\)
Not sufficient

Combining (1) and (2):
From (1): \(a = \frac{3}{2}b\)
From (2): \(b = \frac{4}{3}c\)
Substituting \(b\) in the equation for \(a\):
\(a = \frac{3}{2} \times \frac{4}{3}c = 2c\)
Thus, the contributions become:
- A: \(2a = 4c\)
- B: \(3b = 4c\)
- C: \(4c\)

Since all machines contribute equally (4c each), no single machine filled the greatest part.

Final Answer: (C) Together, both statements are sufficient.