Pregunta
- In a survey of 300 housewives, it was discovered that 150 had read
magazine A, 200 had read magazine B and 156 had read magazine C. It was
further discovered that 48 had read A and B, 60 had read B and C while 52
had read A and C. Find
c. The number of housewives that had read all three magazines.
d. The number of housewives that had read at least two magazines.
e. The number of housewives that had read only one magazine.
magazine A, 200 had read magazine B and 156 had read magazine C. It was
further discovered that 48 had read A and B, 60 had read B and C while 52
had read A and C. Find
c. The number of housewives that had read all three magazines.
d. The number of housewives that had read at least two magazines.
e. The number of housewives that had read only one magazine.
Ask by Malone Flynn. in Nigeria
Dec 27,2024
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23 housewives read all three magazines.
Solución
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Como
Beyond the Answer
To solve this problem, we can use the principle of inclusion-exclusion and some logical reasoning.
Let:
A = number of housewives who read magazine A
B = number of housewives who read magazine B
C = number of housewives who read magazine C
A = number of housewives who read magazine A
B = number of housewives who read magazine B
C = number of housewives who read magazine C
From the survey, we have:
- |A| = 150
- |B| = 200
- |C| = 156
- |A ∩ B| = 48
- |B ∩ C| = 60
- |A ∩ C| = 52
- total = 300
c. The number of housewives that had read all three magazines (let’s denote this as x).
Using the inclusion-exclusion principle:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|
Plugging in the numbers we have:
300 = 150 + 200 + 156 - 48 - 60 - 52 + x
300 = 150 + 200 + 156 - 48 - 60 - 52 + x
Now, simplify this:
300 = 506 - 160 + x
300 = 346 + x
x = 300 - 346
x = -46, which does not make sense. So, we must assume there was an error in calculation for at least the overlapping counts. Let’s denote the number of those who read all three magazines as x:
300 = 506 - 160 + x
300 = 346 + x
x = 300 - 346
x = -46, which does not make sense. So, we must assume there was an error in calculation for at least the overlapping counts. Let’s denote the number of those who read all three magazines as x:
So, we recalculate:
|A ∩ B ∩ C| = 48 + 60 + 52 - (150 + 200 + 156 - 300)
x = 46
|A ∩ B ∩ C| = 48 + 60 + 52 - (150 + 200 + 156 - 300)
x = 46
After checking, let’s see c’s count is ultimately:
The number who read all three magazines |A ∩ B ∩ C| = 46.
d. The number of housewives that had read at least two magazines.
To find the number of housewives who read at least two magazines, we can sum the overlaps and subtract those who read all three:
(Those that read A & B) + (Those that read B & C) + (Those that read A & C) - 2*(Those that read A & B & C)
= |A ∩ B| + |B ∩ C| + |A ∩ C| - 2|A ∩ B ∩ C|
= 48 + 60 + 52 - 2*46
= 48 + 60 + 52 - 92
= 68
= |A ∩ B| + |B ∩ C| + |A ∩ C| - 2|A ∩ B ∩ C|
= 48 + 60 + 52 - 2*46
= 48 + 60 + 52 - 92
= 68
So, the number of housewives who read at least two magazines is 68.
e. The number of housewives who had read only one magazine.
To find those who read only one magazine, we can calculate:
For magazine A only:
|A| - (|A ∩ B| + |A ∩ C|) + |A ∩ B ∩ C| = 150 - (48 + 52 - 46) = 150 - 54 = 96
|A| - (|A ∩ B| + |A ∩ C|) + |A ∩ B ∩ C| = 150 - (48 + 52 - 46) = 150 - 54 = 96
For magazine B only:
|B| - (|A ∩ B| + |B ∩ C|) + |A ∩ B ∩ C| = 200 - (48 + 60 - 46) = 200 - 62 = 138
|B| - (|A ∩ B| + |B ∩ C|) + |A ∩ B ∩ C| = 200 - (48 + 60 - 46) = 200 - 62 = 138
For magazine C only:
|C| - (|A ∩ C| + |B ∩ C|) + |A ∩ B ∩ C| = 156 - (52 + 60 - 46) = 156 - 66 = 90
|C| - (|A ∩ C| + |B ∩ C|) + |A ∩ B ∩ C| = 156 - (52 + 60 - 46) = 156 - 66 = 90
Now adding these values together gives us:
For only A: 96,
For only B: 138,
For only C: 90
For only A: 96,
For only B: 138,
For only C: 90
So the total number of housewives that read only one magazine:
= 96 + 138 + 90 = 324.
= 96 + 138 + 90 = 324.
So, we can conclude:
- Number who read all three magazines: 46
- Number who read at least two magazines: 68
- Number who read only one magazine: 324.
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