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In a survey of 400 households, 220 use natural gas, 180 use electric heating, and 90 use both. How many households use neither natural gas nor electric heating?
70
90
110
130
90
To find the number of households using neither natural gas nor electric heating, we apply the Principle of Inclusion-Exclusion from set theory. Subtracting the intersection of both utilities from the sum of individual users gives the total number of households using at least one utility, which is then subtracted from the universal set.
To find the number of households using neither natural gas nor electric heating, we apply the Principle of Inclusion-Exclusion from set theory. Subtracting the intersection of both utilities from the sum of individual users gives the total number of households using at least one utility, which is then subtracted from the universal set.
Imagine a room where people are eating apples or bananas; to count everyone eating at least one fruit without double-counting those eating both, you add apple-eaters and banana-eaters, then subtract the overlap.
Add the parts, subtract the overlap, then take the remaining gap from the total.
n(AтИкB)=n(A)+n(B)тИТn(AтИйB) тАФ Principle of Inclusion-Exclusion for two sets
n(AтИкB)тА▓=n(U)тИТn(AтИкB) тАФ Complement of the union representing neither set
The total number of households using either natural gas or electric heating is calculated using the formula n(AтИкB)=n(A)+n(B)тИТn(AтИйB). Subtracting this union from the universal set of surveyed households yields those using neither, given by n(U)тИТn(AтИкB).
The universal set n(U) represents the total surveyed households (400).
The intersection n(AтИйB) represents households using both utilities (90).
Households using neither are found by subtracting the union from the total sample size.
Provides a systematic and foolproof method to handle overlapping survey data.
Eliminates double-counting in multi-category problems.
Prone to arithmetic errors if intersection values are mistakenly added instead of subtracted.
Data analysis and demographic surveys
Probability and statistical data interpretation
Total households surveyed: 400
Option B (90) is correct because 400тИТ(220+180тИТ90)=90.
B is correct тАФ 90 households use neither natural gas nor electric heating.
Always verify whether the intersection is explicitly included or excluded when interpreting survey results in quantitative aptitude exams.