In a recent survey (conducted by HLL) of 1,000 houses, washing machine, vacuum cleaners and refrigerators were counted. Each house had at least one of these products. 400 had no refrigerators, 380 had no vacuum cleaners, 542 had no washing machines. 294 had both a vacuum cleaner and washing machines, 277 had both a vacuum cleaner and a refrigerator, and 120 had both a refrigerator and a washing machine. How many had only a vacuum cleaner ?
Answers
In a recent survey (conducted by HLL) of 1,000 houses, washing machine, vacuum cleaners and refrigerators were counted. Each house had at least one of these products. 400 had no refrigerators, 380 had no vacuum cleaners, 542 had no washing machines. 294 had both a vacuum cleaner and washing machines, 277 had both a vacuum cleaner and a refrigerator, and 120 had both a refrigerator and a washing machine. How many had only a vacuum cleaner ?
Answer:
Step-by-step explanation:
The sample space is 1000 houses (black) of which 600 own refrigerators (red), 620 houses own vacuum cleaners (blue), and 458 own washing machines (brown).
We know that 294 own a vacuum cleaner and also a washing machine. We will that the 294 own atleast a vacuum cleaner and a washing machine; some houses also own a refrigerator. We will assume
|V∩W|=294 (purple).
Some houses own both refrigerators and vacuum cleaners (R∩V≠∅) because otherwise there would be 1220 households.
Some houses own both refrigerators and washing machines (R∩W≠∅) because otherwise there would be 1058 households.
Some houses own both vacuum cleaners and washing machines (V∩W≠∅) because otherwise there would be 1078 households.
Some houses own washing machines but do not own vacuum cleaners (W⊈V) because otherwise the purple region would have 458 elements.
Some houses own vacuum cleaners but do not own refrigerators (V⊈R) because |V|>|R|.
Some houses own refrigerators but not washing machines (R⊈W) because |R|>|W|.
Some houses own vacuum cleaners but do not own washing machines (V⊈W) because |V|>|W|.
|R∪V∪W|=|R|+|V|+|W|−|R∩V|−|R∩W|−|V∩W|+|R∩V∩W|≤1000. Thus, 600+620+458−|R∩V|−|R∩W|−|V∩W|+|R∩V∩W|≤1000 and |R∩V∩W|≤|R∩V|+|R∩W|+|V∩W|−678.
Step-by-step explanation:
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