The value of Machine 1 depreciates x% per year and the value of Machine 2 depreciates y% each year. Will...

Understanding the Question

We need to determine whether Machine 1's value will be greater than Machine 2's value after 10 years.

Given Information

• Machine 1 depreciates \(\mathrm{x}\%\) per year
• Machine 2 depreciates \(\mathrm{y}\%\) per year
• We need a definitive YES or NO answer

What We Need to Determine

Will Machine 1 be worth more than Machine 2 after 10 years?

This is a yes/no question, so "sufficient" means we can definitively answer either YES (Machine 1 will be worth more) or NO (Machine 1 will not be worth more) in all cases.

Key Insight

Two competing factors will determine the outcome:

1. Starting value advantage - If Machine 1 starts higher, it has a head start
2. Depreciation rate disadvantage - If Machine 1 depreciates faster, it loses value more quickly

The winner after 10 years depends on which factor dominates.

Analyzing Statement 1

Statement 1 tells us: Machine 1's current value is twice Machine 2's current value.

So Machine 1 starts with a \(2 \times\) advantage, but we know nothing about how fast each machine loses value.

Testing Different Scenarios

Let's explore whether different depreciation rates could lead to different outcomes:

Scenario 1 - Equal depreciation: If both machines depreciate at \(5\%\) per year

• Machine 1 maintains its \(2 \times\) advantage throughout
• After 10 years: Machine 1 still worth more → Answer is YES

Scenario 2 - Extreme difference: If Machine 1 depreciates at \(50\%\) per year, Machine 2 at \(1\%\) per year

• Machine 1 loses most of its value (nearly worthless after 10 years)
• Machine 2 retains most of its value
• The \(2 \times\) starting advantage gets overwhelmed → Answer is NO

Conclusion

Since different depreciation scenarios lead to different answers (YES in some cases, NO in others), Statement 1 alone cannot give us a definitive answer.

Statement 1 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 tells us: \(\mathrm{x} = 2\mathrm{y}\) (Machine 1 depreciates twice as fast as Machine 2)

Now we know the depreciation relationship, but nothing about the starting values.

Testing Different Scenarios

Scenario 1 - Equal starting values: If both machines start at the same value

• Machine 1 loses value twice as fast
• After 10 years: Machine 1 will definitely be worth less → Answer is NO

Scenario 2 - Machine 1 starts much higher: If Machine 1 starts at \(10 \times\) Machine 2's value

• Even losing value twice as fast, Machine 1's huge advantage could persist
• After 10 years: Machine 1 could still be worth more → Answer is YES

Conclusion

Since different starting value scenarios lead to different answers (YES in some cases, NO in others), Statement 2 alone cannot give us a definitive answer.

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

This eliminates choice B.

Combining Statements

Using both statements together, we know:

• Machine 1 starts with \(2 \times\) the value (Statement 1)
• Machine 1 depreciates \(2 \times\) as fast (Statement 2)

The Critical Question

Does a \(2 \times\) starting advantage overcome a \(2 \times\) depreciation disadvantage over 10 years?

The answer depends on the actual depreciation rates!

With low depreciation rates (say \(\mathrm{y} = 2\%\), \(\mathrm{x} = 4\%\)):

• After 10 years, Machine 2 retains about \(82\%\) of its value
• Machine 1 retains about \(66\%\) of its value
• Machine 1's value: \(2 \times 66\% = 132\%\) of Machine 2's starting value
• Machine 2's value: \(82\%\) of its starting value
• Machine 1 stays ahead → Answer is YES

With high depreciation rates (say \(\mathrm{y} = 40\%\), \(\mathrm{x} = 80\%\)):

• Both machines lose most of their value
• Machine 1 loses value much more dramatically
• The depreciation effect dominates the starting advantage → Answer is NO

Conclusion

Even with both statements, we get different answers depending on the actual rate values. The competing factors of "\(2 \times\) starting advantage" versus "\(2 \times\) faster depreciation" play out differently at different rate levels.

The statements together are NOT sufficient. [STOP - Not Sufficient!]

The Answer: E

The statements together are not sufficient because the outcome depends on whether we're dealing with low or high depreciation rates. When rates are low, the starting advantage persists. When rates are high, the depreciation disadvantage dominates.

Answer Choice E: "The statements together are not sufficient."