Last month Grace worked, and was paid for, a total of x hours. Some of the hours were on the...

Understanding the Question

We need to find the exact number of hours Grace worked on the day shift last month.

Given Information

• Total hours worked = \(\mathrm{x}\) hours
• Hours split between day shift and night shift
• Night shift hourly pay is 20% higher than day shift hourly pay

What We Need to Determine

To answer this question, we must find a single, specific value for Grace's day shift hours. If multiple values are possible, the information is not sufficient.

Key Insight

Since night shift pays 20% more per hour than day shift, Grace earns more money per night shift hour. This creates an important connection between how her hours are split and how her total pay is distributed.

Analyzing Statement 1

Statement 1: \(\mathrm{x = 55}\) (total hours worked is 55)

This tells us the total hours, but nothing about how those 55 hours are split between day and night shifts.

Testing Different Scenarios

Let's see what's possible:

• Grace could have worked 40 day hours and 15 night hours
• She could have worked 20 day hours and 35 night hours
• She could have worked 50 day hours and 5 night hours

Each scenario adds up to 55 total hours, but gives a completely different answer to our question.

Conclusion

Since many different values for day shift hours are possible, Statement 1 alone is NOT sufficient.

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

Analyzing Statement 2

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

Statement 2: Grace's day shift pay was exactly 50% of her total pay

This means her night shift pay was also 50% of the total. But remember—night shift pays 20% more per hour.

Logical Analysis

Here's the key insight: If day shift and night shift each account for half the total pay, but night shift pays \(\mathrm{1.2 \times}\) as much per hour, what does this reveal?

Think of it this way: To earn the same total amount at different hourly rates, you need to work more hours at the lower rate. Specifically:

• \(\mathrm{Day\,hours \times (day\,rate) = Night\,hours \times (1.2 \times day\,rate)}\)
• Since both equal half the total pay, we can see that:
• \(\mathrm{Day\,hours = 1.2 \times Night\,hours}\)

We now know the ratio between day and night hours! But without knowing the total hours, we still can't find the actual number of day shift hours.

Examples to Illustrate

• If night hours = 10, then day hours = 12 (total = 22 hours)
• If night hours = 25, then day hours = 30 (total = 55 hours)
• If night hours = 50, then day hours = 60 (total = 110 hours)

Conclusion

Statement 2 gives us the relationship between day and night hours but not the actual values. It is NOT sufficient.

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

Combining Statements

Now let's use both statements together.

Combined Information

• From Statement 1: Total hours = 55
• From Statement 2: \(\mathrm{Day\,hours = 1.2 \times Night\,hours}\)

Why Together They Are Sufficient

Think of this as a parts problem:

• If night shift = 1 part
• Then day shift = 1.2 parts
• Total = 1 + 1.2 = 2.2 parts

Since 2.2 parts = 55 hours:

• 1 part = \(\mathrm{55 \div 2.2 = 25}\) hours
• So night shift = 25 hours
• And day shift = \(\mathrm{1.2 \times 25 = 30}\) hours

Verification

Let's verify this makes sense:

• Day hours: 30
• Night hours: 25
• Total: \(\mathrm{30 + 25 = 55}\) ✓
• Day pay (at rate r): 30r
• Night pay (at rate 1.2r): \(\mathrm{25 \times 1.2r = 30r}\)
• Day pay is indeed 50% of total pay (\(\mathrm{30r\,out\,of\,60r}\)) ✓

We now have a unique answer: Grace worked 30 hours on the day shift.

[STOP - Sufficient!] The statements together are sufficient.

The Answer: C

Both statements together give us exactly what we need—the total hours and the ratio between shifts—allowing us to find the unique number of day shift hours.

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