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A shirt and a jacket are each for sale at a price that is discounted from the item's original price. Which has the lower discounted price?
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.
This is a comparison problem with two competing factors:
The outcome depends on which factor dominates.
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:
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.
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:
The answer depends on the relationship between original prices.
Statement 2 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices B and D.
Now we know both:
Let's see if this pins down the answer. If the shirt originally costs \(\$50\) and gets discount percentage s:
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):
Testing with large discounts (shirt \(30\%\) off, jacket \(90\%\) off):
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.
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."