A shirt and a jacket are each for sale at a price that is discounted from the item's original price....

Understanding the Question

We need to determine which item has the lower discounted price - the shirt or the jacket?

Both items are on sale at discounted prices. We must figure out which final price is lower after applying the discounts.

What makes this sufficient: We need to definitively determine whether the shirt OR the jacket has the lower discounted price. If we can establish that one item will always be cheaper regardless of specific values, or if we can calculate the exact relationship, then we have sufficiency.

Key Insight

This is a comparison problem with two competing factors:

1. The jacket starts at a higher original price (pushing its final price up)
2. The jacket gets a larger discount (pulling its final price down)

The outcome depends on which factor dominates.

Analyzing Statement 1

Statement 1: The jacket's original price was twice the shirt's original price.

So if the shirt costs \(\$50\), the jacket costs \(\$100\). But we know nothing about the discount percentages.

Let's test different scenarios:

• Scenario 1: Both items get the same \(10\%\) discount
  • Shirt: \(\$50 \rightarrow \$45\) (\(10\%\) off)
  • Jacket: \(\$100 \rightarrow \$90\) (\(10\%\) off)
  • Result: Jacket is still more expensive ✓
• Scenario 2: Shirt gets \(20\%\) off, jacket gets \(70\%\) off
  • Shirt: \(\$50 \rightarrow \$40\) (\(20\%\) off)
  • Jacket: \(\$100 \rightarrow \$30\) (\(70\%\) off)
  • Result: Now the jacket is cheaper! ✗

Since we can get different answers depending on the discount percentages, we cannot determine which item has the lower price.

Statement 1 is NOT sufficient.

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

Analyzing Statement 2

Now we forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: The percent of discount on the jacket is three times the percent of discount on the shirt.

So if the shirt gets \(10\%\) off, the jacket gets \(30\%\) off. But we don't know the original prices.

Let's test different scenarios:

• Scenario 1: Shirt costs \(\$50\), jacket costs \(\$200\)
  • Shirt: \(\$50 \rightarrow \$45\) (\(10\%\) off)
  • Jacket: \(\$200 \rightarrow \$140\) (\(30\%\) off)
  • Result: Jacket is still more expensive despite bigger discount ✓
• Scenario 2: Shirt costs \(\$50\), jacket costs \(\$60\)
  • Shirt: \(\$50 \rightarrow \$40\) (\(20\%\) off)
  • Jacket: \(\$60 \rightarrow \$24\) (\(60\%\) off)
  • Result: Now the jacket is cheaper! ✗

The answer depends on the relationship between original prices.

Statement 2 is NOT sufficient.

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

Combining Both Statements

Now we know both:

• Jacket's original price = \(2 \times\) Shirt's original price
• Jacket's discount = \(3 \times\) Shirt's discount

Let's see if this pins down the answer. If the shirt originally costs \(\$50\) and gets discount percentage s:

• Shirt final price: \(\$50 \times (1 - s)\)
• Jacket final price: \(\$100 \times (1 - 3s)\)

The key question: Does the jacket's \(3\times\) bigger discount overcome its \(2\times\) higher starting price?

Testing with small discounts (shirt \(5\%\) off, jacket \(15\%\) off):

• Shirt: \(\$50 \rightarrow \$47.50\)
• Jacket: \(\$100 \rightarrow \$85\)
• Result: Jacket remains more expensive ✓

Testing with large discounts (shirt \(30\%\) off, jacket \(90\%\) off):

• Shirt: \(\$50 \rightarrow \$35\)
• Jacket: \(\$100 \rightarrow \$10\)
• Result: Jacket becomes cheaper! ✗

The critical insight: With small discounts, the \(2\times\) price factor dominates. With large discounts, the \(3\times\) discount factor dominates. Since different discount sizes give us different answers about which item is cheaper, we still cannot determine which has the lower discounted price.

Both statements together are NOT sufficient.

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

The Answer: E

Even with both pieces of information, we cannot determine which item has the lower discounted price because the outcome changes based on the magnitude of the discounts.

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