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CAT 2020 Slot 3 DILR · Set 3 of 5Data InterpretationEasy DifficultyQ11–Q14 · 2 MCQ + 2 TITA

CAT 2020 Slot 3 DILR Set 3: Venn Diagrams (4 Questions) with Solutions

The Venn Diagrams set from the official CAT 2020 Slot 3 DILR paper: the caselet, all 4 questions, the official answer key and a step-by-step solution for each. 100% free, no sign-up.

The Set: Read the Caselet First

CAT 2020 Slot 3 DILR · Questions 11–14
A survey of 600600 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
8080 schools did not have any of the four facilities - F1, F2, F3, F4.
2
4040 schools had all four facilities.
3
The number of schools with only F1, only F2, only F3, and only F4 was 25, 30, 2626 and 2020 respectively.
4
The number of schools with exactly three of the facilities was the same irrespective of which three were considered.
5
313313 schools had F2.
6
2626 schools had only F2 and F3 (but neither F1 nor F4).
7
Among the schools having F4, 2424 had only F3, and 4545 had only F2.
8
162162 schools had both F1 and F2.
9
The number of schools having F1 was the same as the number of schools having F4.

In the exam this set competed with 4 others for 40 minutes. Budget about 8 minutes if you attempt it.

Q11. Venn Diagrams

MCQ · +3 / −1
What was the total number of schools having exactly three of the four facilities?
  1. A) 200
  2. B) 50
  3. C) 64
  4. D) 80
Correct Answer
A) 200
Step-by-step solution
📌 Core Concept
Four-set Venn diagram: split the schools into regions (none, only one, exactly two, exactly three, all four). Use the F2 total (observation 5) and the F1∩F2 total (observation 8) to find the common 'exactly three' value.
🔢 Step-by-Step Solution
1
Let each 'exactly three' region have tt schools (observation 4). Let aa be the number of schools with only F1 and F2.
2
F2 total is 313. The regions inside F2 are: only F2 (30), only F2 and F3 (26), only F2 and F4 (45), only F1 and F2 (aa), three of the facilities including F2 (3t3t), and all four (40).
3
So 30+26+45+a+3t+40=31330+26+45+a+3t+40=313, which gives a+3t=172a+3t=172.
4
F1 and F2 together total 162. The regions inside both are: only F1 and F2 (aa), F1, F2, F3 only (tt), F1, F2, F4 only (tt), and all four (40).
5
So a+2t+40=162a+2t+40=162, which gives a+2t=122a+2t=122.
6
Subtracting step 5 from step 3 gives t=50t=50, and then a=22a=22.
7
Schools with exactly three facilities: 4t=4×50=2004t=4\times 50=200.
Why Not the Others
50 is only one of the four 'exactly three' regions.
64 and 80 do not match 4t4t for t=50t=50.
Correct Answer: A

Q12. Venn Diagrams

MCQ · +3 / −1
What was the number of schools having facilities F2 and F4?
  1. A) 85
  2. B) 95
  3. C) 45
  4. D) 185
Correct Answer
D) 185
Step-by-step solution
📌 Core Concept
Count every region of the Venn diagram that lies inside both F2 and F4.
🔢 Step-by-Step Solution
1
Observation 7 gives the schools with F4 and F2 only, which is 45.
2
From the previous question, each 'exactly three' region has t=50t=50 schools.
3
Two of those three-facility regions contain both F2 and F4: F1, F2, F4 only (50) and F2, F3, F4 only (50).
4
The schools with all four facilities number 40 (observation 2).
5
Total with F2 and F4: 45+50+50+40=18545+50+50+40=185.
Why Not the Others
45 counts only the 'F2 and F4 only' region.
85 leaves out one of the three-facility regions.
95 counts only one of the three-facility regions (50) plus the 45 and leaves out the all-four group.
Correct Answer: D

Q13. Venn Diagrams

TITA · +3 / 0
What was the number of schools having only facilities F1 and F3?
Correct Answer
42
Step-by-step solution
📌 Core Concept
Total the known regions to find the combined size of the 'only F1 and F3' and 'only F1 and F4' regions. Then use observation 9 (F1 = F4) to separate them.
🔢 Step-by-Step Solution
1
From the earlier questions, each 'exactly three' region has 50 schools, and only F1 and F2 has 22.
2
Let bb be the number of schools with only F1 and F3, and cc the number with only F1 and F4.
3
F1 total: 25+22+b+c+3×50+40=237+b+c25+22+b+c+3\times 50+40=237+b+c.
4
F4 total: only F4 (20), only F4 and F1 (cc), only F4 and F2 (45), only F4 and F3 (24), three-facility regions with F4 (3×503\times 50), and all four (40). This gives 20+c+45+24+150+40=279+c20+c+45+24+150+40=279+c.
5
Observation 9 says F1 = F4, so 237+b+c=279+c237+b+c=279+c. The cc cancels, so b=42b=42.
Quick Check
The regions must sum to 600−80=520600-80=520. Singles total 101, exactly-three regions total 200, all four is 40, and the pairs are 22+26+24+45+42+20=17922+26+24+45+42+20=179. Then 101+200+40+179=520101+200+40+179=520. ✓
Correct Answer: 42

Q14. Venn Diagrams

TITA · +3 / 0
What was the number of schools having only facilities F1 and F4?
Correct Answer
20
Step-by-step solution
📌 Core Concept
After finding the F1-and-F3-only region, use the total of 520 schools with at least one facility to find the F1-and-F4-only region.
🔢 Step-by-Step Solution
1
Schools with at least one facility: 600−80=520600-80=520.
2
Only one facility: 25+30+26+20=10125+30+26+20=101.
3
Exactly three facilities: 4×50=2004\times 50=200. All four:
40
4. Exactly two facilities must therefore be 520−101−200−40=179520-101-200-40=179.
5
The known pair regions are: F1 and F2 (22), F2 and F3 (26), F3 and F4 (24), F2 and F4 (45), and F1 and F3 (42, from the previous question). These total 159.
6
Only F1 and F4: 179−159=20179-159=20.
Quick Check
F1 =237+42+20=299=237+42+20=299 and F4 =279+20=299=279+20=299, so F1 = F4 as observation 9 requires. ✓
Correct Answer: 20
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CAT 2020 Slot 3 DILR at a Glance

Sets in this section5
Questions in this section24
Section time40 minutes
Convenor IIMIIM Indore

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