At a certain store, the retail price of a coat was p percent less than its list price. If the...

Understanding the Question

Let's break down what's happening here. A coat goes through two price reductions:

1. List price → Retail price (reduced by \(\mathrm{p\%}\))
2. Retail price → Sale price (reduced by \(\mathrm{r\%}\))

What we need to find: The sale price as a percentage of the list price.

Here's the key insight: When you have two successive percentage reductions, the final result isn't simply \((100 - \mathrm{p} - \mathrm{r})\%\). The reductions interact with each other, creating a compound effect.

For this question to be sufficient, we need to find one specific value for how the sale price relates to the list price.

Analyzing Statement 1

What Statement 1 tells us: \(\mathrm{p} - \mathrm{r} + \frac{\mathrm{pr}}{100} = 10\)

This gives us a relationship between p and r, but notice it's about their difference (adjusted for their interaction). To understand why this might not be enough, let's test some scenarios.

Testing Different Scenarios

Let's see if different values of p and r can satisfy this equation while giving us different final answers:

Scenario 1: \(\mathrm{p} = 20, \mathrm{r} = 12.5\)

• Check: \(20 - 12.5 + \frac{20 \times 12.5}{100} = 7.5 + 2.5 = 10\) ✓
• Sale price calculation: After a \(20\%\) reduction then a \(12.5\%\) reduction, the sale price = \(100\% - 20\% - 12.5\% + 2.5\% = 70\%\) of list price

Scenario 2: \(\mathrm{p} = 30, \mathrm{r} = 23.33...\)

• Check: \(30 - 23.33 + \frac{30 \times 23.33}{100} = 6.67 + 7 \approx 10\) ✓
• Sale price calculation: After a \(30\%\) reduction then a \(23.33\%\) reduction, the sale price = \(100\% - 30\% - 23.33\% + 7\% \approx 53.67\%\) of list price

Different combinations of p and r satisfy Statement 1 but lead to different final percentages (\(70\%\) vs \(53.67\%\)).

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.

What Statement 2 provides: \(\mathrm{p} + \mathrm{r} - \frac{\mathrm{pr}}{100} = 40\)

This is exactly what we need! Here's why: the expression \(\mathrm{p} + \mathrm{r} - \frac{\mathrm{pr}}{100}\) represents the total percentage reduction from list price to sale price.

Why This Works

When you apply two successive reductions of \(\mathrm{p\%}\) and \(\mathrm{r\%}\), the formula for the total reduction is:

• Total reduction = \(\mathrm{p} + \mathrm{r} - \frac{\mathrm{pr}}{100}\)

If Statement 2 tells us this equals 40, then:

• Sale price = \((100 - 40)\% = \)\(60\%\) of list price

We don't need to know the individual values of p and r—knowing their combined effect directly gives us the answer we're looking for.

Statement 2 is sufficient.

[STOP - Sufficient!] This eliminates choices C and E, leaving only choice B.

The Answer: B

Statement 2 alone gives us exactly the information we need to determine the sale price as a percentage of the list price (\(60\%\)), while Statement 1 only gives us a difference relationship that allows multiple solutions.

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