Question 15: Out of the 180 students who appeard in the annual examanination, 120 passed the math test, 90 passed the science test, and 60 passed both the math and science tests.
\((a)\) \(\;\) How many passed either the math or science test?
\((b)\) \(\;\) How many did not pass either of the two tests?
\((c)\) \(\;\) How many passed the science test but not the math test?
\((d)\) \(\;\) How many failed the science test?
Solution:
Let, \(M=\text{Student passed in math}\; ; \; \)\(S=\text{Student passed in science}\)
\(n(M)=120\)
\(n(S)=90\)
\(n(M\cap S)=60\)
\((a)\) \(\;\) How many passed either the math or science test?
\(n(M\cup S)=\ \)\(n(M)+n(S)\ \)\(−\ n(M\cap S)\)
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\ \)\(120+90\ \)\(−\ 60\)
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\ \)\(150\)
\(150\) students passed either the Math or Science test.
\((b)\) \(\;\) How many did not pass either of the two tests?
\(\text{Not passed either}\)\(\ =\ \)\(\text{Total}\)\(\ -\ \)\(\text{Passed either math or science}\)
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\ =\ \)\(180\)\(\ -\ \)\(150\)
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\ =\ \)\(30\)
\(30\) students did not pass either of the two tests.
\((c)\) \(\;\) How many passed the science test but not the math test?
\(n(S- M)=\ \)\(n(S)\ \)\(−\ n(M\cap S)\)
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\ \)\(90-60\ \)
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\ \)\(30\)
\(30\) students passed the Science test but not the math.
\((d)\) \(\;\) How many failed the science test?
\(\text{Failed the science test}\)\(\ =\ \)\(\text{Total}\)\(\ -\ \)\(\text{passed only the Science test.}\)
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\ =\ \)\(180\)\(\ -\ \)\(90\)
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\ =\ \)\(90\)
\(90\) students failed the Science test.