# Logical Reasoning – A survey of 600 schools in India was conducted

A survey of 600 schools in India was conducted to gather information about their online teaching learning processes (OTLP). The following four facilities were studied.
F1: Own software for OTLP
F2: Trained teachers for OTLP
F3: Training materials for OTLP
F4: All students having Laptops
The following observations were summarized from the survey.
1. 80 schools did not have any of the four facilities – F1, F2, F3, F4.
2. 40 schools had all four facilities.
3. The number of schools with only F1, only F2, only F3, and only F4 was 25, 30, 26 and 20 respectively.
4. The number of schools with exactly three of the facilities was the same irrespective of which three were considered.
6. 26 schools had only F2 and F3 (but neither F1 nor F4).
7. Among the schools having F4, 24 had only F3, and 45 had only F2.
8. 162 schools had both F1 and F2.
9. The number of schools having F1 was the same as the number of schools having F4.

Q.1 What was the total number of schools having exactly three of the four facilities?
1. 200
2. 64
3. 50
4. 80

Q.2 What was the number of schools having facilities F2 and F4?
1. 95
2. 45
3. 185
4. 85

Q.3 What was the number of schools having only facilities F1 and F3?

Q.4 What was the number of schools having only facilities F1 and F4?

Solutions:
As in the given questions there are a lot of data given . So it will be easier to make a venn diagram and fill all the given data and then solve the questions for missing values. Given ,
3x + y + 40+45+26+30= 313
Or 3x + y = 172———1)
162 schools had both F1 and F2. So
2x + y + 40 = 162
2x + y = 122 ———-2)
On solving eq 1) and eq 2)
x = 50 and y = 22
Now , The number of schools having F1 was the same as the number of schools having F4. So
3x + y + z + w + 25 + 40 = 3x + w + 40+45 + 24+20
172 + z + 25 = 150 + 89
z = 42
Also , 80 schools did not have any of the four facilities – F1, F2, F3, F4.
Thus number of schools having at least one facility = 600 – 80 = 520
4x + z + w + y + 25+30+26+20+26+40+45+24 = 520
200 + 42+w + 22 + 236 = 520
500+ w = 520
w = 20
now the complete venn-diagram will look like , Now all questions can be answered.

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