Workers A, B, and C, working at their individual constant rates, can work together and complete a piece of work...

Understanding the Question

We need to find: How long will it take for workers B and C together to complete a piece of work?

Given Information

• Workers A, B, and C each work at individual constant rates
• Working together, all three can complete the work in 6 days

What We Need to Determine

We're looking for the time it takes B and C together to finish the job. Since we know the combined rate of all three workers, we need to find what fraction of that combined rate belongs to B and C.

For sufficiency: We need to be able to determine a single, specific value for the time B and C need.

Key Insight

This problem is perfect for ratio thinking. Instead of calculating exact work rates as fractions, we can use the relationships between workers' efficiencies to find what proportion of the total work rate belongs to B and C together.

Analyzing Statement 1

Statement 1: A is twice as efficient as B. B is three times as efficient as C.

This gives us a complete efficiency ratio. Let's visualize this using simple units:

• If C's efficiency = 1 unit
• Then B's efficiency = 3 units (three times C)
• Then A's efficiency = 6 units (twice B's 3 units)

Total efficiency = 6 + 3 + 1 = 10 units

The key insight: B and C together represent 4 units out of 10 total units.

• This means B and C contribute \(\frac{4}{10} = \frac{2}{5}\) of the combined work rate
• Since they do \(\frac{2}{5}\) of the work rate, they would need \(\frac{5}{2}\) times as long working alone

Time for B and C = \(\frac{5}{2} \times 6\) days = 15 days

[STOP - Sufficient!]

Statement 1 is sufficient.

This eliminates choices B, C, and E.

Analyzing Statement 2

Important: Let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: The time taken by A to do 9 units of work equals the time taken by B and C together to do 6 units of work.

This reveals the ratio of their work rates. Think about it this way:

• In the same time period:
  • A completes 9 units
  • B and C together complete 6 units
• Therefore, their rate ratio is \(A : (B+C) = 9 : 6 = 3 : 2\)

In simpler terms: For every 3 parts of work A does, B and C together do 2 parts.

When all three work together:

• A contributes 3 parts of the work rate
• B and C together contribute 2 parts
• Total = 5 parts

Therefore, B and C represent \(\frac{2}{5}\) of the total work rate.

Since B and C have \(\frac{2}{5}\) of the total rate, they need \(\frac{5}{2}\) times as long:
Time for B and C = \(\frac{5}{2} \times 6\) days = 15 days

[STOP - Sufficient!]

Statement 2 is sufficient.

The Answer: D

Both statements independently reveal that B and C together contribute \(\frac{2}{5}\) of the combined work rate. This allows us to calculate that they need 15 days working together.

Notice how both statements, though worded differently, lead us to the same fundamental insight about the proportion of work B and C can do together.

Answer Choice D: Each statement alone is sufficient.