In a survey of the 100 out-patients who reported at the hospital one day , it was found out that 70 complained of fever, 50 complained of stomach ache and 30 were injured.All 100 patients had at least one of the complaints and 44 had exactly two of the complaints.How many patients had all three complaints?
If x had two complaints,
70+50+20 - 44 + x = 100
Review 3-part Venn diagrams.
To find out how many patients had all three complaints, we can use the principle of inclusion-exclusion.
Step 1: Find the total number of patients who reported at least one complaint. In this case, it is given that all 100 patients had at least one complaint.
Step 2: Subtract the number of patients who reported exactly two complaints. It is given that 44 patients reported exactly two complaints.
Step 3: Add back the number of patients who reported all three complaints, since they were subtracted twice in Step 2.
Let's proceed with the calculations:
Step 1: Total number of patients with at least one complaint = 100
Step 2: Patients with exactly two complaints = 44
Step 3: Let x represent the number of patients with all three complaints.
- Patients with fever and stomach ache (70 - x)
- Patients with fever and injury (x)
- Patients with stomach ache and injury (50 - x)
Using the principle of inclusion-exclusion, we can form the equation:
100 = (70 - x) + (x) + (50 - x) - 44 + x
Simplifying this equation:
100 = 120 - 44 + x
100 = 76 + x
x = 100 - 76
x = 24
Therefore, there were 24 patients who had all three complaints.
To determine how many patients had all three complaints, you can use a technique called inclusion-exclusion principle.
First, let's break down the information given:
- 70 patients complained of fever.
- 50 patients complained of a stomach ache.
- 30 patients were injured.
- 100 patients had at least one of the complaints.
- 44 patients had exactly two complaints.
Using this information, we will calculate the number of patients who had all three complaints.
Step 1: Calculate the total number of patients with each individual complaint:
- Fever: 70 patients.
- Stomach ache: 50 patients.
- Injury: 30 patients.
Step 2: Calculate the total number of patients with exactly two complaints:
- 44 patients had exactly two complaints.
Step 3: Calculate the total number of patients who had only one complaint:
To determine this, we subtract the number of patients with exactly two complaints from the number of patients with each individual complaint:
- Fever: 70 - 44 = 26 patients had only fever.
- Stomach ache: 50 - 44 = 6 patients had only a stomach ache.
- Injury: 30 - 44 = 0 patients had only an injury (since there are only 30 patients with an injury in total).
Step 4: Calculate the number of patients who had all three complaints:
To determine this, we subtract the number of patients who had only one or two complaints from the total number of patients:
Total number of patients = 100
- Patients with only fever: 26
- Patients with only a stomach ache: 6
- Patients with only an injury: 0
- Patients with exactly two complaints: 44
Number of patients who had all three complaints = Total number of patients - (Patients with only fever + Patients with only a stomach ache + Patients with only an injury + Patients with exactly two complaints)
Number of patients who had all three complaints = 100 - (26 + 6 + 0 + 44)
Number of patients who had all three complaints = 100 - 76
Number of patients who had all three complaints = 24
Therefore, there were 24 patients who had all three complaints (fever, stomach ache, and injury) in the survey.