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Zorluk: Çok zorResolve the Paradox

In the state of New Lemuria, a strict building code was implemented requiring all new residential constructions to utilize advanced fire-resistant materials. Over the five years following the implementation of this code, the absolute number of newly constructed homes that experienced a residential fire was exactly 20% higher than the number of newly constructed homes that experienced a fire during the five years immediately prior to the code. Nevertheless, the state's fire marshal concluded that the new building code was highly effective in reducing the likelihood that any newly constructed home would experience a fire.

Statement: The apparent discrepancy between the fire incident data and the fire marshal's conclusion is successfully resolved if it is true that the total number of newly constructed homes built in New Lemuria during the five years after the code was implemented was exactly 10% greater than the total number of newly constructed homes built during the five years prior.

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False
The statement is False because it fails to mathematically reconcile the evidence with the conclusion. If the absolute number of fires increased by 20%, but the total number of homes only increased by 10%, the rate (likelihood) of fire per home must have increased. This directly contradicts the fire marshal's conclusion that the likelihood decreased. To resolve the paradox, the percentage increase in total homes would have to be strictly greater than the percentage increase in fires (e.g., a 30% increase in homes).

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1
Identify the two apparently contradictory premises in the stimulus.
Premise 1: The absolute number of fires in new homes increased by 20%. Premise 2: The fire marshal concluded that the likelihood (rate) of a new home catching fire decreased.
Resolving the paradox requires finding a fact that allows both the absolute number of incidents to rise and the rate of incidents to fall simultaneously.
2
Define the mathematical relationship between absolute numbers and likelihood (rates).
The rate of fire is equal to the absolute number of fires divided by the total number of homes built (Rate=FiresHomesRate = \frac{Fires}{Homes}).
This formula connects the evidence (absolute numbers) to the conclusion (likelihood).
3
Evaluate the mathematical impact of the proposed statement on the established relationship.
The statement posits a 10% increase in total homes. Given a 20% increase in fires, the new rate is 1.20×Fires1.10×Homes\frac{1.20 \times Fires}{1.10 \times Homes}, which is approximately 1.091.09 times the old rate.
Calculating the precise impact determines if the statement actually resolves the contradiction.
4
Compare the resulting rate to the fire marshal's conclusion.
The calculation shows the rate of fire increased by about 9%, which directly contradicts the fire marshal's conclusion that the likelihood decreased.
A statement that contradicts a premise or conclusion in the stimulus exacerbates rather than resolves the paradox.

Anahtar Kavram

Distinguishing between absolute numbers and proportional rates, and understanding the mathematical constraints required for a baseline shift to resolve a discrepancy between the two.
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Konular

Resolve the Paradox

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