If two copying machines work simultaneously at their respective constant rates, how many copies do they produce in 5text{ minutes}?...

Understanding the Question

We need to find: If two copying machines work simultaneously at their respective constant rates, how many copies do they produce in 5 minutes?

This is a value question - we need to determine a specific number of copies. For this to be sufficient, we must be able to calculate exactly one unique value for the total copies produced.

Given information:

• Two copying machines working together
• Each has its own constant rate
• Time period: 5 minutes

What we need: The exact total number of copies produced in 5 minutes

To find this total, we need to know both machines' individual rates. Once we have both rates, the calculation is straightforward: Total copies = \(\mathrm{5 \times (rate_1 + rate_2)}\).

Analyzing Statement 1

Statement 1: One of the machines produces copies at the constant rate of 250 copies per minute.

This gives us one machine's rate (250 copies/minute), but we don't know:

• Which machine has this rate (is it Machine 1 or Machine 2?)
• What the other machine's rate is

Without the second machine's rate, we cannot determine the total. Consider these possibilities:

• If the other machine produces 100 copies/min → Total = \(\mathrm{5 \times (250 + 100) = 1,750}\) copies
• If the other machine produces 500 copies/min → Total = \(\mathrm{5 \times (250 + 500) = 3,750}\) copies

Since different values for the unknown rate lead to different totals, Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Now we forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: One of the machines produces copies at twice the constant rate of the other machine.

This gives us a relationship between the rates (one is double the other), but not the actual rates. If we call the slower rate "r," then the faster rate is "2r."

Without knowing the value of r, we cannot determine the total:

• If r = 100 copies/min → Machines produce 100 and 200 copies/min → Total = \(\mathrm{5 \times 300 = 1,500}\) copies
• If r = 150 copies/min → Machines produce 150 and 300 copies/min → Total = \(\mathrm{5 \times 450 = 2,250}\) copies

Different base rates lead to different totals, so Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Statements

Now let's see what happens when we use both statements together:

• From Statement 1: One machine produces 250 copies/minute
• From Statement 2: One machine produces at twice the rate of the other

Here's the critical insight: Even with both pieces of information, we still face ambiguity.

Scenario 1: The 250 copies/minute machine is the faster one

• If 250 is the faster rate, and one machine is twice as fast as the other
• Then: \(\mathrm{250 = 2 \times (slower\ rate)}\)
• So the slower machine produces 125 copies/minute
• Total = \(\mathrm{5 \times (250 + 125) = 1,875}\) copies

Scenario 2: The 250 copies/minute machine is the slower one

• If 250 is the slower rate, and one machine is twice as fast as the other
• Then: \(\mathrm{(faster\ rate) = 2 \times 250}\)
• So the faster machine produces 500 copies/minute
• Total = \(\mathrm{5 \times (250 + 500) = 3,750}\) copies

Since we get two different possible totals (1,875 vs. 3,750) depending on which machine has the 250 copies/minute rate, we cannot determine a unique answer. The statements together are NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice C.

The Answer: E

Since neither statement alone is sufficient, and even combining both statements doesn't give us a unique answer, the answer is E.

Strategic Insight: Sometimes in Data Sufficiency, even when statements give us specific values and relationships, ambiguity about which piece of information applies to which variable can prevent us from finding a unique answer. Always check whether the given information uniquely determines what you're looking for.