In an examination, 53% students passed in Mathematics, 61% passed in P

In an examination, 53% students passed in Mathematics, 61% passed in Physics, 60% passed in Chemistry, 24% passed in Mathematics and Physics, 35% in Physics and Chemistry, 27% in Mathematics and Chemistry and 5% in none. The ratio of percentage of passes in Mathematics and Chemistry but not in Physics in relation to the percentage of passes in Physics and Chemistry but not in Mathematics is:

7:5
5:7
4:5
5:4
This question was previously asked in
UPSC CAPF – 2018
Let M, P, and C represent the sets of students who passed in Mathematics, Physics, and Chemistry, respectively. We are given the following percentages: |M|=53, |P|=61, |C|=60, |M∩P|=24, |P∩C|=35, |M∩C|=27, |None|=5. The percentage of students who passed in at least one subject is |M∪P∪C| = 100 – |None| = 100 – 5 = 95%. Using the principle of inclusion-exclusion: |M∪P∪C| = |M| + |P| + |C| – |M∩P| – |M∩C| – |P∩C| + |M∩P∩C|. So, 95 = 53 + 61 + 60 – 24 – 27 – 35 + |M∩P∩C|. 95 = 174 – 86 + |M∩P∩C|. 95 = 88 + |M∩P∩C|. Thus, |M∩P∩C| = 95 – 88 = 7%.
Percentage of passes in Mathematics and Chemistry but not in Physics is the region (M∩C) excluding the triple intersection (M∩C∩P). This is |M∩C| – |M∩C∩P| = 27% – 7% = 20%.
Percentage of passes in Physics and Chemistry but not in Mathematics is the region (P∩C) excluding the triple intersection (M∩P∩C). This is |P∩C| – |M∩P∩C| = 35% – 7% = 28%.
The ratio of the percentage of passes in Mathematics and Chemistry but not in Physics to the percentage of passes in Physics and Chemistry but not in Mathematics is 20 : 28. Simplifying the ratio by dividing by 4, we get 5 : 7.
This problem requires using the principle of inclusion-exclusion for three sets and calculating the percentages corresponding to specific regions in a Venn diagram (intersections of two sets excluding the third).
Using a Venn diagram helps visualize the different regions. The percentage of students in M only is |M| – (|M∩P| + |M∩C|) + |M∩P∩C|. Similar calculations can be done for P only, C only, M∩P only, P∩C only, and M∩C only. The sum of percentages in all 7 regions plus the ‘none’ percentage should equal 100%.