Each type A machine fills 400 cans per minute, each Type B machine fills 600 cans per minute, and each...

Understanding the Question

Let's clarify what we need to find: How many machines are working in total?

We have three types of machines:

• Type A: fills 400 cans per minute
• Type B: fills 600 cans per minute
• Type C: installs 2,400 lids per minute

The crucial constraint: Every filled can gets exactly one lid (no more, no less). This means:

• Number of cans filled = Number of lids installed

To find the total number of machines, we need to determine exactly how many of each type (A, B, and C) are working.

Key Insight

This is a production balance problem. The lid constraint creates a fundamental relationship between all machine types—they must work together to match every can with exactly one lid.

Analyzing Statement 1

Statement 1: A total of 4,800 cans are filled that minute

This tells us the total production, but not how it's distributed among machine types. Let's explore what different combinations could produce 4,800 cans:

Scenario 1 - Using only Type A machines:

• We could have 12 Type A machines: \(12 \times 400 = 4,800\) cans
• These 4,800 cans need 4,800 lids → 2 Type C machines (\(2 \times 2,400 = 4,800\) lids)
• Total machines: \(12 + 0 + 2 = \)14 machines

Scenario 2 - Using only Type B machines:

• We could have 8 Type B machines: \(8 \times 600 = 4,800\) cans
• Still need 2 Type C machines for 4,800 lids
• Total machines: \(0 + 8 + 2 = \)10 machines

Scenario 3 - Using a mix:

• 6 Type A + 4 Type B: \((6 \times 400) + (4 \times 600) = 2,400 + 2,400 = 4,800\) cans
• Still need 2 Type C machines for lids
• Total machines: \(6 + 4 + 2 = \)12 machines

Since different combinations give us different totals (10, 12, or 14 machines), we can't determine a unique answer.

Statement 1 is NOT sufficient. [STOP - Not Sufficient!]

This eliminates answer choices A and D.

Analyzing Statement 2

Now let's completely forget Statement 1 and analyze Statement 2 independently.

Statement 2: For that minute, there are 2 Type B machines working for every Type C machine working

This gives us a ratio: \(\mathrm{B} = 2\mathrm{C}\). But without knowing the actual number of Type C machines, we can't determine the total. Let's test different scales:

If C = 1 (small scale):

• Then B = 2 (from the 2:1 ratio)
• Type B produces: \(2 \times 600 = 1,200\) cans
• Type C can handle: \(1 \times 2,400 = 2,400\) lids
• To balance production (cans must equal lids), we need: \((2,400 - 1,200) = 1,200\) more cans
• Type A machines needed: \(1,200 \div 400 = 3\) machines
• Total: \(3 + 2 + 1 = \)6 machines

If C = 2 (larger scale):

• Then B = 4 (from the 2:1 ratio)
• Type B produces: \(4 \times 600 = 2,400\) cans
• Type C can handle: \(2 \times 2,400 = 4,800\) lids
• To balance production, we need: \((4,800 - 2,400) = 2,400\) more cans
• Type A machines needed: \(2,400 \div 400 = 6\) machines
• Total: \(6 + 4 + 2 = \)12 machines

The ratio alone doesn't determine the scale—we could have 6 total machines, 12 total machines, or other values.

Statement 2 is NOT sufficient. [STOP - Not Sufficient!]

This eliminates answer choice B (and confirms D is already eliminated).

Combining Both Statements

Now let's use both statements together:

• From Statement 1: Total production = 4,800 cans
• From Statement 2: \(\mathrm{B} = 2\mathrm{C}\) (ratio of Type B to Type C machines)

Step 1: Determine Type C machines
Since we need 4,800 cans filled with 4,800 lids:

• Number of Type C machines = \(4,800 \div 2,400 = \)2 machines

Step 2: Determine Type B machines
With C = 2 and the ratio \(\mathrm{B} = 2\mathrm{C}\):

• Number of Type B machines = \(2 \times 2 = \)4 machines

Step 3: Determine Type A machines

• Type B production: \(4 \times 600 = 2,400\) cans
• Remaining cans needed: \(4,800 - 2,400 = 2,400\) cans
• Number of Type A machines: \(2,400 \div 400 = \)6 machines

Final answer: Total machines = \(6 + 4 + 2 = 12\) machines

The combination uniquely determines all values.

Both statements together are SUFFICIENT. [STOP - Sufficient!]

This eliminates answer choice E.

The Answer: C

Both statements together give us exactly what we need:

• Statement 1 provides the total production (which determines how many Type C machines we need)
• Statement 2 provides the B:C ratio (which then determines Type B, and consequently Type A)

Neither statement alone is sufficient, but together they uniquely determine that 12 machines are working.

Answer Choice C: Both statements together are sufficient, but neither statement alone is sufficient.