Ann, Beth, and Steve each spent some amount of time on a certain project last week. If the average (arithmetic...

Understanding the Question

We need to find the median time spent by Ann, Beth, and Steve on a project.

Given Information

• Three people worked on the project: Ann, Beth, and Steve
• Average time per person = 15 hours
• Since there are 3 people, total time = \(3 \times 15 = 45\) hours

What We Need to Determine

The median is the middle value when the three times are arranged in order. For this question to be sufficient, we need to be able to determine one specific value for the median.

Key Insight

Here's the crucial insight: For any three numbers with a fixed sum (45 hours), if one number equals the average of the other two, that number must be the median. This is because it sits perfectly between them - it can't be the highest or lowest value.

Analyzing Statement 1

Statement 1: Ann and Beth spent a total of 30 hours on the project.

What This Reveals

If \(\mathrm{Ann} + \mathrm{Beth} = 30\) hours, and the total is 45 hours, then Steve must have spent 15 hours.

The Strategic Connection

Notice what's happening here:

• \(\mathrm{Ann} + \mathrm{Beth} = 30\) hours
• Their average = \(30 \div 2 = 15\) hours
• \(\mathrm{Steve} = 15\) hours

Steve's time exactly equals the average of Ann and Beth's times. When one value in a set of three equals the average of the other two, it must be the median - it's guaranteed to be in the middle position regardless of how Ann and Beth's 30 hours are distributed between them.

For example:

• If \(\mathrm{Ann} = 10, \mathrm{Beth} = 20\), then the order is 10, 15, 20 → median = 15
• If \(\mathrm{Ann} = 5, \mathrm{Beth} = 25\), then the order is 5, 15, 25 → median = 15

Conclusion

The median is 15 hours.

[STOP - Statement 1 is SUFFICIENT!]

This eliminates choices B, C, and E.

Analyzing Statement 2

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

Statement 2: Steve spent 15 hours on the project.

What This Reveals

If \(\mathrm{Steve} = 15\) hours, and the total is 45 hours, then \(\mathrm{Ann} + \mathrm{Beth} = 30\) hours.

The Same Pattern Emerges

We've arrived at exactly the same situation as Statement 1:

• \(\mathrm{Steve} = 15\) hours
• \(\mathrm{Ann} + \mathrm{Beth} = 30\) hours (so their average is 15 hours)

Once again, Steve's time equals the average of the other two times, which means Steve's 15 hours must be the median value.

Conclusion

The median is 15 hours.

[STOP - Statement 2 is SUFFICIENT!]

This eliminates choices A and C.

The Answer: D

Both statements independently lead us to the same insight: one person's time equals the average of the other two, making that person's time the median.

Answer Choice D: Each statement alone is sufficient.