An estimate of an actual data value has an error of p percent if p = (100|e - a|)/a, where...

Understanding the Question

Let's break down what we're being asked. Emma made estimates for both her total income and her tutoring income, and both estimates had less than 20% error. The question asks: Was Emma's actual tutoring income at most 45% of her actual total income?

This is a yes/no question - we need to determine if we can definitively answer whether the ratio of actual tutoring to actual total income is ≤ 45%.

What "Sufficient" Means Here

A statement is sufficient if it allows us to definitively answer YES or NO to whether actual tutoring income was at most 45% of actual total income.

Given Information

• Error formula: \(\mathrm{p} = \frac{100|\mathrm{e} - \mathrm{a}|}{\mathrm{a}}\) (where e = estimate, a = actual)
• Both estimates had error < 20%
• This means: \(0.8\mathrm{a} < \mathrm{e} < 1.2\mathrm{a}\) for both values

Key Insight

When estimates have error margins, the actual values can differ from the estimated ratios! We need to think about how these error margins could push the actual ratio higher or lower than the estimated ratio.

Analyzing Statement 1

Statement 1 tells us: Emma's estimated tutoring income was 30% of her estimated total income.

So Emma thought tutoring represented 30% of her total income. But since both estimates could be off by up to 20%, could the actual ratio reach or exceed 45%?

Testing the Extreme Case

To maximize the actual tutoring-to-total ratio, we need:

• Tutoring to be underestimated (actual is higher than estimate)
• Total to be overestimated (actual is lower than estimate)

Let's check the most extreme case:

• If tutoring estimate was 20% too low: actual tutoring = \(1.2 \times \mathrm{estimate}\)
• If total estimate was 20% too high: actual total = \(\frac{\mathrm{estimate}}{1.2}\)

Since estimated tutoring = \(0.3 \times \mathrm{estimated\ total}\):

• Actual ratio = \(\frac{1.2 \times 0.3 \times \mathrm{estimated\ total}}{\mathrm{estimated\ total} \div 1.2}\)
• Actual ratio = \(0.36 \times 1.2 = 0.432 = 43.2\%\)

Even in this absolute extreme scenario, we get 43.2%, which is less than 45%.

Therefore, the answer is definitively YES - Emma's actual tutoring income was at most 45% of her actual total income.

[STOP - Sufficient!]

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 tells us: Emma's estimated total income was $40,000.

This gives us the dollar amount of her total income estimate, but tells us absolutely nothing about her tutoring income - neither the estimate nor the actual value.

Without any information about tutoring income, we cannot determine what percentage it represents of total income. The tutoring income could be:

• $4,000 (10% of total)
• $12,000 (30% of total)
• $20,000 (50% of total)
• Or any other amount

We have no way to answer whether tutoring was at most 45% of total.

Statement 2 is NOT sufficient.

This eliminates choices B and D.

The Answer: A

Since Statement 1 alone allows us to definitively answer the question (YES, tutoring was at most 45% of total), while Statement 2 alone provides no relevant information about the tutoring-to-total ratio, the answer is A.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."